realGEcomputational

**Functions**

subroutine **sgebak** (JOB, SIDE, N, ILO, IHI, SCALE, M, V, LDV, INFO)

SGEBAK

subroutine **sgebal** (JOB, N, A, LDA, ILO, IHI, SCALE, INFO)

SGEBAL

subroutine **sgebd2** (M, N, A, LDA, D, E, TAUQ, TAUP, WORK, INFO) **
SGEBD2** reduces a general matrix to bidiagonal form using an unblocked algorithm.

subroutine

SGEBRD

subroutine

SGECON

subroutine

SGEEQU

subroutine

SGEEQUB

subroutine

SGEHD2

subroutine

SGEHRD

subroutine

SGELQ2

subroutine

SGELQF

subroutine

SGEMQRT

subroutine

SGEQL2

subroutine

SGEQLF

subroutine

SGEQP3

subroutine

SGEQR2

subroutine

SGEQR2P

subroutine

SGEQRF

subroutine

SGEQRFP

subroutine

SGEQRT

subroutine

SGEQRT2

recursive subroutine

SGEQRT3

subroutine

SGERFS

subroutine

SGERFSX

subroutine

SGERQ2

subroutine

SGERQF

subroutine

SGESVJ

subroutine

SGETF2

subroutine

SGETRF

recursive subroutine

SGETRF2

subroutine

SGETRI

subroutine

SGETRS

subroutine

SHGEQZ

subroutine

SLA_GEAMV

real function

SLA_GERCOND

subroutine

SLA_GERFSX_EXTENDED

real function

SLA_GERPVGRW

subroutine

SLAORHR_COL_GETRFNP

recursive subroutine

SLAORHR_COL_GETRFNP2

subroutine

STGEVC

subroutine

STGEXC

This is the group of real computational functions for GE matrices

**subroutine sgebak (character JOB, character SIDE, integer N, integer ILO, integer IHI, real, dimension( * ) SCALE, integer M, real, dimension( ldv, * ) V, integer LDV, integer INFO)
SGEBAK**

**Purpose:**

SGEBAK forms the right or left eigenvectors of a real general matrix

by backward transformation on the computed eigenvectors of the

balanced matrix output by SGEBAL.

**Parameters**

*JOB*

JOB is CHARACTER*1

Specifies the type of backward transformation required:

= ’N’: do nothing, return immediately;

= ’P’: do backward transformation for permutation only;

= ’S’: do backward transformation for scaling only;

= ’B’: do backward transformations for both permutation and

scaling.

JOB must be the same as the argument JOB supplied to SGEBAL.

*SIDE*

SIDE is CHARACTER*1

= ’R’: V contains right eigenvectors;

= ’L’: V contains left eigenvectors.

*N*

N is INTEGER

The number of rows of the matrix V. N >= 0.

*ILO*

ILO is INTEGER

*IHI*

IHI is INTEGER

The integers ILO and IHI determined by SGEBAL.

1 <= ILO <= IHI <= N, if N > 0; ILO=1 and IHI=0, if N=0.

*SCALE*

SCALE is REAL array, dimension (N)

Details of the permutation and scaling factors, as returned

by SGEBAL.

*M*

M is INTEGER

The number of columns of the matrix V. M >= 0.

*V*

V is REAL array, dimension (LDV,M)

On entry, the matrix of right or left eigenvectors to be

transformed, as returned by SHSEIN or STREVC.

On exit, V is overwritten by the transformed eigenvectors.

*LDV*

LDV is INTEGER

The leading dimension of the array V. LDV >= max(1,N).

*INFO*

INFO is INTEGER

= 0: successful exit

< 0: if INFO = -i, the i-th argument had an illegal value.

**Author**

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

December 2016

**subroutine sgebal (character JOB, integer N, real, dimension( lda, * ) A, integer LDA, integer ILO, integer IHI, real, dimension( * ) SCALE, integer INFO)
SGEBAL**

**Purpose:**

SGEBAL balances a general real matrix A. This involves, first,

permuting A by a similarity transformation to isolate eigenvalues

in the first 1 to ILO-1 and last IHI+1 to N elements on the

diagonal; and second, applying a diagonal similarity transformation

to rows and columns ILO to IHI to make the rows and columns as

close in norm as possible. Both steps are optional.

Balancing may reduce the 1-norm of the matrix, and improve the

accuracy of the computed eigenvalues and/or eigenvectors.

**Parameters**

*JOB*

JOB is CHARACTER*1

Specifies the operations to be performed on A:

= ’N’: none: simply set ILO = 1, IHI = N, SCALE(I) = 1.0

for i = 1,...,N;

= ’P’: permute only;

= ’S’: scale only;

= ’B’: both permute and scale.

*N*

N is INTEGER

The order of the matrix A. N >= 0.

*A*

A is REAL array, dimension (LDA,N)

On entry, the input matrix A.

On exit, A is overwritten by the balanced matrix.

If JOB = ’N’, A is not referenced.

See Further Details.

*LDA*

LDA is INTEGER

The leading dimension of the array A. LDA >= max(1,N).

*ILO*

ILO is INTEGER

*IHI*

IHI is INTEGER

ILO and IHI are set to integers such that on exit

A(i,j) = 0 if i > j and j = 1,...,ILO-1 or I = IHI+1,...,N.

If JOB = ’N’ or ’S’, ILO = 1 and IHI = N.

*SCALE*

SCALE is REAL array, dimension (N)

Details of the permutations and scaling factors applied to

A. If P(j) is the index of the row and column interchanged

with row and column j and D(j) is the scaling factor

applied to row and column j, then

SCALE(j) = P(j) for j = 1,...,ILO-1

= D(j) for j = ILO,...,IHI

= P(j) for j = IHI+1,...,N.

The order in which the interchanges are made is N to IHI+1,

then 1 to ILO-1.

*INFO*

INFO is INTEGER

= 0: successful exit.

< 0: if INFO = -i, the i-th argument had an illegal value.

**Author**

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

December 2016

**Further Details:**

The permutations consist of row and column interchanges which put

the matrix in the form

( T1 X Y )

P A P = ( 0 B Z )

( 0 0 T2 )

where T1 and T2 are upper triangular matrices whose eigenvalues lie

along the diagonal. The column indices ILO and IHI mark the starting

and ending columns of the submatrix B. Balancing consists of applying

a diagonal similarity transformation inv(D) * B * D to make the

1-norms of each row of B and its corresponding column nearly equal.

The output matrix is

( T1 X*D Y )

( 0 inv(D)*B*D inv(D)*Z ).

( 0 0 T2 )

Information about the permutations P and the diagonal matrix D is

returned in the vector SCALE.

This subroutine is based on the EISPACK routine BALANC.

Modified by Tzu-Yi Chen, Computer Science Division, University of

California at Berkeley, USA

**subroutine sgebd2 (integer M, integer N, real, dimension( lda, * ) A, integer LDA, real, dimension( * ) D, real, dimension( * ) E, real, dimension( * ) TAUQ, real, dimension( * ) TAUP, real, dimension( * ) WORK, integer INFO)
SGEBD2** reduces a general matrix to bidiagonal form using an unblocked algorithm.

**Purpose:**

SGEBD2 reduces a real general m by n matrix A to upper or lower

bidiagonal form B by an orthogonal transformation: Q**T * A * P = B.

If m >= n, B is upper bidiagonal; if m < n, B is lower bidiagonal.

**Parameters**

*M*

M is INTEGER

The number of rows in the matrix A. M >= 0.

*N*

N is INTEGER

The number of columns in the matrix A. N >= 0.

*A*

A is REAL array, dimension (LDA,N)

On entry, the m by n general matrix to be reduced.

On exit,

if m >= n, the diagonal and the first superdiagonal are

overwritten with the upper bidiagonal matrix B; the

elements below the diagonal, with the array TAUQ, represent

the orthogonal matrix Q as a product of elementary

reflectors, and the elements above the first superdiagonal,

with the array TAUP, represent the orthogonal matrix P as

a product of elementary reflectors;

if m < n, the diagonal and the first subdiagonal are

overwritten with the lower bidiagonal matrix B; the

elements below the first subdiagonal, with the array TAUQ,

represent the orthogonal matrix Q as a product of

elementary reflectors, and the elements above the diagonal,

with the array TAUP, represent the orthogonal matrix P as

a product of elementary reflectors.

See Further Details.

*LDA*

LDA is INTEGER

The leading dimension of the array A. LDA >= max(1,M).

*D*

D is REAL array, dimension (min(M,N))

The diagonal elements of the bidiagonal matrix B:

D(i) = A(i,i).

*E*

E is REAL array, dimension (min(M,N)-1)

The off-diagonal elements of the bidiagonal matrix B:

if m >= n, E(i) = A(i,i+1) for i = 1,2,...,n-1;

if m < n, E(i) = A(i+1,i) for i = 1,2,...,m-1.

*TAUQ*

TAUQ is REAL array, dimension (min(M,N))

The scalar factors of the elementary reflectors which

represent the orthogonal matrix Q. See Further Details.

*TAUP*

TAUP is REAL array, dimension (min(M,N))

The scalar factors of the elementary reflectors which

represent the orthogonal matrix P. See Further Details.

*WORK*

WORK is REAL array, dimension (max(M,N))

*INFO*

INFO is INTEGER

= 0: successful exit.

< 0: if INFO = -i, the i-th argument had an illegal value.

**Author**

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

June 2017

**Further Details:**

The matrices Q and P are represented as products of elementary

reflectors:

If m >= n,

Q = H(1) H(2) . . . H(n) and P = G(1) G(2) . . . G(n-1)

Each H(i) and G(i) has the form:

H(i) = I - tauq * v * v**T and G(i) = I - taup * u * u**T

where tauq and taup are real scalars, and v and u are real vectors;

v(1:i-1) = 0, v(i) = 1, and v(i+1:m) is stored on exit in A(i+1:m,i);

u(1:i) = 0, u(i+1) = 1, and u(i+2:n) is stored on exit in A(i,i+2:n);

tauq is stored in TAUQ(i) and taup in TAUP(i).

If m < n,

Q = H(1) H(2) . . . H(m-1) and P = G(1) G(2) . . . G(m)

Each H(i) and G(i) has the form:

H(i) = I - tauq * v * v**T and G(i) = I - taup * u * u**T

where tauq and taup are real scalars, and v and u are real vectors;

v(1:i) = 0, v(i+1) = 1, and v(i+2:m) is stored on exit in A(i+2:m,i);

u(1:i-1) = 0, u(i) = 1, and u(i+1:n) is stored on exit in A(i,i+1:n);

tauq is stored in TAUQ(i) and taup in TAUP(i).

The contents of A on exit are illustrated by the following examples:

m = 6 and n = 5 (m > n): m = 5 and n = 6 (m < n):

( d e u1 u1 u1 ) ( d u1 u1 u1 u1 u1 )

( v1 d e u2 u2 ) ( e d u2 u2 u2 u2 )

( v1 v2 d e u3 ) ( v1 e d u3 u3 u3 )

( v1 v2 v3 d e ) ( v1 v2 e d u4 u4 )

( v1 v2 v3 v4 d ) ( v1 v2 v3 e d u5 )

( v1 v2 v3 v4 v5 )

where d and e denote diagonal and off-diagonal elements of B, vi

denotes an element of the vector defining H(i), and ui an element of

the vector defining G(i).

**subroutine sgebrd (integer M, integer N, real, dimension( lda, * ) A, integer LDA, real, dimension( * ) D, real, dimension( * ) E, real, dimension( * ) TAUQ, real, dimension( * ) TAUP, real, dimension( * ) WORK, integer LWORK, integer INFO)
SGEBRD**

**Purpose:**

SGEBRD reduces a general real M-by-N matrix A to upper or lower

bidiagonal form B by an orthogonal transformation: Q**T * A * P = B.

If m >= n, B is upper bidiagonal; if m < n, B is lower bidiagonal.

**Parameters**

*M*

M is INTEGER

The number of rows in the matrix A. M >= 0.

*N*

N is INTEGER

The number of columns in the matrix A. N >= 0.

*A*

A is REAL array, dimension (LDA,N)

On entry, the M-by-N general matrix to be reduced.

On exit,

if m >= n, the diagonal and the first superdiagonal are

overwritten with the upper bidiagonal matrix B; the

elements below the diagonal, with the array TAUQ, represent

the orthogonal matrix Q as a product of elementary

reflectors, and the elements above the first superdiagonal,

with the array TAUP, represent the orthogonal matrix P as

a product of elementary reflectors;

if m < n, the diagonal and the first subdiagonal are

overwritten with the lower bidiagonal matrix B; the

elements below the first subdiagonal, with the array TAUQ,

represent the orthogonal matrix Q as a product of

elementary reflectors, and the elements above the diagonal,

with the array TAUP, represent the orthogonal matrix P as

a product of elementary reflectors.

See Further Details.

*LDA*

LDA is INTEGER

The leading dimension of the array A. LDA >= max(1,M).

*D*

D is REAL array, dimension (min(M,N))

The diagonal elements of the bidiagonal matrix B:

D(i) = A(i,i).

*E*

E is REAL array, dimension (min(M,N)-1)

The off-diagonal elements of the bidiagonal matrix B:

if m >= n, E(i) = A(i,i+1) for i = 1,2,...,n-1;

if m < n, E(i) = A(i+1,i) for i = 1,2,...,m-1.

*TAUQ*

TAUQ is REAL array, dimension (min(M,N))

The scalar factors of the elementary reflectors which

represent the orthogonal matrix Q. See Further Details.

*TAUP*

TAUP is REAL array, dimension (min(M,N))

The scalar factors of the elementary reflectors which

represent the orthogonal matrix P. See Further Details.

*WORK*

WORK is REAL array, dimension (MAX(1,LWORK))

On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

*LWORK*

LWORK is INTEGER

The length of the array WORK. LWORK >= max(1,M,N).

For optimum performance LWORK >= (M+N)*NB, where NB

is the optimal blocksize.

If LWORK = -1, then a workspace query is assumed; the routine

only calculates the optimal size of the WORK array, returns

this value as the first entry of the WORK array, and no error

message related to LWORK is issued by XERBLA.

*INFO*

INFO is INTEGER

= 0: successful exit

< 0: if INFO = -i, the i-th argument had an illegal value.

**Author**

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

November 2017

**Further Details:**

The matrices Q and P are represented as products of elementary

reflectors:

If m >= n,

Q = H(1) H(2) . . . H(n) and P = G(1) G(2) . . . G(n-1)

Each H(i) and G(i) has the form:

H(i) = I - tauq * v * v**T and G(i) = I - taup * u * u**T

where tauq and taup are real scalars, and v and u are real vectors;

v(1:i-1) = 0, v(i) = 1, and v(i+1:m) is stored on exit in A(i+1:m,i);

u(1:i) = 0, u(i+1) = 1, and u(i+2:n) is stored on exit in A(i,i+2:n);

tauq is stored in TAUQ(i) and taup in TAUP(i).

If m < n,

Q = H(1) H(2) . . . H(m-1) and P = G(1) G(2) . . . G(m)

Each H(i) and G(i) has the form:

H(i) = I - tauq * v * v**T and G(i) = I - taup * u * u**T

where tauq and taup are real scalars, and v and u are real vectors;

v(1:i) = 0, v(i+1) = 1, and v(i+2:m) is stored on exit in A(i+2:m,i);

u(1:i-1) = 0, u(i) = 1, and u(i+1:n) is stored on exit in A(i,i+1:n);

tauq is stored in TAUQ(i) and taup in TAUP(i).

The contents of A on exit are illustrated by the following examples:

m = 6 and n = 5 (m > n): m = 5 and n = 6 (m < n):

( d e u1 u1 u1 ) ( d u1 u1 u1 u1 u1 )

( v1 d e u2 u2 ) ( e d u2 u2 u2 u2 )

( v1 v2 d e u3 ) ( v1 e d u3 u3 u3 )

( v1 v2 v3 d e ) ( v1 v2 e d u4 u4 )

( v1 v2 v3 v4 d ) ( v1 v2 v3 e d u5 )

( v1 v2 v3 v4 v5 )

where d and e denote diagonal and off-diagonal elements of B, vi

denotes an element of the vector defining H(i), and ui an element of

the vector defining G(i).

**subroutine sgecon (character NORM, integer N, real, dimension( lda, * ) A, integer LDA, real ANORM, real RCOND, real, dimension( * ) WORK, integer, dimension( * ) IWORK, integer INFO)
SGECON**

**Purpose:**

SGECON estimates the reciprocal of the condition number of a general

real matrix A, in either the 1-norm or the infinity-norm, using

the LU factorization computed by SGETRF.

An estimate is obtained for norm(inv(A)), and the reciprocal of the

condition number is computed as

RCOND = 1 / ( norm(A) * norm(inv(A)) ).

**Parameters**

*NORM*

NORM is CHARACTER*1

Specifies whether the 1-norm condition number or the

infinity-norm condition number is required:

= ’1’ or ’O’: 1-norm;

= ’I’: Infinity-norm.

*N*

N is INTEGER

The order of the matrix A. N >= 0.

*A*

A is REAL array, dimension (LDA,N)

The factors L and U from the factorization A = P*L*U

as computed by SGETRF.

*LDA*

LDA is INTEGER

The leading dimension of the array A. LDA >= max(1,N).

*ANORM*

ANORM is REAL

If NORM = ’1’ or ’O’, the 1-norm of the original matrix A.

If NORM = ’I’, the infinity-norm of the original matrix A.

*RCOND*

RCOND is REAL

The reciprocal of the condition number of the matrix A,

computed as RCOND = 1/(norm(A) * norm(inv(A))).

*WORK*

WORK is REAL array, dimension (4*N)

*IWORK*

IWORK is INTEGER array, dimension (N)

*INFO*

INFO is INTEGER

= 0: successful exit

< 0: if INFO = -i, the i-th argument had an illegal value

**Author**

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

December 2016

**subroutine sgeequ (integer M, integer N, real, dimension( lda, * ) A, integer LDA, real, dimension( * ) R, real, dimension( * ) C, real ROWCND, real COLCND, real AMAX, integer INFO)
SGEEQU**

**Purpose:**

SGEEQU computes row and column scalings intended to equilibrate an

M-by-N matrix A and reduce its condition number. R returns the row

scale factors and C the column scale factors, chosen to try to make

the largest element in each row and column of the matrix B with

elements B(i,j)=R(i)*A(i,j)*C(j) have absolute value 1.

R(i) and C(j) are restricted to be between SMLNUM = smallest safe

number and BIGNUM = largest safe number. Use of these scaling

factors is not guaranteed to reduce the condition number of A but

works well in practice.

**Parameters**

*M*

M is INTEGER

The number of rows of the matrix A. M >= 0.

*N*

N is INTEGER

The number of columns of the matrix A. N >= 0.

*A*

A is REAL array, dimension (LDA,N)

The M-by-N matrix whose equilibration factors are

to be computed.

*LDA*

LDA is INTEGER

The leading dimension of the array A. LDA >= max(1,M).

*R*

R is REAL array, dimension (M)

If INFO = 0 or INFO > M, R contains the row scale factors

for A.

*C*

C is REAL array, dimension (N)

If INFO = 0, C contains the column scale factors for A.

*ROWCND*

ROWCND is REAL

If INFO = 0 or INFO > M, ROWCND contains the ratio of the

smallest R(i) to the largest R(i). If ROWCND >= 0.1 and

AMAX is neither too large nor too small, it is not worth

scaling by R.

*COLCND*

COLCND is REAL

If INFO = 0, COLCND contains the ratio of the smallest

C(i) to the largest C(i). If COLCND >= 0.1, it is not

worth scaling by C.

*AMAX*

AMAX is REAL

Absolute value of largest matrix element. If AMAX is very

close to overflow or very close to underflow, the matrix

should be scaled.

*INFO*

INFO is INTEGER

= 0: successful exit

< 0: if INFO = -i, the i-th argument had an illegal value

> 0: if INFO = i, and i is

<= M: the i-th row of A is exactly zero

> M: the (i-M)-th column of A is exactly zero

**Author**

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

December 2016

**subroutine sgeequb (integer M, integer N, real, dimension( lda, * ) A, integer LDA, real, dimension( * ) R, real, dimension( * ) C, real ROWCND, real COLCND, real AMAX, integer INFO)
SGEEQUB**

**Purpose:**

SGEEQUB computes row and column scalings intended to equilibrate an

M-by-N matrix A and reduce its condition number. R returns the row

scale factors and C the column scale factors, chosen to try to make

the largest element in each row and column of the matrix B with

elements B(i,j)=R(i)*A(i,j)*C(j) have an absolute value of at most

the radix.

R(i) and C(j) are restricted to be a power of the radix between

SMLNUM = smallest safe number and BIGNUM = largest safe number. Use

of these scaling factors is not guaranteed to reduce the condition

number of A but works well in practice.

This routine differs from SGEEQU by restricting the scaling factors

to a power of the radix. Barring over- and underflow, scaling by

these factors introduces no additional rounding errors. However, the

scaled entries’ magnitudes are no longer approximately 1 but lie

between sqrt(radix) and 1/sqrt(radix).

**Parameters**

*M*

M is INTEGER

The number of rows of the matrix A. M >= 0.

*N*

N is INTEGER

The number of columns of the matrix A. N >= 0.

*A*

A is REAL array, dimension (LDA,N)

The M-by-N matrix whose equilibration factors are

to be computed.

*LDA*

LDA is INTEGER

The leading dimension of the array A. LDA >= max(1,M).

*R*

R is REAL array, dimension (M)

If INFO = 0 or INFO > M, R contains the row scale factors

for A.

*C*

C is REAL array, dimension (N)

If INFO = 0, C contains the column scale factors for A.

*ROWCND*

ROWCND is REAL

If INFO = 0 or INFO > M, ROWCND contains the ratio of the

smallest R(i) to the largest R(i). If ROWCND >= 0.1 and

AMAX is neither too large nor too small, it is not worth

scaling by R.

*COLCND*

COLCND is REAL

If INFO = 0, COLCND contains the ratio of the smallest

C(i) to the largest C(i). If COLCND >= 0.1, it is not

worth scaling by C.

*AMAX*

AMAX is REAL

Absolute value of largest matrix element. If AMAX is very

close to overflow or very close to underflow, the matrix

should be scaled.

*INFO*

INFO is INTEGER

= 0: successful exit

< 0: if INFO = -i, the i-th argument had an illegal value

> 0: if INFO = i, and i is

<= M: the i-th row of A is exactly zero

> M: the (i-M)-th column of A is exactly zero

**Author**

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

December 2016

**subroutine sgehd2 (integer N, integer ILO, integer IHI, real, dimension( lda, * ) A, integer LDA, real, dimension( * ) TAU, real, dimension( * ) WORK, integer INFO)
SGEHD2** reduces a general square matrix to upper Hessenberg form using an unblocked algorithm.

**Purpose:**

SGEHD2 reduces a real general matrix A to upper Hessenberg form H by

an orthogonal similarity transformation: Q**T * A * Q = H .

**Parameters**

*N*

N is INTEGER

The order of the matrix A. N >= 0.

*ILO*

ILO is INTEGER

*IHI*

IHI is INTEGER

It is assumed that A is already upper triangular in rows

and columns 1:ILO-1 and IHI+1:N. ILO and IHI are normally

set by a previous call to SGEBAL; otherwise they should be

set to 1 and N respectively. See Further Details.

1 <= ILO <= IHI <= max(1,N).

*A*

A is REAL array, dimension (LDA,N)

On entry, the n by n general matrix to be reduced.

On exit, the upper triangle and the first subdiagonal of A

are overwritten with the upper Hessenberg matrix H, and the

elements below the first subdiagonal, with the array TAU,

represent the orthogonal matrix Q as a product of elementary

reflectors. See Further Details.

*LDA*

LDA is INTEGER

The leading dimension of the array A. LDA >= max(1,N).

*TAU*

TAU is REAL array, dimension (N-1)

The scalar factors of the elementary reflectors (see Further

Details).

*WORK*

WORK is REAL array, dimension (N)

*INFO*

INFO is INTEGER

= 0: successful exit.

< 0: if INFO = -i, the i-th argument had an illegal value.

**Author**

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

December 2016

**Further Details:**

The matrix Q is represented as a product of (ihi-ilo) elementary

reflectors

Q = H(ilo) H(ilo+1) . . . H(ihi-1).

Each H(i) has the form

H(i) = I - tau * v * v**T

where tau is a real scalar, and v is a real vector with

v(1:i) = 0, v(i+1) = 1 and v(ihi+1:n) = 0; v(i+2:ihi) is stored on

exit in A(i+2:ihi,i), and tau in TAU(i).

The contents of A are illustrated by the following example, with

n = 7, ilo = 2 and ihi = 6:

on entry, on exit,

( a a a a a a a ) ( a a h h h h a )

( a a a a a a ) ( a h h h h a )

( a a a a a a ) ( h h h h h h )

( a a a a a a ) ( v2 h h h h h )

( a a a a a a ) ( v2 v3 h h h h )

( a a a a a a ) ( v2 v3 v4 h h h )

( a ) ( a )

where a denotes an element of the original matrix A, h denotes a

modified element of the upper Hessenberg matrix H, and vi denotes an

element of the vector defining H(i).

**subroutine sgehrd (integer N, integer ILO, integer IHI, real, dimension( lda, * ) A, integer LDA, real, dimension( * ) TAU, real, dimension( * ) WORK, integer LWORK, integer INFO)
SGEHRD**

**Purpose:**

SGEHRD reduces a real general matrix A to upper Hessenberg form H by

an orthogonal similarity transformation: Q**T * A * Q = H .

**Parameters**

*N*

N is INTEGER

The order of the matrix A. N >= 0.

*ILO*

ILO is INTEGER

*IHI*

IHI is INTEGER

It is assumed that A is already upper triangular in rows

and columns 1:ILO-1 and IHI+1:N. ILO and IHI are normally

set by a previous call to SGEBAL; otherwise they should be

set to 1 and N respectively. See Further Details.

1 <= ILO <= IHI <= N, if N > 0; ILO=1 and IHI=0, if N=0.

*A*

A is REAL array, dimension (LDA,N)

On entry, the N-by-N general matrix to be reduced.

On exit, the upper triangle and the first subdiagonal of A

are overwritten with the upper Hessenberg matrix H, and the

elements below the first subdiagonal, with the array TAU,

represent the orthogonal matrix Q as a product of elementary

reflectors. See Further Details.

*LDA*

LDA is INTEGER

The leading dimension of the array A. LDA >= max(1,N).

*TAU*

TAU is REAL array, dimension (N-1)

The scalar factors of the elementary reflectors (see Further

Details). Elements 1:ILO-1 and IHI:N-1 of TAU are set to

zero.

*WORK*

WORK is REAL array, dimension (LWORK)

On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

*LWORK*

LWORK is INTEGER

The length of the array WORK. LWORK >= max(1,N).

For good performance, LWORK should generally be larger.

If LWORK = -1, then a workspace query is assumed; the routine

only calculates the optimal size of the WORK array, returns

this value as the first entry of the WORK array, and no error

message related to LWORK is issued by XERBLA.

*INFO*

INFO is INTEGER

= 0: successful exit

< 0: if INFO = -i, the i-th argument had an illegal value.

**Author**

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

December 2016

**Further Details:**

The matrix Q is represented as a product of (ihi-ilo) elementary

reflectors

Q = H(ilo) H(ilo+1) . . . H(ihi-1).

Each H(i) has the form

H(i) = I - tau * v * v**T

where tau is a real scalar, and v is a real vector with

v(1:i) = 0, v(i+1) = 1 and v(ihi+1:n) = 0; v(i+2:ihi) is stored on

exit in A(i+2:ihi,i), and tau in TAU(i).

The contents of A are illustrated by the following example, with

n = 7, ilo = 2 and ihi = 6:

on entry, on exit,

( a a a a a a a ) ( a a h h h h a )

( a a a a a a ) ( a h h h h a )

( a a a a a a ) ( h h h h h h )

( a a a a a a ) ( v2 h h h h h )

( a a a a a a ) ( v2 v3 h h h h )

( a a a a a a ) ( v2 v3 v4 h h h )

( a ) ( a )

where a denotes an element of the original matrix A, h denotes a

modified element of the upper Hessenberg matrix H, and vi denotes an

element of the vector defining H(i).

This file is a slight modification of LAPACK-3.0’s DGEHRD

subroutine incorporating improvements proposed by Quintana-Orti and

Van de Geijn (2006). (See DLAHR2.)

**subroutine sgelq2 (integer M, integer N, real, dimension( lda, * ) A, integer LDA, real, dimension( * ) TAU, real, dimension( * ) WORK, integer INFO)
SGELQ2** computes the LQ factorization of a general rectangular matrix using an unblocked algorithm.

**Purpose:**

SGELQ2 computes an LQ factorization of a real m-by-n matrix A:

A = ( L 0 ) * Q

where:

Q is a n-by-n orthogonal matrix;

L is an lower-triangular m-by-m matrix;

0 is a m-by-(n-m) zero matrix, if m < n.

**Parameters**

*M*

M is INTEGER

The number of rows of the matrix A. M >= 0.

*N*

N is INTEGER

The number of columns of the matrix A. N >= 0.

*A*

A is REAL array, dimension (LDA,N)

On entry, the m by n matrix A.

On exit, the elements on and below the diagonal of the array

contain the m by min(m,n) lower trapezoidal matrix L (L is

lower triangular if m <= n); the elements above the diagonal,

with the array TAU, represent the orthogonal matrix Q as a

product of elementary reflectors (see Further Details).

*LDA*

LDA is INTEGER

The leading dimension of the array A. LDA >= max(1,M).

*TAU*

TAU is REAL array, dimension (min(M,N))

The scalar factors of the elementary reflectors (see Further

Details).

*WORK*

WORK is REAL array, dimension (M)

*INFO*

INFO is INTEGER

= 0: successful exit

< 0: if INFO = -i, the i-th argument had an illegal value

**Author**

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

November 2019

**Further Details:**

The matrix Q is represented as a product of elementary reflectors

Q = H(k) . . . H(2) H(1), where k = min(m,n).

Each H(i) has the form

H(i) = I - tau * v * v**T

where tau is a real scalar, and v is a real vector with

v(1:i-1) = 0 and v(i) = 1; v(i+1:n) is stored on exit in A(i,i+1:n),

and tau in TAU(i).

**subroutine sgelqf (integer M, integer N, real, dimension( lda, * ) A, integer LDA, real, dimension( * ) TAU, real, dimension( * ) WORK, integer LWORK, integer INFO)
SGELQF**

**Purpose:**

SGELQF computes an LQ factorization of a real M-by-N matrix A:

A = ( L 0 ) * Q

where:

Q is a N-by-N orthogonal matrix;

L is an lower-triangular M-by-M matrix;

0 is a M-by-(N-M) zero matrix, if M < N.

**Parameters**

*M*

M is INTEGER

The number of rows of the matrix A. M >= 0.

*N*

N is INTEGER

The number of columns of the matrix A. N >= 0.

*A*

A is REAL array, dimension (LDA,N)

On entry, the M-by-N matrix A.

On exit, the elements on and below the diagonal of the array

contain the m-by-min(m,n) lower trapezoidal matrix L (L is

lower triangular if m <= n); the elements above the diagonal,

with the array TAU, represent the orthogonal matrix Q as a

product of elementary reflectors (see Further Details).

*LDA*

LDA is INTEGER

The leading dimension of the array A. LDA >= max(1,M).

*TAU*

TAU is REAL array, dimension (min(M,N))

The scalar factors of the elementary reflectors (see Further

Details).

*WORK*

WORK is REAL array, dimension (MAX(1,LWORK))

On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

*LWORK*

LWORK is INTEGER

The dimension of the array WORK. LWORK >= max(1,M).

For optimum performance LWORK >= M*NB, where NB is the

optimal blocksize.

If LWORK = -1, then a workspace query is assumed; the routine

only calculates the optimal size of the WORK array, returns

this value as the first entry of the WORK array, and no error

message related to LWORK is issued by XERBLA.

*INFO*

INFO is INTEGER

= 0: successful exit

< 0: if INFO = -i, the i-th argument had an illegal value

**Author**

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

November 2019

**Further Details:**

The matrix Q is represented as a product of elementary reflectors

Q = H(k) . . . H(2) H(1), where k = min(m,n).

Each H(i) has the form

H(i) = I - tau * v * v**T

where tau is a real scalar, and v is a real vector with

v(1:i-1) = 0 and v(i) = 1; v(i+1:n) is stored on exit in A(i,i+1:n),

and tau in TAU(i).

**subroutine sgemqrt (character SIDE, character TRANS, integer M, integer N, integer K, integer NB, real, dimension( ldv, * ) V, integer LDV, real, dimension( ldt, * ) T, integer LDT, real, dimension( ldc, * ) C, integer LDC, real, dimension( * ) WORK, integer INFO)
SGEMQRT**

**Purpose:**

SGEMQRT overwrites the general real M-by-N matrix C with

SIDE = ’L’ SIDE = ’R’

TRANS = ’N’: Q C C Q

TRANS = ’T’: Q**T C C Q**T

where Q is a real orthogonal matrix defined as the product of K

elementary reflectors:

Q = H(1) H(2) . . . H(K) = I - V T V**T

generated using the compact WY representation as returned by SGEQRT.

Q is of order M if SIDE = ’L’ and of order N if SIDE = ’R’.

**Parameters**

*SIDE*

SIDE is CHARACTER*1

= ’L’: apply Q or Q**T from the Left;

= ’R’: apply Q or Q**T from the Right.

*TRANS*

TRANS is CHARACTER*1

= ’N’: No transpose, apply Q;

= ’T’: Transpose, apply Q**T.

*M*

M is INTEGER

The number of rows of the matrix C. M >= 0.

*N*

N is INTEGER

The number of columns of the matrix C. N >= 0.

*K*

K is INTEGER

The number of elementary reflectors whose product defines

the matrix Q.

If SIDE = ’L’, M >= K >= 0;

if SIDE = ’R’, N >= K >= 0.

*NB*

NB is INTEGER

The block size used for the storage of T. K >= NB >= 1.

This must be the same value of NB used to generate T

in CGEQRT.

*V*

V is REAL array, dimension (LDV,K)

The i-th column must contain the vector which defines the

elementary reflector H(i), for i = 1,2,...,k, as returned by

CGEQRT in the first K columns of its array argument A.

*LDV*

LDV is INTEGER

The leading dimension of the array V.

If SIDE = ’L’, LDA >= max(1,M);

if SIDE = ’R’, LDA >= max(1,N).

*T*

T is REAL array, dimension (LDT,K)

The upper triangular factors of the block reflectors

as returned by CGEQRT, stored as a NB-by-N matrix.

*LDT*

LDT is INTEGER

The leading dimension of the array T. LDT >= NB.

*C*

C is REAL array, dimension (LDC,N)

On entry, the M-by-N matrix C.

On exit, C is overwritten by Q C, Q**T C, C Q**T or C Q.

*LDC*

LDC is INTEGER

The leading dimension of the array C. LDC >= max(1,M).

*WORK*

WORK is REAL array. The dimension of WORK is

N*NB if SIDE = ’L’, or M*NB if SIDE = ’R’.

*INFO*

INFO is INTEGER

= 0: successful exit

< 0: if INFO = -i, the i-th argument had an illegal value

**Author**

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

December 2016

**subroutine sgeql2 (integer M, integer N, real, dimension( lda, * ) A, integer LDA, real, dimension( * ) TAU, real, dimension( * ) WORK, integer INFO)
SGEQL2** computes the QL factorization of a general rectangular matrix using an unblocked algorithm.

**Purpose:**

SGEQL2 computes a QL factorization of a real m by n matrix A:

A = Q * L.

**Parameters**

*M*

M is INTEGER

The number of rows of the matrix A. M >= 0.

*N*

N is INTEGER

The number of columns of the matrix A. N >= 0.

*A*

A is REAL array, dimension (LDA,N)

On entry, the m by n matrix A.

On exit, if m >= n, the lower triangle of the subarray

A(m-n+1:m,1:n) contains the n by n lower triangular matrix L;

if m <= n, the elements on and below the (n-m)-th

superdiagonal contain the m by n lower trapezoidal matrix L;

the remaining elements, with the array TAU, represent the

orthogonal matrix Q as a product of elementary reflectors

(see Further Details).

*LDA*

LDA is INTEGER

The leading dimension of the array A. LDA >= max(1,M).

*TAU*

TAU is REAL array, dimension (min(M,N))

The scalar factors of the elementary reflectors (see Further

Details).

*WORK*

WORK is REAL array, dimension (N)

*INFO*

INFO is INTEGER

= 0: successful exit

< 0: if INFO = -i, the i-th argument had an illegal value

**Author**

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

December 2016

**Further Details:**

The matrix Q is represented as a product of elementary reflectors

Q = H(k) . . . H(2) H(1), where k = min(m,n).

Each H(i) has the form

H(i) = I - tau * v * v**T

where tau is a real scalar, and v is a real vector with

v(m-k+i+1:m) = 0 and v(m-k+i) = 1; v(1:m-k+i-1) is stored on exit in

A(1:m-k+i-1,n-k+i), and tau in TAU(i).

**subroutine sgeqlf (integer M, integer N, real, dimension( lda, * ) A, integer LDA, real, dimension( * ) TAU, real, dimension( * ) WORK, integer LWORK, integer INFO)
SGEQLF**

**Purpose:**

SGEQLF computes a QL factorization of a real M-by-N matrix A:

A = Q * L.

**Parameters**

*M*

M is INTEGER

The number of rows of the matrix A. M >= 0.

*N*

N is INTEGER

The number of columns of the matrix A. N >= 0.

*A*

A is REAL array, dimension (LDA,N)

On entry, the M-by-N matrix A.

On exit,

if m >= n, the lower triangle of the subarray

A(m-n+1:m,1:n) contains the N-by-N lower triangular matrix L;

if m <= n, the elements on and below the (n-m)-th

superdiagonal contain the M-by-N lower trapezoidal matrix L;

the remaining elements, with the array TAU, represent the

orthogonal matrix Q as a product of elementary reflectors

(see Further Details).

*LDA*

LDA is INTEGER

The leading dimension of the array A. LDA >= max(1,M).

*TAU*

The scalar factors of the elementary reflectors (see Further

Details).

*WORK*

WORK is REAL array, dimension (MAX(1,LWORK))

On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

*LWORK*

LWORK is INTEGER

The dimension of the array WORK. LWORK >= max(1,N).

For optimum performance LWORK >= N*NB, where NB is the

optimal blocksize.

only calculates the optimal size of the WORK array, returns

this value as the first entry of the WORK array, and no error

message related to LWORK is issued by XERBLA.

*INFO*

INFO is INTEGER

= 0: successful exit

< 0: if INFO = -i, the i-th argument had an illegal value

**Author**

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

December 2016

**Further Details:**

The matrix Q is represented as a product of elementary reflectors

Q = H(k) . . . H(2) H(1), where k = min(m,n).

Each H(i) has the form

H(i) = I - tau * v * v**T

where tau is a real scalar, and v is a real vector with

v(m-k+i+1:m) = 0 and v(m-k+i) = 1; v(1:m-k+i-1) is stored on exit in

A(1:m-k+i-1,n-k+i), and tau in TAU(i).

**subroutine sgeqp3 (integer M, integer N, real, dimension( lda, * ) A, integer LDA, integer, dimension( * ) JPVT, real, dimension( * ) TAU, real, dimension( * ) WORK, integer LWORK, integer INFO)
SGEQP3**

**Purpose:**

SGEQP3 computes a QR factorization with column pivoting of a

matrix A: A*P = Q*R using Level 3 BLAS.

**Parameters**

*M*

M is INTEGER

The number of rows of the matrix A. M >= 0.

*N*

N is INTEGER

The number of columns of the matrix A. N >= 0.

*A*

A is REAL array, dimension (LDA,N)

On entry, the M-by-N matrix A.

On exit, the upper triangle of the array contains the

min(M,N)-by-N upper trapezoidal matrix R; the elements below

the diagonal, together with the array TAU, represent the

orthogonal matrix Q as a product of min(M,N) elementary

reflectors.

*LDA*

LDA is INTEGER

The leading dimension of the array A. LDA >= max(1,M).

*JPVT*

JPVT is INTEGER array, dimension (N)

On entry, if JPVT(J).ne.0, the J-th column of A is permuted

to the front of A*P (a leading column); if JPVT(J)=0,

the J-th column of A is a free column.

On exit, if JPVT(J)=K, then the J-th column of A*P was the

the K-th column of A.

*TAU*

TAU is REAL array, dimension (min(M,N))

The scalar factors of the elementary reflectors.

*WORK*

WORK is REAL array, dimension (MAX(1,LWORK))

On exit, if INFO=0, WORK(1) returns the optimal LWORK.

*LWORK*

LWORK is INTEGER

The dimension of the array WORK. LWORK >= 3*N+1.

For optimal performance LWORK >= 2*N+( N+1 )*NB, where NB

is the optimal blocksize.

only calculates the optimal size of the WORK array, returns

this value as the first entry of the WORK array, and no error

message related to LWORK is issued by XERBLA.

*INFO*

INFO is INTEGER

= 0: successful exit.

< 0: if INFO = -i, the i-th argument had an illegal value.

**Author**

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

December 2016

**Further Details:**

The matrix Q is represented as a product of elementary reflectors

Q = H(1) H(2) . . . H(k), where k = min(m,n).

Each H(i) has the form

H(i) = I - tau * v * v**T

where tau is a real scalar, and v is a real/complex vector

with v(1:i-1) = 0 and v(i) = 1; v(i+1:m) is stored on exit in

A(i+1:m,i), and tau in TAU(i).

**Contributors:**

G. Quintana-Orti, Depto. de Informatica, Universidad Jaime I, Spain X. Sun, Computer Science Dept., Duke University, USA

**subroutine sgeqr2 (integer M, integer N, real, dimension( lda, * ) A, integer LDA, real, dimension( * ) TAU, real, dimension( * ) WORK, integer INFO)
SGEQR2** computes the QR factorization of a general rectangular matrix using an unblocked algorithm.

**Purpose:**

SGEQR2 computes a QR factorization of a real m-by-n matrix A:

A = Q * ( R ),

( 0 )

where:

Q is a m-by-m orthogonal matrix;

R is an upper-triangular n-by-n matrix;

0 is a (m-n)-by-n zero matrix, if m > n.

**Parameters**

*M*

M is INTEGER

The number of rows of the matrix A. M >= 0.

*N*

N is INTEGER

The number of columns of the matrix A. N >= 0.

*A*

A is REAL array, dimension (LDA,N)

On entry, the m by n matrix A.

On exit, the elements on and above the diagonal of the array

contain the min(m,n) by n upper trapezoidal matrix R (R is

upper triangular if m >= n); the elements below the diagonal,

with the array TAU, represent the orthogonal matrix Q as a

product of elementary reflectors (see Further Details).

*LDA*

LDA is INTEGER

The leading dimension of the array A. LDA >= max(1,M).

*TAU*

The scalar factors of the elementary reflectors (see Further

Details).

*WORK*

WORK is REAL array, dimension (N)

*INFO*

INFO is INTEGER

= 0: successful exit

< 0: if INFO = -i, the i-th argument had an illegal value

**Author**

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

November 2019

**Further Details:**

The matrix Q is represented as a product of elementary reflectors

Q = H(1) H(2) . . . H(k), where k = min(m,n).

Each H(i) has the form

H(i) = I - tau * v * v**T

where tau is a real scalar, and v is a real vector with

v(1:i-1) = 0 and v(i) = 1; v(i+1:m) is stored on exit in A(i+1:m,i),

and tau in TAU(i).

**subroutine sgeqr2p (integer M, integer N, real, dimension( lda, * ) A, integer LDA, real, dimension( * ) TAU, real, dimension( * ) WORK, integer INFO)
SGEQR2P** computes the QR factorization of a general rectangular matrix with non-negative diagonal elements using an unblocked algorithm.

**Purpose:**

SGEQR2P computes a QR factorization of a real m-by-n matrix A:

A = Q * ( R ),

( 0 )

where:

Q is a m-by-m orthogonal matrix;

R is an upper-triangular n-by-n matrix with nonnegative diagonal

entries;

0 is a (m-n)-by-n zero matrix, if m > n.

**Parameters**

*M*

M is INTEGER

The number of rows of the matrix A. M >= 0.

*N*

N is INTEGER

The number of columns of the matrix A. N >= 0.

*A*

A is REAL array, dimension (LDA,N)

On entry, the m by n matrix A.

On exit, the elements on and above the diagonal of the array

contain the min(m,n) by n upper trapezoidal matrix R (R is

upper triangular if m >= n). The diagonal entries of R

are nonnegative; the elements below the diagonal,

with the array TAU, represent the orthogonal matrix Q as a

product of elementary reflectors (see Further Details).

*LDA*

LDA is INTEGER

The leading dimension of the array A. LDA >= max(1,M).

*TAU*

The scalar factors of the elementary reflectors (see Further

Details).

*WORK*

WORK is REAL array, dimension (N)

*INFO*

INFO is INTEGER

= 0: successful exit

< 0: if INFO = -i, the i-th argument had an illegal value

**Author**

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

November 2019

**Further Details:**

The matrix Q is represented as a product of elementary reflectors

Q = H(1) H(2) . . . H(k), where k = min(m,n).

Each H(i) has the form

H(i) = I - tau * v * v**T

where tau is a real scalar, and v is a real vector with

v(1:i-1) = 0 and v(i) = 1; v(i+1:m) is stored on exit in A(i+1:m,i),

and tau in TAU(i).

See Lapack Working Note 203 for details

**subroutine sgeqrf (integer M, integer N, real, dimension( lda, * ) A, integer LDA, real, dimension( * ) TAU, real, dimension( * ) WORK, integer LWORK, integer INFO)
SGEQRF**

**Purpose:**

SGEQRF computes a QR factorization of a real M-by-N matrix A:

A = Q * ( R ),

( 0 )

where:

Q is a M-by-M orthogonal matrix;

R is an upper-triangular N-by-N matrix;

0 is a (M-N)-by-N zero matrix, if M > N.

**Parameters**

*M*

M is INTEGER

The number of rows of the matrix A. M >= 0.

*N*

N is INTEGER

The number of columns of the matrix A. N >= 0.

*A*

A is REAL array, dimension (LDA,N)

On entry, the M-by-N matrix A.

On exit, the elements on and above the diagonal of the array

contain the min(M,N)-by-N upper trapezoidal matrix R (R is

upper triangular if m >= n); the elements below the diagonal,

with the array TAU, represent the orthogonal matrix Q as a

product of min(m,n) elementary reflectors (see Further

Details).

*LDA*

LDA is INTEGER

The leading dimension of the array A. LDA >= max(1,M).

*TAU*

The scalar factors of the elementary reflectors (see Further

Details).

*WORK*

On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

*LWORK*

LWORK is INTEGER

The dimension of the array WORK. LWORK >= max(1,N).

For optimum performance LWORK >= N*NB, where NB is

the optimal blocksize.

only calculates the optimal size of the WORK array, returns

this value as the first entry of the WORK array, and no error

message related to LWORK is issued by XERBLA.

*INFO*

INFO is INTEGER

= 0: successful exit

< 0: if INFO = -i, the i-th argument had an illegal value

**Author**

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

November 2019

**Further Details:**

The matrix Q is represented as a product of elementary reflectors

Q = H(1) H(2) . . . H(k), where k = min(m,n).

Each H(i) has the form

H(i) = I - tau * v * v**T

where tau is a real scalar, and v is a real vector with

v(1:i-1) = 0 and v(i) = 1; v(i+1:m) is stored on exit in A(i+1:m,i),

and tau in TAU(i).

**subroutine sgeqrfp (integer M, integer N, real, dimension( lda, * ) A, integer LDA, real, dimension( * ) TAU, real, dimension( * ) WORK, integer LWORK, integer INFO)
SGEQRFP**

**Purpose:**

SGEQR2P computes a QR factorization of a real M-by-N matrix A:

A = Q * ( R ),

( 0 )

where:

Q is a M-by-M orthogonal matrix;

R is an upper-triangular N-by-N matrix with nonnegative diagonal

entries;

0 is a (M-N)-by-N zero matrix, if M > N.

**Parameters**

*M*

M is INTEGER

The number of rows of the matrix A. M >= 0.

*N*

N is INTEGER

The number of columns of the matrix A. N >= 0.

*A*

A is REAL array, dimension (LDA,N)

On entry, the M-by-N matrix A.

On exit, the elements on and above the diagonal of the array

contain the min(M,N)-by-N upper trapezoidal matrix R (R is

upper triangular if m >= n). The diagonal entries of R

are nonnegative; the elements below the diagonal,

with the array TAU, represent the orthogonal matrix Q as a

product of min(m,n) elementary reflectors (see Further

Details).

*LDA*

LDA is INTEGER

The leading dimension of the array A. LDA >= max(1,M).

*TAU*

The scalar factors of the elementary reflectors (see Further

Details).

*WORK*

On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

*LWORK*

LWORK is INTEGER

The dimension of the array WORK. LWORK >= max(1,N).

For optimum performance LWORK >= N*NB, where NB is

the optimal blocksize.

only calculates the optimal size of the WORK array, returns

this value as the first entry of the WORK array, and no error

message related to LWORK is issued by XERBLA.

*INFO*

INFO is INTEGER

= 0: successful exit

< 0: if INFO = -i, the i-th argument had an illegal value

**Author**

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

November 2019

**Further Details:**

The matrix Q is represented as a product of elementary reflectors

Q = H(1) H(2) . . . H(k), where k = min(m,n).

Each H(i) has the form

H(i) = I - tau * v * v**T

v(1:i-1) = 0 and v(i) = 1; v(i+1:m) is stored on exit in A(i+1:m,i),

and tau in TAU(i).

See Lapack Working Note 203 for details

**subroutine sgeqrt (integer M, integer N, integer NB, real, dimension( lda, * ) A, integer LDA, real, dimension( ldt, * ) T, integer LDT, real, dimension( * ) WORK, integer INFO)
SGEQRT**

**Purpose:**

SGEQRT computes a blocked QR factorization of a real M-by-N matrix A

using the compact WY representation of Q.

**Parameters**

*M*

M is INTEGER

The number of rows of the matrix A. M >= 0.

*N*

N is INTEGER

The number of columns of the matrix A. N >= 0.

*NB*

NB is INTEGER

The block size to be used in the blocked QR. MIN(M,N) >= NB >= 1.

*A*

A is REAL array, dimension (LDA,N)

On entry, the M-by-N matrix A.

On exit, the elements on and above the diagonal of the array

contain the min(M,N)-by-N upper trapezoidal matrix R (R is

upper triangular if M >= N); the elements below the diagonal

are the columns of V.

*LDA*

LDA is INTEGER

The leading dimension of the array A. LDA >= max(1,M).

*T*

T is REAL array, dimension (LDT,MIN(M,N))

The upper triangular block reflectors stored in compact form

as a sequence of upper triangular blocks. See below

for further details.

*LDT*

LDT is INTEGER

The leading dimension of the array T. LDT >= NB.

*WORK*

WORK is REAL array, dimension (NB*N)

*INFO*

INFO is INTEGER

= 0: successful exit

< 0: if INFO = -i, the i-th argument had an illegal value

**Author**

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

June 2017

**Further Details:**

The matrix V stores the elementary reflectors H(i) in the i-th column

below the diagonal. For example, if M=5 and N=3, the matrix V is

V = ( 1 )

( v1 1 )

( v1 v2 1 )

( v1 v2 v3 )

( v1 v2 v3 )

where the vi’s represent the vectors which define H(i), which are returned

in the matrix A. The 1’s along the diagonal of V are not stored in A.

Let K=MIN(M,N). The number of blocks is B = ceiling(K/NB), where each

block is of order NB except for the last block, which is of order

IB = K - (B-1)*NB. For each of the B blocks, a upper triangular block

reflector factor is computed: T1, T2, ..., TB. The NB-by-NB (and IB-by-IB

for the last block) T’s are stored in the NB-by-K matrix T as

T = (T1 T2 ... TB).

**subroutine sgeqrt2 (integer M, integer N, real, dimension( lda, * ) A, integer LDA, real, dimension( ldt, * ) T, integer LDT, integer INFO)
SGEQRT2** computes a QR factorization of a general real or complex matrix using the compact WY representation of Q.

**Purpose:**

SGEQRT2 computes a QR factorization of a real M-by-N matrix A,

using the compact WY representation of Q.

**Parameters**

*M*

M is INTEGER

The number of rows of the matrix A. M >= N.

*N*

N is INTEGER

The number of columns of the matrix A. N >= 0.

*A*

A is REAL array, dimension (LDA,N)

On entry, the real M-by-N matrix A. On exit, the elements on and

above the diagonal contain the N-by-N upper triangular matrix R; the

elements below the diagonal are the columns of V. See below for

further details.

*LDA*

LDA is INTEGER

The leading dimension of the array A. LDA >= max(1,M).

*T*

T is REAL array, dimension (LDT,N)

The N-by-N upper triangular factor of the block reflector.

The elements on and above the diagonal contain the block

reflector T; the elements below the diagonal are not used.

See below for further details.

*LDT*

LDT is INTEGER

The leading dimension of the array T. LDT >= max(1,N).

*INFO*

INFO is INTEGER

= 0: successful exit

< 0: if INFO = -i, the i-th argument had an illegal value

**Author**

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

December 2016

**Further Details:**

The matrix V stores the elementary reflectors H(i) in the i-th column

below the diagonal. For example, if M=5 and N=3, the matrix V is

V = ( 1 )

( v1 1 )

( v1 v2 1 )

( v1 v2 v3 )

( v1 v2 v3 )

where the vi’s represent the vectors which define H(i), which are returned

in the matrix A. The 1’s along the diagonal of V are not stored in A. The

block reflector H is then given by

H = I - V * T * V**T

where V**T is the transpose of V.

**recursive subroutine sgeqrt3 (integer M, integer N, real, dimension( lda, * ) A, integer LDA, real, dimension( ldt, * ) T, integer LDT, integer INFO)
SGEQRT3** recursively computes a QR factorization of a general real or complex matrix using the compact WY representation of Q.

**Purpose:**

SGEQRT3 recursively computes a QR factorization of a real M-by-N

matrix A, using the compact WY representation of Q.

Based on the algorithm of Elmroth and Gustavson,

IBM J. Res. Develop. Vol 44 No. 4 July 2000.

**Parameters**

*M*

M is INTEGER

The number of rows of the matrix A. M >= N.

*N*

N is INTEGER

The number of columns of the matrix A. N >= 0.

*A*

A is REAL array, dimension (LDA,N)

On entry, the real M-by-N matrix A. On exit, the elements on and

above the diagonal contain the N-by-N upper triangular matrix R; the

elements below the diagonal are the columns of V. See below for

further details.

*LDA*

LDA is INTEGER

The leading dimension of the array A. LDA >= max(1,M).

*T*

T is REAL array, dimension (LDT,N)

The N-by-N upper triangular factor of the block reflector.

The elements on and above the diagonal contain the block

reflector T; the elements below the diagonal are not used.

See below for further details.

*LDT*

LDT is INTEGER

The leading dimension of the array T. LDT >= max(1,N).

*INFO*

INFO is INTEGER

= 0: successful exit

< 0: if INFO = -i, the i-th argument had an illegal value

**Author**

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

June 2016

**Further Details:**

The matrix V stores the elementary reflectors H(i) in the i-th column

below the diagonal. For example, if M=5 and N=3, the matrix V is

V = ( 1 )

( v1 1 )

( v1 v2 1 )

( v1 v2 v3 )

( v1 v2 v3 )

where the vi’s represent the vectors which define H(i), which are returned

in the matrix A. The 1’s along the diagonal of V are not stored in A. The

block reflector H is then given by

H = I - V * T * V**T

where V**T is the transpose of V.

For details of the algorithm, see Elmroth and Gustavson (cited above).

**subroutine sgerfs (character TRANS, integer N, integer NRHS, real, dimension( lda, * ) A, integer LDA, real, dimension( ldaf, * ) AF, integer LDAF, integer, dimension( * ) IPIV, real, dimension( ldb, * ) B, integer LDB, real, dimension( ldx, * ) X, integer LDX, real, dimension( * ) FERR, real, dimension( * ) BERR, real, dimension( * ) WORK, integer, dimension( * ) IWORK, integer INFO)
SGERFS**

**Purpose:**

SGERFS improves the computed solution to a system of linear

equations and provides error bounds and backward error estimates for

the solution.

**Parameters**

*TRANS*

TRANS is CHARACTER*1

Specifies the form of the system of equations:

= ’N’: A * X = B (No transpose)

= ’T’: A**T * X = B (Transpose)

= ’C’: A**H * X = B (Conjugate transpose = Transpose)

*N*

N is INTEGER

The order of the matrix A. N >= 0.

*NRHS*

NRHS is INTEGER

The number of right hand sides, i.e., the number of columns

of the matrices B and X. NRHS >= 0.

*A*

A is REAL array, dimension (LDA,N)

The original N-by-N matrix A.

*LDA*

LDA is INTEGER

The leading dimension of the array A. LDA >= max(1,N).

*AF*

AF is REAL array, dimension (LDAF,N)

The factors L and U from the factorization A = P*L*U

as computed by SGETRF.

*LDAF*

LDAF is INTEGER

The leading dimension of the array AF. LDAF >= max(1,N).

*IPIV*

IPIV is INTEGER array, dimension (N)

The pivot indices from SGETRF; for 1<=i<=N, row i of the

matrix was interchanged with row IPIV(i).

*B*

B is REAL array, dimension (LDB,NRHS)

The right hand side matrix B.

*LDB*

LDB is INTEGER

The leading dimension of the array B. LDB >= max(1,N).

*X*

X is REAL array, dimension (LDX,NRHS)

On entry, the solution matrix X, as computed by SGETRS.

On exit, the improved solution matrix X.

*LDX*

LDX is INTEGER

The leading dimension of the array X. LDX >= max(1,N).

*FERR*

FERR is REAL array, dimension (NRHS)

The estimated forward error bound for each solution vector

X(j) (the j-th column of the solution matrix X).

If XTRUE is the true solution corresponding to X(j), FERR(j)

is an estimated upper bound for the magnitude of the largest

element in (X(j) - XTRUE) divided by the magnitude of the

largest element in X(j). The estimate is as reliable as

the estimate for RCOND, and is almost always a slight

overestimate of the true error.

*BERR*

BERR is REAL array, dimension (NRHS)

The componentwise relative backward error of each solution

vector X(j) (i.e., the smallest relative change in

any element of A or B that makes X(j) an exact solution).

*WORK*

WORK is REAL array, dimension (3*N)

*IWORK*

IWORK is INTEGER array, dimension (N)

*INFO*

INFO is INTEGER

= 0: successful exit

< 0: if INFO = -i, the i-th argument had an illegal value

**Internal Parameters:**

ITMAX is the maximum number of steps of iterative refinement.

**Author**

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

December 2016

**subroutine sgerfsx (character TRANS, character EQUED, integer N, integer NRHS, real, dimension( lda, * ) A, integer LDA, real, dimension( ldaf, * ) AF, integer LDAF, integer, dimension( * ) IPIV, real, dimension( * ) R, real, dimension( * ) C, real, dimension( ldb, * ) B, integer LDB, real, dimension( ldx , * ) X, integer LDX, real RCOND, real, dimension( * ) BERR, integer N_ERR_BNDS, real, dimension( nrhs, * ) ERR_BNDS_NORM, real, dimension( nrhs, * ) ERR_BNDS_COMP, integer NPARAMS, real, dimension( * ) PARAMS, real, dimension( * ) WORK, integer, dimension( * ) IWORK, integer INFO)
SGERFSX**

**Purpose:**

SGERFSX improves the computed solution to a system of linear

equations and provides error bounds and backward error estimates

for the solution. In addition to normwise error bound, the code

provides maximum componentwise error bound if possible. See

comments for ERR_BNDS_NORM and ERR_BNDS_COMP for details of the

error bounds.

The original system of linear equations may have been equilibrated

before calling this routine, as described by arguments EQUED, R

and C below. In this case, the solution and error bounds returned

are for the original unequilibrated system.

Some optional parameters are bundled in the PARAMS array. These

settings determine how refinement is performed, but often the

defaults are acceptable. If the defaults are acceptable, users

can pass NPARAMS = 0 which prevents the source code from accessing

the PARAMS argument.

**Parameters**

*TRANS*

TRANS is CHARACTER*1

Specifies the form of the system of equations:

= ’N’: A * X = B (No transpose)

= ’T’: A**T * X = B (Transpose)

= ’C’: A**H * X = B (Conjugate transpose = Transpose)

*EQUED*

EQUED is CHARACTER*1

Specifies the form of equilibration that was done to A

before calling this routine. This is needed to compute

the solution and error bounds correctly.

= ’N’: No equilibration

= ’R’: Row equilibration, i.e., A has been premultiplied by

diag(R).

= ’C’: Column equilibration, i.e., A has been postmultiplied

by diag(C).

= ’B’: Both row and column equilibration, i.e., A has been

replaced by diag(R) * A * diag(C).

The right hand side B has been changed accordingly.

*N*

N is INTEGER

The order of the matrix A. N >= 0.

*NRHS*

NRHS is INTEGER

The number of right hand sides, i.e., the number of columns

of the matrices B and X. NRHS >= 0.

*A*

A is REAL array, dimension (LDA,N)

The original N-by-N matrix A.

*LDA*

LDA is INTEGER

The leading dimension of the array A. LDA >= max(1,N).

*AF*

AF is REAL array, dimension (LDAF,N)

The factors L and U from the factorization A = P*L*U

as computed by SGETRF.

*LDAF*

LDAF is INTEGER

The leading dimension of the array AF. LDAF >= max(1,N).

*IPIV*

IPIV is INTEGER array, dimension (N)

The pivot indices from SGETRF; for 1<=i<=N, row i of the

matrix was interchanged with row IPIV(i).

*R*

R is REAL array, dimension (N)

The row scale factors for A. If EQUED = ’R’ or ’B’, A is

multiplied on the left by diag(R); if EQUED = ’N’ or ’C’, R

is not accessed.

If R is accessed, each element of R should be a power of the radix

to ensure a reliable solution and error estimates. Scaling by

powers of the radix does not cause rounding errors unless the

result underflows or overflows. Rounding errors during scaling

lead to refining with a matrix that is not equivalent to the

input matrix, producing error estimates that may not be

reliable.

*C*

C is REAL array, dimension (N)

The column scale factors for A. If EQUED = ’C’ or ’B’, A is

multiplied on the right by diag(C); if EQUED = ’N’ or ’R’, C

is not accessed.

If C is accessed, each element of C should be a power of the radix

to ensure a reliable solution and error estimates. Scaling by

powers of the radix does not cause rounding errors unless the

result underflows or overflows. Rounding errors during scaling

lead to refining with a matrix that is not equivalent to the

input matrix, producing error estimates that may not be

reliable.

*B*

B is REAL array, dimension (LDB,NRHS)

The right hand side matrix B.

*LDB*

LDB is INTEGER

The leading dimension of the array B. LDB >= max(1,N).

*X*

X is REAL array, dimension (LDX,NRHS)

On entry, the solution matrix X, as computed by SGETRS.

On exit, the improved solution matrix X.

*LDX*

LDX is INTEGER

The leading dimension of the array X. LDX >= max(1,N).

*RCOND*

RCOND is REAL

Reciprocal scaled condition number. This is an estimate of the

reciprocal Skeel condition number of the matrix A after

equilibration (if done). If this is less than the machine

precision (in particular, if it is zero), the matrix is singular

to working precision. Note that the error may still be small even

if this number is very small and the matrix appears ill-

conditioned.

*BERR*

BERR is REAL array, dimension (NRHS)

Componentwise relative backward error. This is the

componentwise relative backward error of each solution vector X(j)

(i.e., the smallest relative change in any element of A or B that

makes X(j) an exact solution).

*N_ERR_BNDS*

N_ERR_BNDS is INTEGER

Number of error bounds to return for each right hand side

and each type (normwise or componentwise). See ERR_BNDS_NORM and

ERR_BNDS_COMP below.

*ERR_BNDS_NORM*

ERR_BNDS_NORM is REAL array, dimension (NRHS, N_ERR_BNDS)

For each right-hand side, this array contains information about

various error bounds and condition numbers corresponding to the

normwise relative error, which is defined as follows:

Normwise relative error in the ith solution vector:

max_j (abs(XTRUE(j,i) - X(j,i)))

------------------------------

max_j abs(X(j,i))

The array is indexed by the type of error information as described

below. There currently are up to three pieces of information

returned.

The first index in ERR_BNDS_NORM(i,:) corresponds to the ith

right-hand side.

The second index in ERR_BNDS_NORM(:,err) contains the following

three fields:

err = 1 "Trust/don’t trust" boolean. Trust the answer if the

reciprocal condition number is less than the threshold

sqrt(n) * slamch(’Epsilon’).

err = 2 "Guaranteed" error bound: The estimated forward error,

almost certainly within a factor of 10 of the true error

so long as the next entry is greater than the threshold

sqrt(n) * slamch(’Epsilon’). This error bound should only

be trusted if the previous boolean is true.

err = 3 Reciprocal condition number: Estimated normwise

reciprocal condition number. Compared with the threshold

sqrt(n) * slamch(’Epsilon’) to determine if the error

estimate is "guaranteed". These reciprocal condition

numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some

appropriately scaled matrix Z.

Let Z = S*A, where S scales each row by a power of the

radix so all absolute row sums of Z are approximately 1.

See Lapack Working Note 165 for further details and extra

cautions.

*ERR_BNDS_COMP*

ERR_BNDS_COMP is REAL array, dimension (NRHS, N_ERR_BNDS)

For each right-hand side, this array contains information about

various error bounds and condition numbers corresponding to the

componentwise relative error, which is defined as follows:

Componentwise relative error in the ith solution vector:

abs(XTRUE(j,i) - X(j,i))

max_j ----------------------

abs(X(j,i))

The array is indexed by the right-hand side i (on which the

componentwise relative error depends), and the type of error

information as described below. There currently are up to three

pieces of information returned for each right-hand side. If

componentwise accuracy is not requested (PARAMS(3) = 0.0), then

ERR_BNDS_COMP is not accessed. If N_ERR_BNDS < 3, then at most

the first (:,N_ERR_BNDS) entries are returned.

The first index in ERR_BNDS_COMP(i,:) corresponds to the ith

right-hand side.

The second index in ERR_BNDS_COMP(:,err) contains the following

three fields:

err = 1 "Trust/don’t trust" boolean. Trust the answer if the

reciprocal condition number is less than the threshold

sqrt(n) * slamch(’Epsilon’).

err = 2 "Guaranteed" error bound: The estimated forward error,

almost certainly within a factor of 10 of the true error

so long as the next entry is greater than the threshold

sqrt(n) * slamch(’Epsilon’). This error bound should only

be trusted if the previous boolean is true.

err = 3 Reciprocal condition number: Estimated componentwise

reciprocal condition number. Compared with the threshold

sqrt(n) * slamch(’Epsilon’) to determine if the error

estimate is "guaranteed". These reciprocal condition

numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some

appropriately scaled matrix Z.

Let Z = S*(A*diag(x)), where x is the solution for the

current right-hand side and S scales each row of

A*diag(x) by a power of the radix so all absolute row

sums of Z are approximately 1.

See Lapack Working Note 165 for further details and extra

cautions.

*NPARAMS*

NPARAMS is INTEGER

Specifies the number of parameters set in PARAMS. If <= 0, the

PARAMS array is never referenced and default values are used.

*PARAMS*

PARAMS is REAL array, dimension NPARAMS

Specifies algorithm parameters. If an entry is < 0.0, then

that entry will be filled with default value used for that

parameter. Only positions up to NPARAMS are accessed; defaults

are used for higher-numbered parameters.

PARAMS(LA_LINRX_ITREF_I = 1) : Whether to perform iterative

refinement or not.

Default: 1.0

= 0.0: No refinement is performed, and no error bounds are

computed.

= 1.0: Use the double-precision refinement algorithm,

possibly with doubled-single computations if the

compilation environment does not support DOUBLE

PRECISION.

(other values are reserved for future use)

PARAMS(LA_LINRX_ITHRESH_I = 2) : Maximum number of residual

computations allowed for refinement.

Default: 10

Aggressive: Set to 100 to permit convergence using approximate

factorizations or factorizations other than LU. If

the factorization uses a technique other than

Gaussian elimination, the guarantees in

err_bnds_norm and err_bnds_comp may no longer be

trustworthy.

PARAMS(LA_LINRX_CWISE_I = 3) : Flag determining if the code

will attempt to find a solution with small componentwise

relative error in the double-precision algorithm. Positive

is true, 0.0 is false.

Default: 1.0 (attempt componentwise convergence)

*WORK*

WORK is REAL array, dimension (4*N)

*IWORK*

IWORK is INTEGER array, dimension (N)

*INFO*

INFO is INTEGER

= 0: Successful exit. The solution to every right-hand side is

guaranteed.

< 0: If INFO = -i, the i-th argument had an illegal value

> 0 and <= N: U(INFO,INFO) is exactly zero. The factorization

has been completed, but the factor U is exactly singular, so

the solution and error bounds could not be computed. RCOND = 0

is returned.

= N+J: The solution corresponding to the Jth right-hand side is

not guaranteed. The solutions corresponding to other right-

hand sides K with K > J may not be guaranteed as well, but

only the first such right-hand side is reported. If a small

componentwise error is not requested (PARAMS(3) = 0.0) then

the Jth right-hand side is the first with a normwise error

bound that is not guaranteed (the smallest J such

that ERR_BNDS_NORM(J,1) = 0.0). By default (PARAMS(3) = 1.0)

the Jth right-hand side is the first with either a normwise or

componentwise error bound that is not guaranteed (the smallest

J such that either ERR_BNDS_NORM(J,1) = 0.0 or

ERR_BNDS_COMP(J,1) = 0.0). See the definition of

ERR_BNDS_NORM(:,1) and ERR_BNDS_COMP(:,1). To get information

about all of the right-hand sides check ERR_BNDS_NORM or

ERR_BNDS_COMP.

**Author**

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

December 2016

**subroutine sgerq2 (integer M, integer N, real, dimension( lda, * ) A, integer LDA, real, dimension( * ) TAU, real, dimension( * ) WORK, integer INFO)
SGERQ2** computes the RQ factorization of a general rectangular matrix using an unblocked algorithm.

**Purpose:**

SGERQ2 computes an RQ factorization of a real m by n matrix A:

A = R * Q.

**Parameters**

*M*

M is INTEGER

The number of rows of the matrix A. M >= 0.

*N*

N is INTEGER

The number of columns of the matrix A. N >= 0.

*A*

A is REAL array, dimension (LDA,N)

On entry, the m by n matrix A.

On exit, if m <= n, the upper triangle of the subarray

A(1:m,n-m+1:n) contains the m by m upper triangular matrix R;

if m >= n, the elements on and above the (m-n)-th subdiagonal

contain the m by n upper trapezoidal matrix R; the remaining

elements, with the array TAU, represent the orthogonal matrix

Q as a product of elementary reflectors (see Further

Details).

*LDA*

LDA is INTEGER

The leading dimension of the array A. LDA >= max(1,M).

*TAU*

The scalar factors of the elementary reflectors (see Further

Details).

*WORK*

WORK is REAL array, dimension (M)

*INFO*

INFO is INTEGER

= 0: successful exit

< 0: if INFO = -i, the i-th argument had an illegal value

**Author**

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

December 2016

**Further Details:**

The matrix Q is represented as a product of elementary reflectors

Q = H(1) H(2) . . . H(k), where k = min(m,n).

Each H(i) has the form

H(i) = I - tau * v * v**T

where tau is a real scalar, and v is a real vector with

v(n-k+i+1:n) = 0 and v(n-k+i) = 1; v(1:n-k+i-1) is stored on exit in

A(m-k+i,1:n-k+i-1), and tau in TAU(i).

**subroutine sgerqf (integer M, integer N, real, dimension( lda, * ) A, integer LDA, real, dimension( * ) TAU, real, dimension( * ) WORK, integer LWORK, integer INFO)
SGERQF**

**Purpose:**

SGERQF computes an RQ factorization of a real M-by-N matrix A:

A = R * Q.

**Parameters**

*M*

M is INTEGER

The number of rows of the matrix A. M >= 0.

*N*

N is INTEGER

The number of columns of the matrix A. N >= 0.

*A*

A is REAL array, dimension (LDA,N)

On entry, the M-by-N matrix A.

On exit,

if m <= n, the upper triangle of the subarray

A(1:m,n-m+1:n) contains the M-by-M upper triangular matrix R;

if m >= n, the elements on and above the (m-n)-th subdiagonal

contain the M-by-N upper trapezoidal matrix R;

the remaining elements, with the array TAU, represent the

orthogonal matrix Q as a product of min(m,n) elementary

reflectors (see Further Details).

*LDA*

LDA is INTEGER

The leading dimension of the array A. LDA >= max(1,M).

*TAU*

The scalar factors of the elementary reflectors (see Further

Details).

*WORK*

On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

*LWORK*

LWORK is INTEGER

The dimension of the array WORK. LWORK >= max(1,M).

For optimum performance LWORK >= M*NB, where NB is

the optimal blocksize.

only calculates the optimal size of the WORK array, returns

this value as the first entry of the WORK array, and no error

message related to LWORK is issued by XERBLA.

*INFO*

INFO is INTEGER

= 0: successful exit

< 0: if INFO = -i, the i-th argument had an illegal value

**Author**

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

December 2016

**Further Details:**

The matrix Q is represented as a product of elementary reflectors

Q = H(1) H(2) . . . H(k), where k = min(m,n).

Each H(i) has the form

H(i) = I - tau * v * v**T

where tau is a real scalar, and v is a real vector with

v(n-k+i+1:n) = 0 and v(n-k+i) = 1; v(1:n-k+i-1) is stored on exit in

A(m-k+i,1:n-k+i-1), and tau in TAU(i).

**subroutine sgesvj (character*1 JOBA, character*1 JOBU, character*1 JOBV, integer M, integer N, real, dimension( lda, * ) A, integer LDA, real, dimension( n ) SVA, integer MV, real, dimension( ldv, * ) V, integer LDV, real, dimension( lwork ) WORK, integer LWORK, integer INFO)
SGESVJ**

**Purpose:**

SGESVJ computes the singular value decomposition (SVD) of a real

M-by-N matrix A, where M >= N. The SVD of A is written as

[++] [xx] [x0] [xx]

A = U * SIGMA * V^t, [++] = [xx] * [ox] * [xx]

[++] [xx]

where SIGMA is an N-by-N diagonal matrix, U is an M-by-N orthonormal

matrix, and V is an N-by-N orthogonal matrix. The diagonal elements

of SIGMA are the singular values of A. The columns of U and V are the

left and the right singular vectors of A, respectively.

SGESVJ can sometimes compute tiny singular values and their singular vectors much

more accurately than other SVD routines, see below under Further Details.

**Parameters**

*JOBA*

JOBA is CHARACTER*1

Specifies the structure of A.

= ’L’: The input matrix A is lower triangular;

= ’U’: The input matrix A is upper triangular;

= ’G’: The input matrix A is general M-by-N matrix, M >= N.

*JOBU*

JOBU is CHARACTER*1

Specifies whether to compute the left singular vectors

(columns of U):

= ’U’: The left singular vectors corresponding to the nonzero

singular values are computed and returned in the leading

columns of A. See more details in the description of A.

The default numerical orthogonality threshold is set to

approximately TOL=CTOL*EPS, CTOL=SQRT(M), EPS=SLAMCH(’E’).

= ’C’: Analogous to JOBU=’U’, except that user can control the

level of numerical orthogonality of the computed left

singular vectors. TOL can be set to TOL = CTOL*EPS, where

CTOL is given on input in the array WORK.

No CTOL smaller than ONE is allowed. CTOL greater

than 1 / EPS is meaningless. The option ’C’

can be used if M*EPS is satisfactory orthogonality

of the computed left singular vectors, so CTOL=M could

save few sweeps of Jacobi rotations.

See the descriptions of A and WORK(1).

= ’N’: The matrix U is not computed. However, see the

description of A.

*JOBV*

JOBV is CHARACTER*1

Specifies whether to compute the right singular vectors, that

is, the matrix V:

= ’V’: the matrix V is computed and returned in the array V

= ’A’: the Jacobi rotations are applied to the MV-by-N

array V. In other words, the right singular vector

matrix V is not computed explicitly; instead it is

applied to an MV-by-N matrix initially stored in the

first MV rows of V.

= ’N’: the matrix V is not computed and the array V is not

referenced

*M*

M is INTEGER

The number of rows of the input matrix A. 1/SLAMCH(’E’) > M >= 0.

*N*

N is INTEGER

The number of columns of the input matrix A.

M >= N >= 0.

*A*

A is REAL array, dimension (LDA,N)

On entry, the M-by-N matrix A.

On exit,

If JOBU = ’U’ .OR. JOBU = ’C’:

If INFO = 0:

RANKA orthonormal columns of U are returned in the

leading RANKA columns of the array A. Here RANKA <= N

is the number of computed singular values of A that are

above the underflow threshold SLAMCH(’S’). The singular

vectors corresponding to underflowed or zero singular

values are not computed. The value of RANKA is returned

in the array WORK as RANKA=NINT(WORK(2)). Also see the

descriptions of SVA and WORK. The computed columns of U

are mutually numerically orthogonal up to approximately

TOL=SQRT(M)*EPS (default); or TOL=CTOL*EPS (JOBU = ’C’),

see the description of JOBU.

If INFO > 0,

the procedure SGESVJ did not converge in the given number

of iterations (sweeps). In that case, the computed

columns of U may not be orthogonal up to TOL. The output

U (stored in A), SIGMA (given by the computed singular

values in SVA(1:N)) and V is still a decomposition of the

input matrix A in the sense that the residual

||A-SCALE*U*SIGMA*V^T||_2 / ||A||_2 is small.

If JOBU = ’N’:

If INFO = 0:

Note that the left singular vectors are ’for free’ in the

one-sided Jacobi SVD algorithm. However, if only the

singular values are needed, the level of numerical

orthogonality of U is not an issue and iterations are

stopped when the columns of the iterated matrix are

numerically orthogonal up to approximately M*EPS. Thus,

on exit, A contains the columns of U scaled with the

corresponding singular values.

If INFO > 0:

the procedure SGESVJ did not converge in the given number

of iterations (sweeps).

*LDA*

LDA is INTEGER

The leading dimension of the array A. LDA >= max(1,M).

*SVA*

SVA is REAL array, dimension (N)

On exit,

If INFO = 0 :

depending on the value SCALE = WORK(1), we have:

If SCALE = ONE:

SVA(1:N) contains the computed singular values of A.

During the computation SVA contains the Euclidean column

norms of the iterated matrices in the array A.

If SCALE .NE. ONE:

The singular values of A are SCALE*SVA(1:N), and this

factored representation is due to the fact that some of the

singular values of A might underflow or overflow.

If INFO > 0 :

the procedure SGESVJ did not converge in the given number of

iterations (sweeps) and SCALE*SVA(1:N) may not be accurate.

*MV*

MV is INTEGER

If JOBV = ’A’, then the product of Jacobi rotations in SGESVJ

is applied to the first MV rows of V. See the description of JOBV.

*V*

V is REAL array, dimension (LDV,N)

If JOBV = ’V’, then V contains on exit the N-by-N matrix of

the right singular vectors;

If JOBV = ’A’, then V contains the product of the computed right

singular vector matrix and the initial matrix in

the array V.

If JOBV = ’N’, then V is not referenced.

*LDV*

LDV is INTEGER

The leading dimension of the array V, LDV >= 1.

If JOBV = ’V’, then LDV >= max(1,N).

If JOBV = ’A’, then LDV >= max(1,MV) .

*WORK*

WORK is REAL array, dimension (LWORK)

On entry,

If JOBU = ’C’ :

WORK(1) = CTOL, where CTOL defines the threshold for convergence.

The process stops if all columns of A are mutually

orthogonal up to CTOL*EPS, EPS=SLAMCH(’E’).

It is required that CTOL >= ONE, i.e. it is not

allowed to force the routine to obtain orthogonality

below EPSILON.

On exit,

WORK(1) = SCALE is the scaling factor such that SCALE*SVA(1:N)

are the computed singular vcalues of A.

(See description of SVA().)

WORK(2) = NINT(WORK(2)) is the number of the computed nonzero

singular values.

WORK(3) = NINT(WORK(3)) is the number of the computed singular

values that are larger than the underflow threshold.

WORK(4) = NINT(WORK(4)) is the number of sweeps of Jacobi

rotations needed for numerical convergence.

WORK(5) = max_{i.NE.j} |COS(A(:,i),A(:,j))| in the last sweep.

This is useful information in cases when SGESVJ did

not converge, as it can be used to estimate whether

the output is still useful and for post festum analysis.

WORK(6) = the largest absolute value over all sines of the

Jacobi rotation angles in the last sweep. It can be

useful for a post festum analysis.

*LWORK*

LWORK is INTEGER

length of WORK, WORK >= MAX(6,M+N)

*INFO*

INFO is INTEGER

= 0: successful exit.

< 0: if INFO = -i, then the i-th argument had an illegal value

> 0: SGESVJ did not converge in the maximal allowed number (30)

of sweeps. The output may still be useful. See the

description of WORK.

**Author**

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

June 2017

**Further Details:**

The orthogonal N-by-N matrix V is obtained as a product of Jacobi plane rotations. The rotations are implemented as fast scaled rotations of Anda and Park [1]. In the case of underflow of the Jacobi angle, a modified Jacobi transformation of Drmac [4] is used. Pivot strategy uses column interchanges of de Rijk [2]. The relative accuracy of the computed singular values and the accuracy of the computed singular vectors (in angle metric) is as guaranteed by the theory of Demmel and Veselic [3]. The condition number that determines the accuracy in the full rank case is essentially min_{D=diag} kappa(A*D), where kappa(.) is the spectral condition number. The best performance of this Jacobi SVD procedure is achieved if used in an accelerated version of Drmac and Veselic [5,6], and it is the kernel routine in the SIGMA library [7]. Some tunning parameters (marked with [TP]) are available for the implementer.

The computational range for the nonzero singular values is the machine number interval ( UNDERFLOW , OVERFLOW ). In extreme cases, even denormalized singular values can be computed with the corresponding gradual loss of accurate digits.

**Contributors:**

Zlatko Drmac (Zagreb, Croatia) and Kresimir Veselic (Hagen, Germany)

**References:**

[1] A. A. Anda and H. Park: Fast plane rotations with dynamic scaling.

SIAM J. matrix Anal. Appl., Vol. 15 (1994), pp. 162-174.

[2] P. P. M. De Rijk: A one-sided Jacobi algorithm for computing the singular value decomposition on a vector computer.

SIAM J. Sci. Stat. Comp., Vol. 10 (1998), pp. 359-371.

[3] J. Demmel and K. Veselic: Jacobi method is more accurate than QR.

[4] Z. Drmac: Implementation of Jacobi rotations for accurate singular value computation in floating point arithmetic.

SIAM J. Sci. Comp., Vol. 18 (1997), pp. 1200-1222.

[5] Z. Drmac and K. Veselic: New fast and accurate Jacobi SVD algorithm I.

SIAM J. Matrix Anal. Appl. Vol. 35, No. 2 (2008), pp. 1322-1342.

LAPACK Working note 169.

[6] Z. Drmac and K. Veselic: New fast and accurate Jacobi SVD algorithm II.

SIAM J. Matrix Anal. Appl. Vol. 35, No. 2 (2008), pp. 1343-1362.

LAPACK Working note 170.

[7] Z. Drmac: SIGMA - mathematical software library for accurate SVD, PSV, QSVD, (H,K)-SVD computations.

Department of Mathematics, University of Zagreb, 2008.

**Bugs, Examples and Comments:**

Please report all bugs and send interesting test examples and comments to drmac AT math DOT hr. Thank you.

**subroutine sgetf2 (integer M, integer N, real, dimension( lda, * ) A, integer LDA, integer, dimension( * ) IPIV, integer INFO)
SGETF2** computes the LU factorization of a general m-by-n matrix using partial pivoting with row interchanges (unblocked algorithm).

**Purpose:**

SGETF2 computes an LU factorization of a general m-by-n matrix A

using partial pivoting with row interchanges.

The factorization has the form

A = P * L * U

where P is a permutation matrix, L is lower triangular with unit

diagonal elements (lower trapezoidal if m > n), and U is upper

triangular (upper trapezoidal if m < n).

This is the right-looking Level 2 BLAS version of the algorithm.

**Parameters**

*M*

M is INTEGER

The number of rows of the matrix A. M >= 0.

*N*

N is INTEGER

The number of columns of the matrix A. N >= 0.

*A*

A is REAL array, dimension (LDA,N)

On entry, the m by n matrix to be factored.

On exit, the factors L and U from the factorization

A = P*L*U; the unit diagonal elements of L are not stored.

*LDA*

LDA is INTEGER

The leading dimension of the array A. LDA >= max(1,M).

*IPIV*

IPIV is INTEGER array, dimension (min(M,N))

The pivot indices; for 1 <= i <= min(M,N), row i of the

matrix was interchanged with row IPIV(i).

*INFO*

INFO is INTEGER

= 0: successful exit

< 0: if INFO = -k, the k-th argument had an illegal value

> 0: if INFO = k, U(k,k) is exactly zero. The factorization

has been completed, but the factor U is exactly

singular, and division by zero will occur if it is used

to solve a system of equations.

**Author**

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

December 2016

**subroutine sgetrf (integer M, integer N, real, dimension( lda, * ) A, integer LDA, integer, dimension( * ) IPIV, integer INFO)
SGETRF SGETRF** VARIANT: iterative version of Sivan Toledo’s recursive LU algorithm

**SGETRF** VARIANT: left-looking Level 3 BLAS version of the algorithm.

**Purpose:**

SGETRF computes an LU factorization of a general M-by-N matrix A

using partial pivoting with row interchanges.

The factorization has the form

A = P * L * U

where P is a permutation matrix, L is lower triangular with unit

diagonal elements (lower trapezoidal if m > n), and U is upper

triangular (upper trapezoidal if m < n).

This is the right-looking Level 3 BLAS version of the algorithm.

**Parameters**

*M*

M is INTEGER

The number of rows of the matrix A. M >= 0.

*N*

N is INTEGER

The number of columns of the matrix A. N >= 0.

*A*

A is REAL array, dimension (LDA,N)

On entry, the M-by-N matrix to be factored.

On exit, the factors L and U from the factorization

A = P*L*U; the unit diagonal elements of L are not stored.

*LDA*

LDA is INTEGER

The leading dimension of the array A. LDA >= max(1,M).

*IPIV*

IPIV is INTEGER array, dimension (min(M,N))

The pivot indices; for 1 <= i <= min(M,N), row i of the

matrix was interchanged with row IPIV(i).

*INFO*

INFO is INTEGER

= 0: successful exit

< 0: if INFO = -i, the i-th argument had an illegal value

> 0: if INFO = i, U(i,i) is exactly zero. The factorization

has been completed, but the factor U is exactly

singular, and division by zero will occur if it is used

to solve a system of equations.

**Author**

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

December 2016

**Purpose:**

SGETRF computes an LU factorization of a general M-by-N matrix A

using partial pivoting with row interchanges.

The factorization has the form

A = P * L * U

where P is a permutation matrix, L is lower triangular with unit

diagonal elements (lower trapezoidal if m > n), and U is upper

triangular (upper trapezoidal if m < n).

This is the left-looking Level 3 BLAS version of the algorithm.

**Parameters**

*M*

M is INTEGER

The number of rows of the matrix A. M >= 0.

*N*

N is INTEGER

The number of columns of the matrix A. N >= 0.

*A*

A is REAL array, dimension (LDA,N)

On entry, the M-by-N matrix to be factored.

On exit, the factors L and U from the factorization

A = P*L*U; the unit diagonal elements of L are not stored.

*LDA*

LDA is INTEGER

The leading dimension of the array A. LDA >= max(1,M).

*IPIV*

IPIV is INTEGER array, dimension (min(M,N))

The pivot indices; for 1 <= i <= min(M,N), row i of the

matrix was interchanged with row IPIV(i).

*INFO*

INFO is INTEGER

= 0: successful exit

< 0: if INFO = -i, the i-th argument had an illegal value

> 0: if INFO = i, U(i,i) is exactly zero. The factorization

has been completed, but the factor U is exactly

singular, and division by zero will occur if it is used

to solve a system of equations.

**Author**

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

December 2016

**Purpose:**

SGETRF computes an LU factorization of a general M-by-N matrix A

using partial pivoting with row interchanges.

A = P * L * U

where P is a permutation matrix, L is lower triangular with unit

diagonal elements (lower trapezoidal if m > n), and U is upper

triangular (upper trapezoidal if m < n).

This code implements an iterative version of Sivan Toledo’s recursive

LU algorithm[1]. For square matrices, this iterative versions should

be within a factor of two of the optimum number of memory transfers.

The pattern is as follows, with the large blocks of U being updated

in one call to STRSM, and the dotted lines denoting sections that

have had all pending permutations applied:

1 2 3 4 5 6 7 8

+-+-+---+-------+------

| |1| | |

|.+-+ 2 | |

| | | | |

|.|.+-+-+ 4 |

| | | |1| |

| | |.+-+ |

| | | | | |

|.|.|.|.+-+-+---+ 8

| | | | | |1| |

| | | | |.+-+ 2 |

| | | | | | | |

| | | | |.|.+-+-+

| | | | | | | |1|

| | | | | | |.+-+

| | | | | | | | |

|.|.|.|.|.|.|.|.+-----

| | | | | | | | |

The 1-2-1-4-1-2-1-8-... pattern is the position of the last 1 bit in

the binary expansion of the current column. Each Schur update is

applied as soon as the necessary portion of U is available.

[1] Toledo, S. 1997. Locality of Reference in LU Decomposition with

Partial Pivoting. SIAM J. Matrix Anal. Appl. 18, 4 (Oct. 1997),

1065-1081. http://dx.doi.org/10.1137/S0895479896297744

**Parameters**

*M*

M is INTEGER

The number of rows of the matrix A. M >= 0.

*N*

N is INTEGER

The number of columns of the matrix A. N >= 0.

*A*

A is REAL array, dimension (LDA,N)

On entry, the M-by-N matrix to be factored.

On exit, the factors L and U from the factorization

A = P*L*U; the unit diagonal elements of L are not stored.

*LDA*

LDA is INTEGER

The leading dimension of the array A. LDA >= max(1,M).

*IPIV*

The pivot indices; for 1 <= i <= min(M,N), row i of the

matrix was interchanged with row IPIV(i).

*INFO*

INFO is INTEGER

= 0: successful exit

< 0: if INFO = -i, the i-th argument had an illegal value

> 0: if INFO = i, U(i,i) is exactly zero. The factorization

has been completed, but the factor U is exactly

singular, and division by zero will occur if it is used

to solve a system of equations.

**Author**

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

December 2016

**recursive subroutine sgetrf2 (integer M, integer N, real, dimension( lda, * ) A, integer LDA, integer, dimension( * ) IPIV, integer INFO)
SGETRF2**

**Purpose:**

SGETRF2 computes an LU factorization of a general M-by-N matrix A

using partial pivoting with row interchanges.

A = P * L * U

where P is a permutation matrix, L is lower triangular with unit

diagonal elements (lower trapezoidal if m > n), and U is upper

triangular (upper trapezoidal if m < n).

This is the recursive version of the algorithm. It divides

the matrix into four submatrices:

[ A11 | A12 ] where A11 is n1 by n1 and A22 is n2 by n2

A = [ -----|----- ] with n1 = min(m,n)/2

[ A21 | A22 ] n2 = n-n1

[ A11 ]

The subroutine calls itself to factor [ --- ],

[ A12 ]

[ A12 ]

do the swaps on [ --- ], solve A12, update A22,

[ A22 ]

then calls itself to factor A22 and do the swaps on A21.

**Parameters**

*M*

M is INTEGER

The number of rows of the matrix A. M >= 0.

*N*

N is INTEGER

The number of columns of the matrix A. N >= 0.

*A*

On entry, the M-by-N matrix to be factored.

On exit, the factors L and U from the factorization

A = P*L*U; the unit diagonal elements of L are not stored.

*LDA*

LDA is INTEGER

The leading dimension of the array A. LDA >= max(1,M).

*IPIV*

The pivot indices; for 1 <= i <= min(M,N), row i of the

matrix was interchanged with row IPIV(i).

*INFO*

= 0: successful exit

< 0: if INFO = -i, the i-th argument had an illegal value

> 0: if INFO = i, U(i,i) is exactly zero. The factorization

has been completed, but the factor U is exactly

singular, and division by zero will occur if it is used

to solve a system of equations.

**Author**

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

June 2016

**subroutine sgetri (integer N, real, dimension( lda, * ) A, integer LDA, integer, dimension( * ) IPIV, real, dimension( * ) WORK, integer LWORK, integer INFO)
SGETRI**

**Purpose:**

SGETRI computes the inverse of a matrix using the LU factorization

computed by SGETRF.

This method inverts U and then computes inv(A) by solving the system

inv(A)*L = inv(U) for inv(A).

**Parameters**

*N*

N is INTEGER

The order of the matrix A. N >= 0.

*A*

A is REAL array, dimension (LDA,N)

On entry, the factors L and U from the factorization

A = P*L*U as computed by SGETRF.

On exit, if INFO = 0, the inverse of the original matrix A.

*LDA*

LDA is INTEGER

The leading dimension of the array A. LDA >= max(1,N).

*IPIV*

IPIV is INTEGER array, dimension (N)

The pivot indices from SGETRF; for 1<=i<=N, row i of the

matrix was interchanged with row IPIV(i).

*WORK*

WORK is REAL array, dimension (MAX(1,LWORK))

On exit, if INFO=0, then WORK(1) returns the optimal LWORK.

*LWORK*

LWORK is INTEGER

The dimension of the array WORK. LWORK >= max(1,N).

For optimal performance LWORK >= N*NB, where NB is

the optimal blocksize returned by ILAENV.

only calculates the optimal size of the WORK array, returns

this value as the first entry of the WORK array, and no error

message related to LWORK is issued by XERBLA.

*INFO*

INFO is INTEGER

= 0: successful exit

< 0: if INFO = -i, the i-th argument had an illegal value

> 0: if INFO = i, U(i,i) is exactly zero; the matrix is

singular and its inverse could not be computed.

**Author**

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

December 2016

**subroutine sgetrs (character TRANS, integer N, integer NRHS, real, dimension( lda, * ) A, integer LDA, integer, dimension( * ) IPIV, real, dimension( ldb, * ) B, integer LDB, integer INFO)
SGETRS**

**Purpose:**

SGETRS solves a system of linear equations

A * X = B or A**T * X = B

with a general N-by-N matrix A using the LU factorization computed

by SGETRF.

**Parameters**

*TRANS*

TRANS is CHARACTER*1

Specifies the form of the system of equations:

= ’N’: A * X = B (No transpose)

= ’T’: A**T* X = B (Transpose)

= ’C’: A**T* X = B (Conjugate transpose = Transpose)

*N*

N is INTEGER

The order of the matrix A. N >= 0.

*NRHS*

NRHS is INTEGER

The number of right hand sides, i.e., the number of columns

of the matrix B. NRHS >= 0.

*A*

A is REAL array, dimension (LDA,N)

The factors L and U from the factorization A = P*L*U

as computed by SGETRF.

*LDA*

LDA is INTEGER

The leading dimension of the array A. LDA >= max(1,N).

*IPIV*

The pivot indices from SGETRF; for 1<=i<=N, row i of the

matrix was interchanged with row IPIV(i).

*B*

B is REAL array, dimension (LDB,NRHS)

On entry, the right hand side matrix B.

On exit, the solution matrix X.

*LDB*

LDB is INTEGER

The leading dimension of the array B. LDB >= max(1,N).

*INFO*

INFO is INTEGER

= 0: successful exit

< 0: if INFO = -i, the i-th argument had an illegal value

**Author**

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

December 2016

**subroutine shgeqz (character JOB, character COMPQ, character COMPZ, integer N, integer ILO, integer IHI, real, dimension( ldh, * ) H, integer LDH, real, dimension( ldt, * ) T, integer LDT, real, dimension( * ) ALPHAR, real, dimension( * ) ALPHAI, real, dimension( * ) BETA, real, dimension( ldq, * ) Q, integer LDQ, real, dimension( ldz, * ) Z, integer LDZ, real, dimension( * ) WORK, integer LWORK, integer INFO)
SHGEQZ**

**Purpose:**

SHGEQZ computes the eigenvalues of a real matrix pair (H,T),

where H is an upper Hessenberg matrix and T is upper triangular,

using the double-shift QZ method.

Matrix pairs of this type are produced by the reduction to

generalized upper Hessenberg form of a real matrix pair (A,B):

A = Q1*H*Z1**T, B = Q1*T*Z1**T,

as computed by SGGHRD.

If JOB=’S’, then the Hessenberg-triangular pair (H,T) is

also reduced to generalized Schur form,

H = Q*S*Z**T, T = Q*P*Z**T,

where Q and Z are orthogonal matrices, P is an upper triangular

matrix, and S is a quasi-triangular matrix with 1-by-1 and 2-by-2

diagonal blocks.

The 1-by-1 blocks correspond to real eigenvalues of the matrix pair

(H,T) and the 2-by-2 blocks correspond to complex conjugate pairs of

eigenvalues.

Additionally, the 2-by-2 upper triangular diagonal blocks of P

corresponding to 2-by-2 blocks of S are reduced to positive diagonal

form, i.e., if S(j+1,j) is non-zero, then P(j+1,j) = P(j,j+1) = 0,

P(j,j) > 0, and P(j+1,j+1) > 0.

Optionally, the orthogonal matrix Q from the generalized Schur

factorization may be postmultiplied into an input matrix Q1, and the

orthogonal matrix Z may be postmultiplied into an input matrix Z1.

If Q1 and Z1 are the orthogonal matrices from SGGHRD that reduced

the matrix pair (A,B) to generalized upper Hessenberg form, then the

output matrices Q1*Q and Z1*Z are the orthogonal factors from the

generalized Schur factorization of (A,B):

A = (Q1*Q)*S*(Z1*Z)**T, B = (Q1*Q)*P*(Z1*Z)**T.

To avoid overflow, eigenvalues of the matrix pair (H,T) (equivalently,

of (A,B)) are computed as a pair of values (alpha,beta), where alpha is

complex and beta real.

If beta is nonzero, lambda = alpha / beta is an eigenvalue of the

generalized nonsymmetric eigenvalue problem (GNEP)

A*x = lambda*B*x

and if alpha is nonzero, mu = beta / alpha is an eigenvalue of the

alternate form of the GNEP

mu*A*y = B*y.

Real eigenvalues can be read directly from the generalized Schur

form:

alpha = S(i,i), beta = P(i,i).

Ref: C.B. Moler & G.W. Stewart, "An Algorithm for Generalized Matrix

Eigenvalue Problems", SIAM J. Numer. Anal., 10(1973),

pp. 241--256.

**Parameters**

*JOB*

JOB is CHARACTER*1

= ’E’: Compute eigenvalues only;

= ’S’: Compute eigenvalues and the Schur form.

*COMPQ*

COMPQ is CHARACTER*1

= ’N’: Left Schur vectors (Q) are not computed;

= ’I’: Q is initialized to the unit matrix and the matrix Q

of left Schur vectors of (H,T) is returned;

= ’V’: Q must contain an orthogonal matrix Q1 on entry and

the product Q1*Q is returned.

*COMPZ*

COMPZ is CHARACTER*1

= ’N’: Right Schur vectors (Z) are not computed;

= ’I’: Z is initialized to the unit matrix and the matrix Z

of right Schur vectors of (H,T) is returned;

= ’V’: Z must contain an orthogonal matrix Z1 on entry and

the product Z1*Z is returned.

*N*

N is INTEGER

The order of the matrices H, T, Q, and Z. N >= 0.

*ILO*

ILO is INTEGER

*IHI*

IHI is INTEGER

ILO and IHI mark the rows and columns of H which are in

Hessenberg form. It is assumed that A is already upper

triangular in rows and columns 1:ILO-1 and IHI+1:N.

If N > 0, 1 <= ILO <= IHI <= N; if N = 0, ILO=1 and IHI=0.

*H*

H is REAL array, dimension (LDH, N)

On entry, the N-by-N upper Hessenberg matrix H.

On exit, if JOB = ’S’, H contains the upper quasi-triangular

matrix S from the generalized Schur factorization.

If JOB = ’E’, the diagonal blocks of H match those of S, but

the rest of H is unspecified.

*LDH*

LDH is INTEGER

The leading dimension of the array H. LDH >= max( 1, N ).

*T*

T is REAL array, dimension (LDT, N)

On entry, the N-by-N upper triangular matrix T.

On exit, if JOB = ’S’, T contains the upper triangular

matrix P from the generalized Schur factorization;

2-by-2 diagonal blocks of P corresponding to 2-by-2 blocks of S

are reduced to positive diagonal form, i.e., if H(j+1,j) is

non-zero, then T(j+1,j) = T(j,j+1) = 0, T(j,j) > 0, and

T(j+1,j+1) > 0.

If JOB = ’E’, the diagonal blocks of T match those of P, but

the rest of T is unspecified.

*LDT*

LDT is INTEGER

The leading dimension of the array T. LDT >= max( 1, N ).

*ALPHAR*

ALPHAR is REAL array, dimension (N)

The real parts of each scalar alpha defining an eigenvalue

of GNEP.

*ALPHAI*

ALPHAI is REAL array, dimension (N)

The imaginary parts of each scalar alpha defining an

eigenvalue of GNEP.

If ALPHAI(j) is zero, then the j-th eigenvalue is real; if

positive, then the j-th and (j+1)-st eigenvalues are a

complex conjugate pair, with ALPHAI(j+1) = -ALPHAI(j).

*BETA*

BETA is REAL array, dimension (N)

The scalars beta that define the eigenvalues of GNEP.

Together, the quantities alpha = (ALPHAR(j),ALPHAI(j)) and

beta = BETA(j) represent the j-th eigenvalue of the matrix

pair (A,B), in one of the forms lambda = alpha/beta or

mu = beta/alpha. Since either lambda or mu may overflow,

they should not, in general, be computed.

*Q*

Q is REAL array, dimension (LDQ, N)

On entry, if COMPQ = ’V’, the orthogonal matrix Q1 used in

the reduction of (A,B) to generalized Hessenberg form.

On exit, if COMPQ = ’I’, the orthogonal matrix of left Schur

vectors of (H,T), and if COMPQ = ’V’, the orthogonal matrix

of left Schur vectors of (A,B).

Not referenced if COMPQ = ’N’.

*LDQ*

LDQ is INTEGER

The leading dimension of the array Q. LDQ >= 1.

If COMPQ=’V’ or ’I’, then LDQ >= N.

*Z*

Z is REAL array, dimension (LDZ, N)

On entry, if COMPZ = ’V’, the orthogonal matrix Z1 used in

the reduction of (A,B) to generalized Hessenberg form.

On exit, if COMPZ = ’I’, the orthogonal matrix of

right Schur vectors of (H,T), and if COMPZ = ’V’, the

orthogonal matrix of right Schur vectors of (A,B).

Not referenced if COMPZ = ’N’.

*LDZ*

LDZ is INTEGER

The leading dimension of the array Z. LDZ >= 1.

If COMPZ=’V’ or ’I’, then LDZ >= N.

*WORK*

WORK is REAL array, dimension (MAX(1,LWORK))

On exit, if INFO >= 0, WORK(1) returns the optimal LWORK.

*LWORK*

LWORK is INTEGER

The dimension of the array WORK. LWORK >= max(1,N).

only calculates the optimal size of the WORK array, returns

this value as the first entry of the WORK array, and no error

message related to LWORK is issued by XERBLA.

*INFO*

INFO is INTEGER

= 0: successful exit

< 0: if INFO = -i, the i-th argument had an illegal value

= 1,...,N: the QZ iteration did not converge. (H,T) is not

in Schur form, but ALPHAR(i), ALPHAI(i), and

BETA(i), i=INFO+1,...,N should be correct.

= N+1,...,2*N: the shift calculation failed. (H,T) is not

in Schur form, but ALPHAR(i), ALPHAI(i), and

BETA(i), i=INFO-N+1,...,N should be correct.

**Author**

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

June 2016

**Further Details:**

Iteration counters:

JITER -- counts iterations.

IITER -- counts iterations run since ILAST was last

changed. This is therefore reset only when a 1-by-1 or

2-by-2 block deflates off the bottom.

**subroutine sla_geamv (integer TRANS, integer M, integer N, real ALPHA, real, dimension( lda, * ) A, integer LDA, real, dimension( * ) X, integer INCX, real BETA, real, dimension( * ) Y, integer INCY)
SLA_GEAMV** computes a matrix-vector product using a general matrix to calculate error bounds.

**Purpose:**

SLA_GEAMV performs one of the matrix-vector operations

y := alpha*abs(A)*abs(x) + beta*abs(y),

or y := alpha*abs(A)**T*abs(x) + beta*abs(y),

where alpha and beta are scalars, x and y are vectors and A is an

m by n matrix.

This function is primarily used in calculating error bounds.

To protect against underflow during evaluation, components in

the resulting vector are perturbed away from zero by (N+1)

times the underflow threshold. To prevent unnecessarily large

errors for block-structure embedded in general matrices,

"symbolically" zero components are not perturbed. A zero

entry is considered "symbolic" if all multiplications involved

in computing that entry have at least one zero multiplicand.

**Parameters**

*TRANS*

TRANS is INTEGER

On entry, TRANS specifies the operation to be performed as

follows:

BLAS_NO_TRANS y := alpha*abs(A)*abs(x) + beta*abs(y)

BLAS_TRANS y := alpha*abs(A**T)*abs(x) + beta*abs(y)

BLAS_CONJ_TRANS y := alpha*abs(A**T)*abs(x) + beta*abs(y)

Unchanged on exit.

*M*

M is INTEGER

On entry, M specifies the number of rows of the matrix A.

M must be at least zero.

Unchanged on exit.

*N*

N is INTEGER

On entry, N specifies the number of columns of the matrix A.

N must be at least zero.

Unchanged on exit.

*ALPHA*

ALPHA is REAL

On entry, ALPHA specifies the scalar alpha.

Unchanged on exit.

*A*

A is REAL array, dimension ( LDA, n )

Before entry, the leading m by n part of the array A must

contain the matrix of coefficients.

Unchanged on exit.

*LDA*

LDA is INTEGER

On entry, LDA specifies the first dimension of A as declared

in the calling (sub) program. LDA must be at least

max( 1, m ).

Unchanged on exit.

*X*

X is REAL array, dimension

( 1 + ( n - 1 )*abs( INCX ) ) when TRANS = ’N’ or ’n’

and at least

( 1 + ( m - 1 )*abs( INCX ) ) otherwise.

Before entry, the incremented array X must contain the

vector x.

Unchanged on exit.

*INCX*

INCX is INTEGER

On entry, INCX specifies the increment for the elements of

X. INCX must not be zero.

Unchanged on exit.

*BETA*

BETA is REAL

On entry, BETA specifies the scalar beta. When BETA is

supplied as zero then Y need not be set on input.

Unchanged on exit.

*Y*

Y is REAL array,

dimension at least

( 1 + ( m - 1 )*abs( INCY ) ) when TRANS = ’N’ or ’n’

and at least

( 1 + ( n - 1 )*abs( INCY ) ) otherwise.

Before entry with BETA non-zero, the incremented array Y

must contain the vector y. On exit, Y is overwritten by the

updated vector y.

*INCY*

INCY is INTEGER

On entry, INCY specifies the increment for the elements of

Y. INCY must not be zero.

Unchanged on exit.

Level 2 Blas routine.

**Author**

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

June 2017

**real function sla_gercond (character TRANS, integer N, real, dimension( lda, * ) A, integer LDA, real, dimension( ldaf, * ) AF, integer LDAF, integer, dimension( * ) IPIV, integer CMODE, real, dimension( * ) C, integer INFO, real, dimension( * ) WORK, integer, dimension( * ) IWORK)
SLA_GERCOND** estimates the Skeel condition number for a general matrix.

**Purpose:**

SLA_GERCOND estimates the Skeel condition number of op(A) * op2(C)

where op2 is determined by CMODE as follows

CMODE = 1 op2(C) = C

CMODE = 0 op2(C) = I

CMODE = -1 op2(C) = inv(C)

The Skeel condition number cond(A) = norminf( |inv(A)||A| )

is computed by computing scaling factors R such that

diag(R)*A*op2(C) is row equilibrated and computing the standard

infinity-norm condition number.

**Parameters**

*TRANS*

TRANS is CHARACTER*1

Specifies the form of the system of equations:

= ’N’: A * X = B (No transpose)

= ’T’: A**T * X = B (Transpose)

= ’C’: A**H * X = B (Conjugate Transpose = Transpose)

*N*

N is INTEGER

The number of linear equations, i.e., the order of the

matrix A. N >= 0.

*A*

A is REAL array, dimension (LDA,N)

On entry, the N-by-N matrix A.

*LDA*

LDA is INTEGER

The leading dimension of the array A. LDA >= max(1,N).

*AF*

AF is REAL array, dimension (LDAF,N)

The factors L and U from the factorization

A = P*L*U as computed by SGETRF.

*LDAF*

LDAF is INTEGER

The leading dimension of the array AF. LDAF >= max(1,N).

*IPIV*

IPIV is INTEGER array, dimension (N)

The pivot indices from the factorization A = P*L*U

as computed by SGETRF; row i of the matrix was interchanged

with row IPIV(i).

*CMODE*

CMODE is INTEGER

Determines op2(C) in the formula op(A) * op2(C) as follows:

CMODE = 1 op2(C) = C

CMODE = 0 op2(C) = I

CMODE = -1 op2(C) = inv(C)

*C*

C is REAL array, dimension (N)

The vector C in the formula op(A) * op2(C).

*INFO*

INFO is INTEGER

= 0: Successful exit.

i > 0: The ith argument is invalid.

*WORK*

WORK is REAL array, dimension (3*N).

Workspace.

*IWORK*

IWORK is INTEGER array, dimension (N).

Workspace.2

**Author**

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

December 2016

**subroutine sla_gerfsx_extended (integer PREC_TYPE, integer TRANS_TYPE, integer N, integer NRHS, real, dimension( lda, * ) A, integer LDA, real, dimension( ldaf, * ) AF, integer LDAF, integer, dimension( * ) IPIV, logical COLEQU, real, dimension( * ) C, real, dimension( ldb, * ) B, integer LDB, real, dimension( ldy, * ) Y, integer LDY, real, dimension( * ) BERR_OUT, integer N_NORMS, real, dimension( nrhs, * ) ERRS_N, real, dimension( nrhs, * ) ERRS_C, real, dimension( * ) RES, real, dimension( * ) AYB, real, dimension( * ) DY, real, dimension( * ) Y_TAIL, real RCOND, integer ITHRESH, real RTHRESH, real DZ_UB, logical IGNORE_CWISE, integer INFO)
SLA_GERFSX_EXTENDED** improves the computed solution to a system of linear equations for general matrices by performing extra-precise iterative refinement and provides error bounds and backward error estimates for the solution.

**Purpose:**

SLA_GERFSX_EXTENDED improves the computed solution to a system of

linear equations by performing extra-precise iterative refinement

and provides error bounds and backward error estimates for the solution.

This subroutine is called by SGERFSX to perform iterative refinement.

In addition to normwise error bound, the code provides maximum

componentwise error bound if possible. See comments for ERRS_N

and ERRS_C for details of the error bounds. Note that this

subroutine is only resonsible for setting the second fields of

ERRS_N and ERRS_C.

**Parameters**

*PREC_TYPE*

PREC_TYPE is INTEGER

Specifies the intermediate precision to be used in refinement.

The value is defined by ILAPREC(P) where P is a CHARACTER and P

= ’S’: Single

= ’D’: Double

= ’I’: Indigenous

= ’X’ or ’E’: Extra

*TRANS_TYPE*

TRANS_TYPE is INTEGER

Specifies the transposition operation on A.

The value is defined by ILATRANS(T) where T is a CHARACTER and T

= ’N’: No transpose

= ’T’: Transpose

= ’C’: Conjugate transpose

*N*

N is INTEGER

The number of linear equations, i.e., the order of the

matrix A. N >= 0.

*NRHS*

NRHS is INTEGER

The number of right-hand-sides, i.e., the number of columns of the

matrix B.

*A*

A is REAL array, dimension (LDA,N)

On entry, the N-by-N matrix A.

*LDA*

LDA is INTEGER

The leading dimension of the array A. LDA >= max(1,N).

*AF*

The factors L and U from the factorization

A = P*L*U as computed by SGETRF.

*LDAF*

LDAF is INTEGER

The leading dimension of the array AF. LDAF >= max(1,N).

*IPIV*

IPIV is INTEGER array, dimension (N)

The pivot indices from the factorization A = P*L*U

as computed by SGETRF; row i of the matrix was interchanged

with row IPIV(i).

*COLEQU*

COLEQU is LOGICAL

If .TRUE. then column equilibration was done to A before calling

this routine. This is needed to compute the solution and error

bounds correctly.

*C*

C is REAL array, dimension (N)

The column scale factors for A. If COLEQU = .FALSE., C

is not accessed. If C is input, each element of C should be a power

of the radix to ensure a reliable solution and error estimates.

Scaling by powers of the radix does not cause rounding errors unless

the result underflows or overflows. Rounding errors during scaling

lead to refining with a matrix that is not equivalent to the

input matrix, producing error estimates that may not be

reliable.

*B*

B is REAL array, dimension (LDB,NRHS)

The right-hand-side matrix B.

*LDB*

LDB is INTEGER

The leading dimension of the array B. LDB >= max(1,N).

*Y*

Y is REAL array, dimension (LDY,NRHS)

On entry, the solution matrix X, as computed by SGETRS.

On exit, the improved solution matrix Y.

*LDY*

LDY is INTEGER

The leading dimension of the array Y. LDY >= max(1,N).

*BERR_OUT*

BERR_OUT is REAL array, dimension (NRHS)

On exit, BERR_OUT(j) contains the componentwise relative backward

error for right-hand-side j from the formula

max(i) ( abs(RES(i)) / ( abs(op(A_s))*abs(Y) + abs(B_s) )(i) )

where abs(Z) is the componentwise absolute value of the matrix

or vector Z. This is computed by SLA_LIN_BERR.

*N_NORMS*

N_NORMS is INTEGER

Determines which error bounds to return (see ERRS_N

and ERRS_C).

If N_NORMS >= 1 return normwise error bounds.

If N_NORMS >= 2 return componentwise error bounds.

*ERRS_N*

ERRS_N is REAL array, dimension (NRHS, N_ERR_BNDS)

For each right-hand side, this array contains information about

various error bounds and condition numbers corresponding to the

normwise relative error, which is defined as follows:

Normwise relative error in the ith solution vector:

max_j (abs(XTRUE(j,i) - X(j,i)))

------------------------------

max_j abs(X(j,i))

The array is indexed by the type of error information as described

below. There currently are up to three pieces of information

returned.

The first index in ERRS_N(i,:) corresponds to the ith

right-hand side.

The second index in ERRS_N(:,err) contains the following

three fields:

err = 1 "Trust/don’t trust" boolean. Trust the answer if the

reciprocal condition number is less than the threshold

sqrt(n) * slamch(’Epsilon’).

err = 2 "Guaranteed" error bound: The estimated forward error,

almost certainly within a factor of 10 of the true error

so long as the next entry is greater than the threshold

sqrt(n) * slamch(’Epsilon’). This error bound should only

be trusted if the previous boolean is true.

err = 3 Reciprocal condition number: Estimated normwise

reciprocal condition number. Compared with the threshold

sqrt(n) * slamch(’Epsilon’) to determine if the error

estimate is "guaranteed". These reciprocal condition

numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some

appropriately scaled matrix Z.

Let Z = S*A, where S scales each row by a power of the

radix so all absolute row sums of Z are approximately 1.

This subroutine is only responsible for setting the second field

above.

See Lapack Working Note 165 for further details and extra

cautions.

*ERRS_C*

ERRS_C is REAL array, dimension (NRHS, N_ERR_BNDS)

For each right-hand side, this array contains information about

various error bounds and condition numbers corresponding to the

componentwise relative error, which is defined as follows:

Componentwise relative error in the ith solution vector:

abs(XTRUE(j,i) - X(j,i))

max_j ----------------------

abs(X(j,i))

The array is indexed by the right-hand side i (on which the

componentwise relative error depends), and the type of error

information as described below. There currently are up to three

pieces of information returned for each right-hand side. If

componentwise accuracy is not requested (PARAMS(3) = 0.0), then

ERRS_C is not accessed. If N_ERR_BNDS < 3, then at most

the first (:,N_ERR_BNDS) entries are returned.

The first index in ERRS_C(i,:) corresponds to the ith

right-hand side.

The second index in ERRS_C(:,err) contains the following

three fields:

err = 1 "Trust/don’t trust" boolean. Trust the answer if the

reciprocal condition number is less than the threshold

sqrt(n) * slamch(’Epsilon’).

almost certainly within a factor of 10 of the true error

so long as the next entry is greater than the threshold

sqrt(n) * slamch(’Epsilon’). This error bound should only

be trusted if the previous boolean is true.

err = 3 Reciprocal condition number: Estimated componentwise

reciprocal condition number. Compared with the threshold

sqrt(n) * slamch(’Epsilon’) to determine if the error

estimate is "guaranteed". These reciprocal condition

numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some

appropriately scaled matrix Z.

Let Z = S*(A*diag(x)), where x is the solution for the

current right-hand side and S scales each row of

A*diag(x) by a power of the radix so all absolute row

sums of Z are approximately 1.

This subroutine is only responsible for setting the second field

above.

See Lapack Working Note 165 for further details and extra

cautions.

*RES*

RES is REAL array, dimension (N)

Workspace to hold the intermediate residual.

*AYB*

AYB is REAL array, dimension (N)

Workspace. This can be the same workspace passed for Y_TAIL.

*DY*

DY is REAL array, dimension (N)

Workspace to hold the intermediate solution.

*Y_TAIL*

Y_TAIL is REAL array, dimension (N)

Workspace to hold the trailing bits of the intermediate solution.

*RCOND*

RCOND is REAL

Reciprocal scaled condition number. This is an estimate of the

reciprocal Skeel condition number of the matrix A after

equilibration (if done). If this is less than the machine

precision (in particular, if it is zero), the matrix is singular

to working precision. Note that the error may still be small even

if this number is very small and the matrix appears ill-

conditioned.

*ITHRESH*

ITHRESH is INTEGER

The maximum number of residual computations allowed for

refinement. The default is 10. For ’aggressive’ set to 100 to

permit convergence using approximate factorizations or

factorizations other than LU. If the factorization uses a

technique other than Gaussian elimination, the guarantees in

ERRS_N and ERRS_C may no longer be trustworthy.

*RTHRESH*

RTHRESH is REAL

Determines when to stop refinement if the error estimate stops

decreasing. Refinement will stop when the next solution no longer

satisfies norm(dx_{i+1}) < RTHRESH * norm(dx_i) where norm(Z) is

the infinity norm of Z. RTHRESH satisfies 0 < RTHRESH <= 1. The

default value is 0.5. For ’aggressive’ set to 0.9 to permit

convergence on extremely ill-conditioned matrices. See LAWN 165

for more details.

*DZ_UB*

DZ_UB is REAL

Determines when to start considering componentwise convergence.

Componentwise convergence is only considered after each component

of the solution Y is stable, which we definte as the relative

change in each component being less than DZ_UB. The default value

is 0.25, requiring the first bit to be stable. See LAWN 165 for

more details.

*IGNORE_CWISE*

IGNORE_CWISE is LOGICAL

If .TRUE. then ignore componentwise convergence. Default value

is .FALSE..

*INFO*

INFO is INTEGER

= 0: Successful exit.

< 0: if INFO = -i, the ith argument to SGETRS had an illegal

value

**Author**

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

December 2016

**real function sla_gerpvgrw (integer N, integer NCOLS, real, dimension( lda, * ) A, integer LDA, real, dimension( ldaf, * ) AF, integer LDAF)
SLA_GERPVGRW**

**Purpose:**

SLA_GERPVGRW computes the reciprocal pivot growth factor

norm(A)/norm(U). The "max absolute element" norm is used. If this is

much less than 1, the stability of the LU factorization of the

(equilibrated) matrix A could be poor. This also means that the

solution X, estimated condition numbers, and error bounds could be

unreliable.

**Parameters**

*N*

N is INTEGER

The number of linear equations, i.e., the order of the

matrix A. N >= 0.

*NCOLS*

NCOLS is INTEGER

The number of columns of the matrix A. NCOLS >= 0.

*A*

A is REAL array, dimension (LDA,N)

On entry, the N-by-N matrix A.

*LDA*

LDA is INTEGER

The leading dimension of the array A. LDA >= max(1,N).

*AF*

The factors L and U from the factorization

A = P*L*U as computed by SGETRF.

*LDAF*

LDAF is INTEGER

The leading dimension of the array AF. LDAF >= max(1,N).

**Author**

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

December 2016

**subroutine slaorhr_col_getrfnp (integer M, integer N, real, dimension( lda, * ) A, integer LDA, real, dimension( * ) D, integer INFO)
SLAORHR_COL_GETRFNP**

**Purpose:**

SLAORHR_COL_GETRFNP computes the modified LU factorization without

pivoting of a real general M-by-N matrix A. The factorization has

the form:

A - S = L * U,

where:

S is a m-by-n diagonal sign matrix with the diagonal D, so that

D(i) = S(i,i), 1 <= i <= min(M,N). The diagonal D is constructed

as D(i)=-SIGN(A(i,i)), where A(i,i) is the value after performing

i-1 steps of Gaussian elimination. This means that the diagonal

element at each step of "modified" Gaussian elimination is

at least one in absolute value (so that division-by-zero not

not possible during the division by the diagonal element);

L is a M-by-N lower triangular matrix with unit diagonal elements

(lower trapezoidal if M > N);

and U is a M-by-N upper triangular matrix

(upper trapezoidal if M < N).

This routine is an auxiliary routine used in the Householder

reconstruction routine SORHR_COL. In SORHR_COL, this routine is

applied to an M-by-N matrix A with orthonormal columns, where each

element is bounded by one in absolute value. With the choice of

the matrix S above, one can show that the diagonal element at each

step of Gaussian elimination is the largest (in absolute value) in

the column on or below the diagonal, so that no pivoting is required

for numerical stability [1].

For more details on the Householder reconstruction algorithm,

including the modified LU factorization, see [1].

This is the blocked right-looking version of the algorithm,

calling Level 3 BLAS to update the submatrix. To factorize a block,

this routine calls the recursive routine SLAORHR_COL_GETRFNP2.

[1] "Reconstructing Householder vectors from tall-skinny QR",

G. Ballard, J. Demmel, L. Grigori, M. Jacquelin, H.D. Nguyen,

E. Solomonik, J. Parallel Distrib. Comput.,

vol. 85, pp. 3-31, 2015.

**Parameters**

*M*

M is INTEGER

The number of rows of the matrix A. M >= 0.

*N*

N is INTEGER

The number of columns of the matrix A. N >= 0.

*A*

A is REAL array, dimension (LDA,N)

On entry, the M-by-N matrix to be factored.

On exit, the factors L and U from the factorization

A-S=L*U; the unit diagonal elements of L are not stored.

*LDA*

LDA is INTEGER

The leading dimension of the array A. LDA >= max(1,M).

*D*

D is REAL array, dimension min(M,N)

The diagonal elements of the diagonal M-by-N sign matrix S,

D(i) = S(i,i), where 1 <= i <= min(M,N). The elements can

be only plus or minus one.

*INFO*

INFO is INTEGER

= 0: successful exit

< 0: if INFO = -i, the i-th argument had an illegal value

**Author**

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

November 2019

**Contributors:**

November 2019, Igor Kozachenko,

Computer Science Division,

University of California, Berkeley

**recursive subroutine slaorhr_col_getrfnp2 (integer M, integer N, real, dimension( lda, * ) A, integer LDA, real, dimension( * ) D, integer INFO)
SLAORHR_COL_GETRFNP2**

**Purpose:**

SLAORHR_COL_GETRFNP2 computes the modified LU factorization without

pivoting of a real general M-by-N matrix A. The factorization has

the form:

A - S = L * U,

where:

S is a m-by-n diagonal sign matrix with the diagonal D, so that

D(i) = S(i,i), 1 <= i <= min(M,N). The diagonal D is constructed

as D(i)=-SIGN(A(i,i)), where A(i,i) is the value after performing

i-1 steps of Gaussian elimination. This means that the diagonal

element at each step of "modified" Gaussian elimination is at

least one in absolute value (so that division-by-zero not

possible during the division by the diagonal element);

L is a M-by-N lower triangular matrix with unit diagonal elements

(lower trapezoidal if M > N);

and U is a M-by-N upper triangular matrix

(upper trapezoidal if M < N).

This routine is an auxiliary routine used in the Householder

reconstruction routine SORHR_COL. In SORHR_COL, this routine is

applied to an M-by-N matrix A with orthonormal columns, where each

element is bounded by one in absolute value. With the choice of

the matrix S above, one can show that the diagonal element at each

step of Gaussian elimination is the largest (in absolute value) in

the column on or below the diagonal, so that no pivoting is required

for numerical stability [1].

For more details on the Householder reconstruction algorithm,

including the modified LU factorization, see [1].

This is the recursive version of the LU factorization algorithm.

Denote A - S by B. The algorithm divides the matrix B into four

submatrices:

[ B11 | B12 ] where B11 is n1 by n1,

B = [ --—|--— ] B21 is (m-n1) by n1,

[ B21 | B22 ] B12 is n1 by n2,

B22 is (m-n1) by n2,

with n1 = min(m,n)/2, n2 = n-n1.

The subroutine calls itself to factor B11, solves for B21,

solves for B12, updates B22, then calls itself to factor B22.

For more details on the recursive LU algorithm, see [2].

SLAORHR_COL_GETRFNP2 is called to factorize a block by the blocked

routine SLAORHR_COL_GETRFNP, which uses blocked code calling

is self-sufficient and can be used without SLAORHR_COL_GETRFNP.

[1] "Reconstructing Householder vectors from tall-skinny QR",

G. Ballard, J. Demmel, L. Grigori, M. Jacquelin, H.D. Nguyen,

E. Solomonik, J. Parallel Distrib. Comput.,

vol. 85, pp. 3-31, 2015.

[2] "Recursion leads to automatic variable blocking for dense linear

algebra algorithms", F. Gustavson, IBM J. of Res. and Dev.,

vol. 41, no. 6, pp. 737-755, 1997.

**Parameters**

*M*

M is INTEGER

The number of rows of the matrix A. M >= 0.

*N*

N is INTEGER

The number of columns of the matrix A. N >= 0.

*A*

A is REAL array, dimension (LDA,N)

On entry, the M-by-N matrix to be factored.

On exit, the factors L and U from the factorization

A-S=L*U; the unit diagonal elements of L are not stored.

*LDA*

LDA is INTEGER

The leading dimension of the array A. LDA >= max(1,M).

*D*

D is REAL array, dimension min(M,N)

The diagonal elements of the diagonal M-by-N sign matrix S,

D(i) = S(i,i), where 1 <= i <= min(M,N). The elements can

be only plus or minus one.

*INFO*

INFO is INTEGER

= 0: successful exit

< 0: if INFO = -i, the i-th argument had an illegal value

**Author**

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

November 2019

**Contributors:**

November 2019, Igor Kozachenko,

Computer Science Division,

University of California, Berkeley

**subroutine stgevc (character SIDE, character HOWMNY, logical, dimension( * ) SELECT, integer N, real, dimension( lds, * ) S, integer LDS, real, dimension( ldp, * ) P, integer LDP, real, dimension( ldvl, * ) VL, integer LDVL, real, dimension( ldvr, * ) VR, integer LDVR, integer MM, integer M, real, dimension( * ) WORK, integer INFO)
STGEVC**

**Purpose:**

STGEVC computes some or all of the right and/or left eigenvectors of

a pair of real matrices (S,P), where S is a quasi-triangular matrix

and P is upper triangular. Matrix pairs of this type are produced by

the generalized Schur factorization of a matrix pair (A,B):

A = Q*S*Z**T, B = Q*P*Z**T

as computed by SGGHRD + SHGEQZ.

The right eigenvector x and the left eigenvector y of (S,P)

corresponding to an eigenvalue w are defined by:

S*x = w*P*x, (y**H)*S = w*(y**H)*P,

where y**H denotes the conjugate tranpose of y.

The eigenvalues are not input to this routine, but are computed

directly from the diagonal blocks of S and P.

This routine returns the matrices X and/or Y of right and left

eigenvectors of (S,P), or the products Z*X and/or Q*Y,

where Z and Q are input matrices.

If Q and Z are the orthogonal factors from the generalized Schur

factorization of a matrix pair (A,B), then Z*X and Q*Y

are the matrices of right and left eigenvectors of (A,B).

**Parameters**

*SIDE*

SIDE is CHARACTER*1

= ’R’: compute right eigenvectors only;

= ’L’: compute left eigenvectors only;

= ’B’: compute both right and left eigenvectors.

*HOWMNY*

HOWMNY is CHARACTER*1

= ’A’: compute all right and/or left eigenvectors;

= ’B’: compute all right and/or left eigenvectors,

backtransformed by the matrices in VR and/or VL;

= ’S’: compute selected right and/or left eigenvectors,

specified by the logical array SELECT.

*SELECT*

SELECT is LOGICAL array, dimension (N)

If HOWMNY=’S’, SELECT specifies the eigenvectors to be

computed. If w(j) is a real eigenvalue, the corresponding

real eigenvector is computed if SELECT(j) is .TRUE..

If w(j) and w(j+1) are the real and imaginary parts of a

complex eigenvalue, the corresponding complex eigenvector

is computed if either SELECT(j) or SELECT(j+1) is .TRUE.,

and on exit SELECT(j) is set to .TRUE. and SELECT(j+1) is

set to .FALSE..

Not referenced if HOWMNY = ’A’ or ’B’.

*N*

N is INTEGER

The order of the matrices S and P. N >= 0.

*S*

S is REAL array, dimension (LDS,N)

The upper quasi-triangular matrix S from a generalized Schur

factorization, as computed by SHGEQZ.

*LDS*

LDS is INTEGER

The leading dimension of array S. LDS >= max(1,N).

*P*

P is REAL array, dimension (LDP,N)

The upper triangular matrix P from a generalized Schur

factorization, as computed by SHGEQZ.

2-by-2 diagonal blocks of P corresponding to 2-by-2 blocks

of S must be in positive diagonal form.

*LDP*

LDP is INTEGER

The leading dimension of array P. LDP >= max(1,N).

*VL*

VL is REAL array, dimension (LDVL,MM)

On entry, if SIDE = ’L’ or ’B’ and HOWMNY = ’B’, VL must

contain an N-by-N matrix Q (usually the orthogonal matrix Q

of left Schur vectors returned by SHGEQZ).

On exit, if SIDE = ’L’ or ’B’, VL contains:

if HOWMNY = ’A’, the matrix Y of left eigenvectors of (S,P);

if HOWMNY = ’B’, the matrix Q*Y;

if HOWMNY = ’S’, the left eigenvectors of (S,P) specified by

SELECT, stored consecutively in the columns of

VL, in the same order as their eigenvalues.

A complex eigenvector corresponding to a complex eigenvalue

is stored in two consecutive columns, the first holding the

real part, and the second the imaginary part.

Not referenced if SIDE = ’R’.

*LDVL*

LDVL is INTEGER

The leading dimension of array VL. LDVL >= 1, and if

SIDE = ’L’ or ’B’, LDVL >= N.

*VR*

VR is REAL array, dimension (LDVR,MM)

On entry, if SIDE = ’R’ or ’B’ and HOWMNY = ’B’, VR must

contain an N-by-N matrix Z (usually the orthogonal matrix Z

of right Schur vectors returned by SHGEQZ).

On exit, if SIDE = ’R’ or ’B’, VR contains:

if HOWMNY = ’A’, the matrix X of right eigenvectors of (S,P);

if HOWMNY = ’B’ or ’b’, the matrix Z*X;

if HOWMNY = ’S’ or ’s’, the right eigenvectors of (S,P)

specified by SELECT, stored consecutively in the

columns of VR, in the same order as their

eigenvalues.

A complex eigenvector corresponding to a complex eigenvalue

is stored in two consecutive columns, the first holding the

real part and the second the imaginary part.

Not referenced if SIDE = ’L’.

*LDVR*

LDVR is INTEGER

The leading dimension of the array VR. LDVR >= 1, and if

SIDE = ’R’ or ’B’, LDVR >= N.

*MM*

MM is INTEGER

The number of columns in the arrays VL and/or VR. MM >= M.

*M*

M is INTEGER

The number of columns in the arrays VL and/or VR actually

used to store the eigenvectors. If HOWMNY = ’A’ or ’B’, M

is set to N. Each selected real eigenvector occupies one

column and each selected complex eigenvector occupies two

columns.

*WORK*

WORK is REAL array, dimension (6*N)

*INFO*

INFO is INTEGER

= 0: successful exit.

< 0: if INFO = -i, the i-th argument had an illegal value.

> 0: the 2-by-2 block (INFO:INFO+1) does not have a complex

eigenvalue.

**Author**

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

December 2016

**Further Details:**

Allocation of workspace:

---------- -- ---------

WORK( j ) = 1-norm of j-th column of A, above the diagonal

WORK( N+j ) = 1-norm of j-th column of B, above the diagonal

WORK( 2*N+1:3*N ) = real part of eigenvector

WORK( 3*N+1:4*N ) = imaginary part of eigenvector

WORK( 4*N+1:5*N ) = real part of back-transformed eigenvector

WORK( 5*N+1:6*N ) = imaginary part of back-transformed eigenvector

Rowwise vs. columnwise solution methods:

------- -- ---------- -------- -------

Finding a generalized eigenvector consists basically of solving the

singular triangular system

(A - w B) x = 0 (for right) or: (A - w B)**H y = 0 (for left)

Consider finding the i-th right eigenvector (assume all eigenvalues

are real). The equation to be solved is:

n i

0 = sum C(j,k) v(k) = sum C(j,k) v(k) for j = i,. . .,1

k=j k=j

where C = (A - w B) (The components v(i+1:n) are 0.)

The "rowwise" method is:

(1) v(i) := 1

for j = i-1,. . .,1:

i

(2) compute s = - sum C(j,k) v(k) and

k=j+1

(3) v(j) := s / C(j,j)

Step 2 is sometimes called the "dot product" step, since it is an

inner product between the j-th row and the portion of the eigenvector

that has been computed so far.

The "columnwise" method consists basically in doing the sums

for all the rows in parallel. As each v(j) is computed, the

contribution of v(j) times the j-th column of C is added to the

partial sums. Since FORTRAN arrays are stored columnwise, this has

the advantage that at each step, the elements of C that are accessed

are adjacent to one another, whereas with the rowwise method, the

elements accessed at a step are spaced LDS (and LDP) words apart.

When finding left eigenvectors, the matrix in question is the

transpose of the one in storage, so the rowwise method then

actually accesses columns of A and B at each step, and so is the

preferred method.

**subroutine stgexc (logical WANTQ, logical WANTZ, integer N, real, dimension( lda, * ) A, integer LDA, real, dimension( ldb, * ) B, integer LDB, real, dimension( ldq, * ) Q, integer LDQ, real, dimension( ldz, * ) Z, integer LDZ, integer IFST, integer ILST, real, dimension( * ) WORK, integer LWORK, integer INFO)
STGEXC**

**Purpose:**

STGEXC reorders the generalized real Schur decomposition of a real

matrix pair (A,B) using an orthogonal equivalence transformation

(A, B) = Q * (A, B) * Z**T,

to row ILST.

(A, B) must be in generalized real Schur canonical form (as returned

by SGGES), i.e. A is block upper triangular with 1-by-1 and 2-by-2

diagonal blocks. B is upper triangular.

Optionally, the matrices Q and Z of generalized Schur vectors are

updated.

Q(in) * A(in) * Z(in)**T = Q(out) * A(out) * Z(out)**T

Q(in) * B(in) * Z(in)**T = Q(out) * B(out) * Z(out)**T

**Parameters**

*WANTQ*

WANTQ is LOGICAL

.TRUE. : update the left transformation matrix Q;

.FALSE.: do not update Q.

*WANTZ*

WANTZ is LOGICAL

.TRUE. : update the right transformation matrix Z;

.FALSE.: do not update Z.

*N*

N is INTEGER

The order of the matrices A and B. N >= 0.

*A*

A is REAL array, dimension (LDA,N)

On entry, the matrix A in generalized real Schur canonical

form.

On exit, the updated matrix A, again in generalized

real Schur canonical form.

*LDA*

LDA is INTEGER

The leading dimension of the array A. LDA >= max(1,N).

*B*

B is REAL array, dimension (LDB,N)

On entry, the matrix B in generalized real Schur canonical

form (A,B).

On exit, the updated matrix B, again in generalized

real Schur canonical form (A,B).

*LDB*

LDB is INTEGER

The leading dimension of the array B. LDB >= max(1,N).

*Q*

Q is REAL array, dimension (LDQ,N)

On entry, if WANTQ = .TRUE., the orthogonal matrix Q.

On exit, the updated matrix Q.

If WANTQ = .FALSE., Q is not referenced.

*LDQ*

LDQ is INTEGER

The leading dimension of the array Q. LDQ >= 1.

If WANTQ = .TRUE., LDQ >= N.

*Z*

Z is REAL array, dimension (LDZ,N)

On entry, if WANTZ = .TRUE., the orthogonal matrix Z.

On exit, the updated matrix Z.

If WANTZ = .FALSE., Z is not referenced.

*LDZ*

LDZ is INTEGER

The leading dimension of the array Z. LDZ >= 1.

If WANTZ = .TRUE., LDZ >= N.

*IFST*

IFST is INTEGER

*ILST*

ILST is INTEGER

Specify the reordering of the diagonal blocks of (A, B).

The block with row index IFST is moved to row ILST, by a

sequence of swapping between adjacent blocks.

On exit, if IFST pointed on entry to the second row of

a 2-by-2 block, it is changed to point to the first row;

ILST always points to the first row of the block in its

final position (which may differ from its input value by

+1 or -1). 1 <= IFST, ILST <= N.

*WORK*

On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

*LWORK*

LWORK is INTEGER

The dimension of the array WORK.

LWORK >= 1 when N <= 1, otherwise LWORK >= 4*N + 16.

only calculates the optimal size of the WORK array, returns

this value as the first entry of the WORK array, and no error

message related to LWORK is issued by XERBLA.

*INFO*

INFO is INTEGER

=0: successful exit.

<0: if INFO = -i, the i-th argument had an illegal value.

=1: The transformed matrix pair (A, B) would be too far

from generalized Schur form; the problem is ill-

conditioned. (A, B) may have been partially reordered,

and ILST points to the first row of the current

position of the block being moved.

**Author**

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

June 2017

**Contributors:**

Bo Kagstrom and Peter Poromaa, Department of Computing Science, Umea University, S-901 87 Umea, Sweden.

**References:**

[1] B. Kagstrom; A Direct Method for Reordering Eigenvalues in the

Generalized Real Schur Form of a Regular Matrix Pair (A, B), in

M.S. Moonen et al (eds), Linear Algebra for Large Scale and

Real-Time Applications, Kluwer Academic Publ. 1993, pp 195-218.

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