stgsen.f −

**Functions/Subroutines**

subroutine **stgsen** (IJOB, WANTQ, WANTZ, SELECT, N, A, LDA, B, LDB, ALPHAR, ALPHAI, BETA, Q, LDQ, Z, LDZ, M, PL, PR, DIF, WORK, LWORK, IWORK, LIWORK, INFO)

STGSEN

**subroutine stgsen (integerIJOB, logicalWANTQ, logicalWANTZ, logical, dimension( * )SELECT, integerN, real, dimension( lda, * )A, integerLDA, real, dimension( ldb, * )B, integerLDB, real, dimension( * )ALPHAR, real, dimension( * )ALPHAI, real, dimension( * )BETA, real, dimension( ldq, * )Q, integerLDQ, real, dimension( ldz, * )Z, integerLDZ, integerM, realPL, realPR, real, dimension( * )DIF, real, dimension( * )WORK, integerLWORK, integer, dimension( * )IWORK, integerLIWORK, integerINFO)
STGSEN**

**Purpose:**

STGSEN reorders the generalized real Schur decomposition of a real

matrix pair (A, B) (in terms of an orthonormal equivalence trans-

formation Q**T * (A, B) * Z), so that a selected cluster of eigenvalues

appears in the leading diagonal blocks of the upper quasi-triangular

matrix A and the upper triangular B. The leading columns of Q and

Z form orthonormal bases of the corresponding left and right eigen-

spaces (deflating subspaces). (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.

STGSEN also computes the generalized eigenvalues

w(j) = (ALPHAR(j) + i*ALPHAI(j))/BETA(j)

of the reordered matrix pair (A, B).

Optionally, STGSEN computes the estimates of reciprocal condition

numbers for eigenvalues and eigenspaces. These are Difu[(A11,B11),

(A22,B22)] and Difl[(A11,B11), (A22,B22)], i.e. the separation(s)

between the matrix pairs (A11, B11) and (A22,B22) that correspond to

the selected cluster and the eigenvalues outside the cluster, resp.,

and norms of "projections" onto left and right eigenspaces w.r.t.

the selected cluster in the (1,1)-block.

**Parameters:**

*IJOB*

IJOB is INTEGER

Specifies whether condition numbers are required for the

cluster of eigenvalues (PL and PR) or the deflating subspaces

(Difu and Difl):

=0: Only reorder w.r.t. SELECT. No extras.

=1: Reciprocal of norms of "projections" onto left and right

eigenspaces w.r.t. the selected cluster (PL and PR).

=2: Upper bounds on Difu and Difl. F-norm-based estimate

(DIF(1:2)).

=3: Estimate of Difu and Difl. 1-norm-based estimate

(DIF(1:2)).

About 5 times as expensive as IJOB = 2.

=4: Compute PL, PR and DIF (i.e. 0, 1 and 2 above): Economic

version to get it all.

=5: Compute PL, PR and DIF (i.e. 0, 1 and 3 above)

*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.

*SELECT*

SELECT is LOGICAL array, dimension (N)

SELECT specifies the eigenvalues in the selected cluster.

To select a real eigenvalue w(j), SELECT(j) must be set to

.TRUE.. To select a complex conjugate pair of eigenvalues

w(j) and w(j+1), corresponding to a 2-by-2 diagonal block,

either SELECT(j) or SELECT(j+1) or both must be set to

.TRUE.; a complex conjugate pair of eigenvalues must be

either both included in the cluster or both excluded.

*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 upper quasi-triangular matrix A, with (A, B) in

generalized real Schur canonical form.

On exit, A is overwritten by the reordered matrix A.

*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 upper triangular matrix B, with (A, B) in

generalized real Schur canonical form.

On exit, B is overwritten by the reordered matrix B.

*LDB*

LDB is INTEGER

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

*ALPHAR*

ALPHAR is REAL array, dimension (N)

*ALPHAI*

ALPHAI is REAL array, dimension (N)

*BETA*

BETA is REAL array, dimension (N)

On exit, (ALPHAR(j) + ALPHAI(j)*i)/BETA(j), j=1,...,N, will

be the generalized eigenvalues. ALPHAR(j) + ALPHAI(j)*i

and BETA(j),j=1,...,N are the diagonals of the complex Schur

form (S,T) that would result if the 2-by-2 diagonal blocks of

the real generalized Schur form of (A,B) were further reduced

to triangular form using complex unitary transformations.

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) negative.

*Q*

Q is REAL array, dimension (LDQ,N)

On entry, if WANTQ = .TRUE., Q is an N-by-N matrix.

On exit, Q has been postmultiplied by the left orthogonal

transformation matrix which reorder (A, B); The leading M

columns of Q form orthonormal bases for the specified pair of

left eigenspaces (deflating subspaces).

If WANTQ = .FALSE., Q is not referenced.

*LDQ*

LDQ is INTEGER

The leading dimension of the array Q. LDQ >= 1;

and if WANTQ = .TRUE., LDQ >= N.

*Z*

Z is REAL array, dimension (LDZ,N)

On entry, if WANTZ = .TRUE., Z is an N-by-N matrix.

On exit, Z has been postmultiplied by the left orthogonal

transformation matrix which reorder (A, B); The leading M

columns of Z form orthonormal bases for the specified pair of

left eigenspaces (deflating subspaces).

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.

*M*

M is INTEGER

The dimension of the specified pair of left and right eigen-

spaces (deflating subspaces). 0 <= M <= N.

*PL*

PL is REAL

*PR*

PR is REAL

If IJOB = 1, 4 or 5, PL, PR are lower bounds on the

reciprocal of the norm of "projections" onto left and right

eigenspaces with respect to the selected cluster.

0 < PL, PR <= 1.

If M = 0 or M = N, PL = PR = 1.

If IJOB = 0, 2 or 3, PL and PR are not referenced.

*DIF*

DIF is REAL array, dimension (2).

If IJOB >= 2, DIF(1:2) store the estimates of Difu and Difl.

If IJOB = 2 or 4, DIF(1:2) are F-norm-based upper bounds on

Difu and Difl. If IJOB = 3 or 5, DIF(1:2) are 1-norm-based

estimates of Difu and Difl.

If M = 0 or N, DIF(1:2) = F-norm([A, B]).

If IJOB = 0 or 1, DIF is not referenced.

*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 >= 4*N+16.

If IJOB = 1, 2 or 4, LWORK >= MAX(4*N+16, 2*M*(N-M)).

If IJOB = 3 or 5, LWORK >= MAX(4*N+16, 4*M*(N-M)).

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.

*IWORK*

IWORK is INTEGER array, dimension (MAX(1,LIWORK))

On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.

*LIWORK*

LIWORK is INTEGER

The dimension of the array IWORK. LIWORK >= 1.

If IJOB = 1, 2 or 4, LIWORK >= N+6.

If IJOB = 3 or 5, LIWORK >= MAX(2*M*(N-M), N+6).

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

routine only calculates the optimal size of the IWORK array,

returns this value as the first entry of the IWORK array, and

no error message related to LIWORK is issued by XERBLA.

*INFO*

INFO is INTEGER

=0: Successful exit.

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

=1: Reordering of (A, B) failed because the transformed

matrix pair (A, B) would be too far from generalized

Schur form; the problem is very ill-conditioned.

(A, B) may have been partially reordered.

If requested, 0 is returned in DIF(*), PL and PR.

**Author:**

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date:**

November 2011

**Further Details:**

STGSEN first collects the selected eigenvalues by computing

orthogonal U and W that move them to the top left corner of (A, B).

In other words, the selected eigenvalues are the eigenvalues of

(A11, B11) in:

U**T*(A, B)*W = (A11 A12) (B11 B12) n1

( 0 A22),( 0 B22) n2

n1 n2 n1 n2

where N = n1+n2 and U**T means the transpose of U. The first n1 columns

of U and W span the specified pair of left and right eigenspaces

(deflating subspaces) of (A, B).

If (A, B) has been obtained from the generalized real Schur

decomposition of a matrix pair (C, D) = Q*(A, B)*Z**T, then the

reordered generalized real Schur form of (C, D) is given by

(C, D) = (Q*U)*(U**T*(A, B)*W)*(Z*W)**T,

and the first n1 columns of Q*U and Z*W span the corresponding

deflating subspaces of (C, D) (Q and Z store Q*U and Z*W, resp.).

Note that if the selected eigenvalue is sufficiently ill-conditioned,

then its value may differ significantly from its value before

reordering.

The reciprocal condition numbers of the left and right eigenspaces

spanned by the first n1 columns of U and W (or Q*U and Z*W) may

be returned in DIF(1:2), corresponding to Difu and Difl, resp.

The Difu and Difl are defined as:

Difu[(A11, B11), (A22, B22)] = sigma-min( Zu )

and

Difl[(A11, B11), (A22, B22)] = Difu[(A22, B22), (A11, B11)],

where sigma-min(Zu) is the smallest singular value of the

(2*n1*n2)-by-(2*n1*n2) matrix

Zu = [ kron(In2, A11) -kron(A22**T, In1) ]

[ kron(In2, B11) -kron(B22**T, In1) ].

Here, Inx is the identity matrix of size nx and A22**T is the

transpose of A22. kron(X, Y) is the Kronecker product between

the matrices X and Y.

When DIF(2) is small, small changes in (A, B) can cause large changes

in the deflating subspace. An approximate (asymptotic) bound on the

maximum angular error in the computed deflating subspaces is

EPS * norm((A, B)) / DIF(2),

where EPS is the machine precision.

The reciprocal norm of the projectors on the left and right

eigenspaces associated with (A11, B11) may be returned in PL and PR.

They are computed as follows. First we compute L and R so that

P*(A, B)*Q is block diagonal, where

P = ( I -L ) n1 Q = ( I R ) n1

( 0 I ) n2 and ( 0 I ) n2

n1 n2 n1 n2

and (L, R) is the solution to the generalized Sylvester equation

A11*R - L*A22 = -A12

B11*R - L*B22 = -B12

Then PL = (F-norm(L)**2+1)**(-1/2) and PR = (F-norm(R)**2+1)**(-1/2).

An approximate (asymptotic) bound on the average absolute error of

the selected eigenvalues is

EPS * norm((A, B)) / PL.

There are also global error bounds which valid for perturbations up

to a certain restriction: A lower bound (x) on the smallest

F-norm(E,F) for which an eigenvalue of (A11, B11) may move and

coalesce with an eigenvalue of (A22, B22) under perturbation (E,F),

(i.e. (A + E, B + F), is

x = min(Difu,Difl)/((1/(PL*PL)+1/(PR*PR))**(1/2)+2*max(1/PL,1/PR)).

An approximate bound on x can be computed from DIF(1:2), PL and PR.

If y = ( F-norm(E,F) / x) <= 1, the angles between the perturbed

(L’, R’) and unperturbed (L, R) left and right deflating subspaces

associated with the selected cluster in the (1,1)-blocks can be

bounded as

max-angle(L, L’) <= arctan( y * PL / (1 - y * (1 - PL * PL)**(1/2))

max-angle(R, R’) <= arctan( y * PR / (1 - y * (1 - PR * PR)**(1/2))

See LAPACK User’s Guide section 4.11 or the following references

for more information.

Note that if the default method for computing the Frobenius-norm-

based estimate DIF is not wanted (see SLATDF), then the parameter

IDIFJB (see below) should be changed from 3 to 4 (routine SLATDF

(IJOB = 2 will be used)). See STGSYL for more details.

**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.

[2] B. Kagstrom and P. Poromaa; Computing Eigenspaces with Specified

Eigenvalues of a Regular Matrix Pair (A, B) and Condition

Estimation: Theory, Algorithms and Software,

Report UMINF - 94.04, Department of Computing Science, Umea

University, S-901 87 Umea, Sweden, 1994. Also as LAPACK Working

Note 87. To appear in Numerical Algorithms, 1996.

[3] B. Kagstrom and P. Poromaa, LAPACK-Style Algorithms and Software

for Solving the Generalized Sylvester Equation and Estimating the

Separation between Regular Matrix Pairs, Report UMINF - 93.23,

Department of Computing Science, Umea University, S-901 87 Umea,

Sweden, December 1993, Revised April 1994, Also as LAPACK Working

Note 75. To appear in ACM Trans. on Math. Software, Vol 22, No 1,

1996.

Definition at line 450 of file stgsen.f.

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