realGBauxiliary

**Functions**

real function **slangb** (NORM, N, KL, KU, AB, LDAB, WORK) **
SLANGB** returns the value of the 1-norm, Frobenius norm, infinity-norm, or the largest absolute value of any element of general band matrix.

subroutine

SLAQGB

This is the group of real auxiliary functions for GB matrices

**real function slangb (character NORM, integer N, integer KL, integer KU, real, dimension( ldab, * ) AB, integer LDAB, real, dimension( * ) WORK)
SLANGB** returns the value of the 1-norm, Frobenius norm, infinity-norm, or the largest absolute value of any element of general band matrix.

**Purpose:**

SLANGB returns the value of the one norm, or the Frobenius norm, or

the infinity norm, or the element of largest absolute value of an

n by n band matrix A, with kl sub-diagonals and ku super-diagonals.

**Returns**

SLANGB

SLANGB = ( max(abs(A(i,j))), NORM = ’M’ or ’m’

(

( norm1(A), NORM = ’1’, ’O’ or ’o’

(

( normI(A), NORM = ’I’ or ’i’

(

( normF(A), NORM = ’F’, ’f’, ’E’ or ’e’

where norm1 denotes the one norm of a matrix (maximum column sum),

normI denotes the infinity norm of a matrix (maximum row sum) and

normF denotes the Frobenius norm of a matrix (square root of sum of

squares). Note that max(abs(A(i,j))) is not a consistent matrix norm.

**Parameters**

*NORM*

NORM is CHARACTER*1

Specifies the value to be returned in SLANGB as described

above.

*N*

N is INTEGER

The order of the matrix A. N >= 0. When N = 0, SLANGB is

set to zero.

*KL*

KL is INTEGER

The number of sub-diagonals of the matrix A. KL >= 0.

*KU*

KU is INTEGER

The number of super-diagonals of the matrix A. KU >= 0.

*AB*

AB is REAL array, dimension (LDAB,N)

The band matrix A, stored in rows 1 to KL+KU+1. The j-th

column of A is stored in the j-th column of the array AB as

follows:

AB(ku+1+i-j,j) = A(i,j) for max(1,j-ku)<=i<=min(n,j+kl).

*LDAB*

LDAB is INTEGER

The leading dimension of the array AB. LDAB >= KL+KU+1.

*WORK*

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

where LWORK >= N when NORM = ’I’; otherwise, WORK is not

referenced.

**Author**

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

December 2016

**subroutine slaqgb (integer M, integer N, integer KL, integer KU, real, dimension( ldab, * ) AB, integer LDAB, real, dimension( * ) R, real, dimension( * ) C, real ROWCND, real COLCND, real AMAX, character EQUED)
SLAQGB** scales a general band matrix, using row and column scaling factors computed by sgbequ.

**Purpose:**

SLAQGB equilibrates a general M by N band matrix A with KL

subdiagonals and KU superdiagonals using the row and scaling factors

in the vectors R and C.

**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.

*KL*

KL is INTEGER

The number of subdiagonals within the band of A. KL >= 0.

*KU*

KU is INTEGER

The number of superdiagonals within the band of A. KU >= 0.

*AB*

AB is REAL array, dimension (LDAB,N)

On entry, the matrix A in band storage, in rows 1 to KL+KU+1.

The j-th column of A is stored in the j-th column of the

array AB as follows:

AB(ku+1+i-j,j) = A(i,j) for max(1,j-ku)<=i<=min(m,j+kl)

On exit, the equilibrated matrix, in the same storage format

as A. See EQUED for the form of the equilibrated matrix.

*LDAB*

LDAB is INTEGER

The leading dimension of the array AB. LDA >= KL+KU+1.

*R*

R is REAL array, dimension (M)

The row scale factors for A.

*C*

C is REAL array, dimension (N)

The column scale factors for A.

*ROWCND*

ROWCND is REAL

Ratio of the smallest R(i) to the largest R(i).

*COLCND*

COLCND is REAL

Ratio of the smallest C(i) to the largest C(i).

*AMAX*

AMAX is REAL

Absolute value of largest matrix entry.

*EQUED*

EQUED is CHARACTER*1

Specifies the form of equilibration that was done.

= ’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).

**Internal Parameters:**

THRESH is a threshold value used to decide if row or column scaling

should be done based on the ratio of the row or column scaling

factors. If ROWCND < THRESH, row scaling is done, and if

COLCND < THRESH, column scaling is done.

LARGE and SMALL are threshold values used to decide if row scaling

should be done based on the absolute size of the largest matrix

element. If AMAX > LARGE or AMAX < SMALL, row scaling is done.

**Author**

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

December 2016

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