realSYauxiliary(3) LAPACK realSYauxiliary(3)

realSYauxiliary

# SYNOPSIS¶

## Functions¶

real function slansy (NORM, UPLO, N, A, LDA, WORK)
SLANSY returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a real symmetric matrix. subroutine slaqsy (UPLO, N, A, LDA, S, SCOND, AMAX, EQUED)
SLAQSY scales a symmetric/Hermitian matrix, using scaling factors computed by spoequ. subroutine slasy2 (LTRANL, LTRANR, ISGN, N1, N2, TL, LDTL, TR, LDTR, B, LDB, SCALE, X, LDX, XNORM, INFO)
SLASY2 solves the Sylvester matrix equation where the matrices are of order 1 or 2. subroutine ssyswapr (UPLO, N, A, LDA, I1, I2)
SSYSWAPR applies an elementary permutation on the rows and columns of a symmetric matrix. subroutine stgsy2 (TRANS, IJOB, M, N, A, LDA, B, LDB, C, LDC, D, LDD, E, LDE, F, LDF, SCALE, RDSUM, RDSCAL, IWORK, PQ, INFO)
STGSY2 solves the generalized Sylvester equation (unblocked algorithm).

# Detailed Description¶

This is the group of real auxiliary functions for SY matrices

# Function Documentation¶

## real function slansy (character NORM, character UPLO, integer N, real, dimension( lda, * ) A, integer LDA, real, dimension( * ) WORK)¶

SLANSY returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a real symmetric matrix.

Purpose:

```
SLANSY  returns the value of the one norm,  or the Frobenius norm, or

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

real symmetric matrix A.```

Returns

SLANSY

```
SLANSY = ( 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 SLANSY as described

above.```

UPLO

```
UPLO is CHARACTER*1

Specifies whether the upper or lower triangular part of the

symmetric matrix A is to be referenced.

= 'U':  Upper triangular part of A is referenced

= 'L':  Lower triangular part of A is referenced```

N

```
N is INTEGER

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

set to zero.```

A

```
A is REAL array, dimension (LDA,N)

The symmetric matrix A.  If UPLO = 'U', the leading n by n

upper triangular part of A contains the upper triangular part

of the matrix A, and the strictly lower triangular part of A

is not referenced.  If UPLO = 'L', the leading n by n lower

triangular part of A contains the lower triangular part of

the matrix A, and the strictly upper triangular part of A is

not referenced.```

LDA

```
LDA is INTEGER

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

WORK

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

where LWORK >= N when NORM = 'I' or '1' or 'O'; otherwise,

WORK is not referenced.```

Author

Univ. of Tennessee

Univ. of California Berkeley

NAG Ltd.

Date

December 2016

## subroutine slaqsy (character UPLO, integer N, real, dimension( lda, * ) A, integer LDA, real, dimension( * ) S, real SCOND, real AMAX, character EQUED)¶

SLAQSY scales a symmetric/Hermitian matrix, using scaling factors computed by spoequ.

Purpose:

```
SLAQSY equilibrates a symmetric matrix A using the scaling factors

in the vector S.```

Parameters

UPLO

```
UPLO is CHARACTER*1

Specifies whether the upper or lower triangular part of the

symmetric matrix A is stored.

= 'U':  Upper triangular

= 'L':  Lower triangular```

N

```
N is INTEGER

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

A

```
A is REAL array, dimension (LDA,N)

On entry, the symmetric matrix A.  If UPLO = 'U', the leading

n by n upper triangular part of A contains the upper

triangular part of the matrix A, and the strictly lower

triangular part of A is not referenced.  If UPLO = 'L', the

leading n by n lower triangular part of A contains the lower

triangular part of the matrix A, and the strictly upper

triangular part of A is not referenced.

On exit, if EQUED = 'Y', the equilibrated matrix:

diag(S) * A * diag(S).```

LDA

```
LDA is INTEGER

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

S

```
S is REAL array, dimension (N)

The scale factors for A.```

SCOND

```
SCOND is REAL

Ratio of the smallest S(i) to the largest S(i).```

AMAX

```
AMAX is REAL

Absolute value of largest matrix entry.```

EQUED

```
EQUED is CHARACTER*1

Specifies whether or not equilibration was done.

= 'N':  No equilibration.

= 'Y':  Equilibration was done, i.e., A has been replaced by

diag(S) * A * diag(S).```

Internal Parameters:

```
THRESH is a threshold value used to decide if scaling should be done

based on the ratio of the scaling factors.  If SCOND < THRESH,

scaling is done.

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

be done based on the absolute size of the largest matrix element.

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

Author

Univ. of Tennessee

Univ. of California Berkeley

NAG Ltd.

Date

December 2016

## subroutine slasy2 (logical LTRANL, logical LTRANR, integer ISGN, integer N1, integer N2, real, dimension( ldtl, * ) TL, integer LDTL, real, dimension( ldtr, * ) TR, integer LDTR, real, dimension( ldb, * ) B, integer LDB, real SCALE, real, dimension( ldx, * ) X, integer LDX, real XNORM, integer INFO)¶

SLASY2 solves the Sylvester matrix equation where the matrices are of order 1 or 2.

Purpose:

```
SLASY2 solves for the N1 by N2 matrix X, 1 <= N1,N2 <= 2, in

op(TL)*X + ISGN*X*op(TR) = SCALE*B,

where TL is N1 by N1, TR is N2 by N2, B is N1 by N2, and ISGN = 1 or

-1.  op(T) = T or T**T, where T**T denotes the transpose of T.```

Parameters

LTRANL

```
LTRANL is LOGICAL

On entry, LTRANL specifies the op(TL):

= .FALSE., op(TL) = TL,

= .TRUE., op(TL) = TL**T.```

LTRANR

```
LTRANR is LOGICAL

On entry, LTRANR specifies the op(TR):

= .FALSE., op(TR) = TR,

= .TRUE., op(TR) = TR**T.```

ISGN

```
ISGN is INTEGER

On entry, ISGN specifies the sign of the equation

as described before. ISGN may only be 1 or -1.```

N1

```
N1 is INTEGER

On entry, N1 specifies the order of matrix TL.

N1 may only be 0, 1 or 2.```

N2

```
N2 is INTEGER

On entry, N2 specifies the order of matrix TR.

N2 may only be 0, 1 or 2.```

TL

```
TL is REAL array, dimension (LDTL,2)

On entry, TL contains an N1 by N1 matrix.```

LDTL

```
LDTL is INTEGER

The leading dimension of the matrix TL. LDTL >= max(1,N1).```

TR

```
TR is REAL array, dimension (LDTR,2)

On entry, TR contains an N2 by N2 matrix.```

LDTR

```
LDTR is INTEGER

The leading dimension of the matrix TR. LDTR >= max(1,N2).```

B

```
B is REAL array, dimension (LDB,2)

On entry, the N1 by N2 matrix B contains the right-hand

side of the equation.```

LDB

```
LDB is INTEGER

The leading dimension of the matrix B. LDB >= max(1,N1).```

SCALE

```
SCALE is REAL

On exit, SCALE contains the scale factor. SCALE is chosen

less than or equal to 1 to prevent the solution overflowing.```

X

```
X is REAL array, dimension (LDX,2)

On exit, X contains the N1 by N2 solution.```

LDX

```
LDX is INTEGER

The leading dimension of the matrix X. LDX >= max(1,N1).```

XNORM

```
XNORM is REAL

On exit, XNORM is the infinity-norm of the solution.```

INFO

```
INFO is INTEGER

On exit, INFO is set to

0: successful exit.

1: TL and TR have too close eigenvalues, so TL or

TR is perturbed to get a nonsingular equation.

NOTE: In the interests of speed, this routine does not

check the inputs for errors.```

Author

Univ. of Tennessee

Univ. of California Berkeley

NAG Ltd.

Date

June 2016

## subroutine ssyswapr (character UPLO, integer N, real, dimension( lda, n ) A, integer LDA, integer I1, integer I2)¶

SSYSWAPR applies an elementary permutation on the rows and columns of a symmetric matrix.

Purpose:

```
SSYSWAPR applies an elementary permutation on the rows and the columns of

a symmetric matrix.```

Parameters

UPLO

```
UPLO is CHARACTER*1

Specifies whether the details of the factorization are stored

as an upper or lower triangular matrix.

= 'U':  Upper triangular, form is A = U*D*U**T;

= 'L':  Lower triangular, form is A = L*D*L**T.```

N

```
N is INTEGER

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

A

```
A is REAL array, dimension (LDA,N)

On entry, the NB diagonal matrix D and the multipliers

used to obtain the factor U or L as computed by SSYTRF.

On exit, if INFO = 0, the (symmetric) inverse of the original

matrix.  If UPLO = 'U', the upper triangular part of the

inverse is formed and the part of A below the diagonal is not

referenced; if UPLO = 'L' the lower triangular part of the

inverse is formed and the part of A above the diagonal is

not referenced.```

LDA

```
LDA is INTEGER

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

I1

```
I1 is INTEGER

Index of the first row to swap```

I2

```
I2 is INTEGER

Index of the second row to swap```

Author

Univ. of Tennessee

Univ. of California Berkeley

NAG Ltd.

Date

December 2016

## subroutine stgsy2 (character TRANS, integer IJOB, integer M, integer N, real, dimension( lda, * ) A, integer LDA, real, dimension( ldb, * ) B, integer LDB, real, dimension( ldc, * ) C, integer LDC, real, dimension( ldd, * ) D, integer LDD, real, dimension( lde, * ) E, integer LDE, real, dimension( ldf, * ) F, integer LDF, real SCALE, real RDSUM, real RDSCAL, integer, dimension( * ) IWORK, integer PQ, integer INFO)¶

STGSY2 solves the generalized Sylvester equation (unblocked algorithm).

Purpose:

```
STGSY2 solves the generalized Sylvester equation:

A * R - L * B = scale * C                (1)

D * R - L * E = scale * F,

using Level 1 and 2 BLAS. where R and L are unknown M-by-N matrices,

(A, D), (B, E) and (C, F) are given matrix pairs of size M-by-M,

N-by-N and M-by-N, respectively, with real entries. (A, D) and (B, E)

must be in generalized Schur canonical form, i.e. A, B are upper

quasi triangular and D, E are upper triangular. The solution (R, L)

overwrites (C, F). 0 <= SCALE <= 1 is an output scaling factor

chosen to avoid overflow.

In matrix notation solving equation (1) corresponds to solve

Z*x = scale*b, where Z is defined as

Z = [ kron(In, A)  -kron(B**T, Im) ]             (2)

[ kron(In, D)  -kron(E**T, Im) ],

Ik is the identity matrix of size k and X**T is the transpose of X.

kron(X, Y) is the Kronecker product between the matrices X and Y.

In the process of solving (1), we solve a number of such systems

where Dim(In), Dim(In) = 1 or 2.

If TRANS = 'T', solve the transposed system Z**T*y = scale*b for y,

which is equivalent to solve for R and L in

A**T * R  + D**T * L   = scale * C           (3)

R  * B**T + L  * E**T  = scale * -F

This case is used to compute an estimate of Dif[(A, D), (B, E)] =

sigma_min(Z) using reverse communication with SLACON.

STGSY2 also (IJOB >= 1) contributes to the computation in STGSYL

of an upper bound on the separation between to matrix pairs. Then

the input (A, D), (B, E) are sub-pencils of the matrix pair in

STGSYL. See STGSYL for details.```

Parameters

TRANS

```
TRANS is CHARACTER*1

= 'N': solve the generalized Sylvester equation (1).

= 'T': solve the 'transposed' system (3).```

IJOB

```
IJOB is INTEGER

Specifies what kind of functionality to be performed.

= 0: solve (1) only.

= 1: A contribution from this subsystem to a Frobenius

norm-based estimate of the separation between two matrix

pairs is computed. (look ahead strategy is used).

= 2: A contribution from this subsystem to a Frobenius

norm-based estimate of the separation between two matrix

pairs is computed. (SGECON on sub-systems is used.)

Not referenced if TRANS = 'T'.```

M

```
M is INTEGER

On entry, M specifies the order of A and D, and the row

dimension of C, F, R and L.```

N

```
N is INTEGER

On entry, N specifies the order of B and E, and the column

dimension of C, F, R and L.```

A

```
A is REAL array, dimension (LDA, M)

On entry, A contains an upper quasi triangular matrix.```

LDA

```
LDA is INTEGER

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

B

```
B is REAL array, dimension (LDB, N)

On entry, B contains an upper quasi triangular matrix.```

LDB

```
LDB is INTEGER

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

C

```
C is REAL array, dimension (LDC, N)

On entry, C contains the right-hand-side of the first matrix

equation in (1).

On exit, if IJOB = 0, C has been overwritten by the

solution R.```

LDC

```
LDC is INTEGER

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

D

```
D is REAL array, dimension (LDD, M)

On entry, D contains an upper triangular matrix.```

LDD

```
LDD is INTEGER

The leading dimension of the matrix D. LDD >= max(1, M).```

E

```
E is REAL array, dimension (LDE, N)

On entry, E contains an upper triangular matrix.```

LDE

```
LDE is INTEGER

The leading dimension of the matrix E. LDE >= max(1, N).```

F

```
F is REAL array, dimension (LDF, N)

On entry, F contains the right-hand-side of the second matrix

equation in (1).

On exit, if IJOB = 0, F has been overwritten by the

solution L.```

LDF

```
LDF is INTEGER

The leading dimension of the matrix F. LDF >= max(1, M).```

SCALE

```
SCALE is REAL

On exit, 0 <= SCALE <= 1. If 0 < SCALE < 1, the solutions

R and L (C and F on entry) will hold the solutions to a

slightly perturbed system but the input matrices A, B, D and

E have not been changed. If SCALE = 0, R and L will hold the

solutions to the homogeneous system with C = F = 0. Normally,

SCALE = 1.```

RDSUM

```
RDSUM is REAL

On entry, the sum of squares of computed contributions to

the Dif-estimate under computation by STGSYL, where the

scaling factor RDSCAL (see below) has been factored out.

On exit, the corresponding sum of squares updated with the

contributions from the current sub-system.

If TRANS = 'T' RDSUM is not touched.

NOTE: RDSUM only makes sense when STGSY2 is called by STGSYL.```

RDSCAL

```
RDSCAL is REAL

On entry, scaling factor used to prevent overflow in RDSUM.

On exit, RDSCAL is updated w.r.t. the current contributions

in RDSUM.

If TRANS = 'T', RDSCAL is not touched.

NOTE: RDSCAL only makes sense when STGSY2 is called by

STGSYL.```

IWORK

```
IWORK is INTEGER array, dimension (M+N+2)```

PQ

```
PQ is INTEGER

On exit, the number of subsystems (of size 2-by-2, 4-by-4 and

8-by-8) solved by this routine.```

INFO

```
INFO is INTEGER

On exit, if INFO is set to

=0: Successful exit

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

>0: The matrix pairs (A, D) and (B, E) have common or very

close eigenvalues.```

Author

Univ. of Tennessee

Univ. of California Berkeley

NAG Ltd.

Date

December 2016

Contributors:

Bo Kagstrom and Peter Poromaa, Department of Computing Science, Umea University, S-901 87 Umea, Sweden.

# Author¶

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