# realSYauxiliary - Man Page

real

## 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.

Definition at line 121 of file slansy.f.

### 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.

Definition at line 132 of file slaqsy.f.

### 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.

Definition at line 172 of file slasy2.f.

### 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.

Definition at line 101 of file ssyswapr.f.

### 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.

Contributors:

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

Definition at line 271 of file stgsy2.f.

## Author

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## Referenced By

The man pages slansy(3), slaqsy(3), slasy2(3), ssyswapr(3) and stgsy2(3) are aliases of realSYauxiliary(3).

Thu Apr 1 2021 Version 3.9.1 LAPACK