complexSYauxiliary(3) LAPACK complexSYauxiliary(3)

# NAME¶

complexSYauxiliary

# SYNOPSIS¶

## Functions¶

subroutine claesy (A, B, C, RT1, RT2, EVSCAL, CS1, SN1)
CLAESY computes the eigenvalues and eigenvectors of a 2-by-2 complex symmetric matrix. real function clansy (NORM, UPLO, N, A, LDA, WORK)
CLANSY returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a complex symmetric matrix. subroutine claqsy (UPLO, N, A, LDA, S, SCOND, AMAX, EQUED)
CLAQSY scales a symmetric/Hermitian matrix, using scaling factors computed by spoequ. subroutine csymv (UPLO, N, ALPHA, A, LDA, X, INCX, BETA, Y, INCY)
CSYMV computes a matrix-vector product for a complex symmetric matrix. subroutine csyr (UPLO, N, ALPHA, X, INCX, A, LDA)
CSYR performs the symmetric rank-1 update of a complex symmetric matrix. subroutine csyswapr (UPLO, N, A, LDA, I1, I2)
CSYSWAPR subroutine ctgsy2 (TRANS, IJOB, M, N, A, LDA, B, LDB, C, LDC, D, LDD, E, LDE, F, LDF, SCALE, RDSUM, RDSCAL, INFO)
CTGSY2 solves the generalized Sylvester equation (unblocked algorithm).

# Detailed Description¶

This is the group of complex auxiliary functions for SY matrices

# Function Documentation¶

## subroutine claesy (complex A, complex B, complex C, complex RT1, complex RT2, complex EVSCAL, complex CS1, complex SN1)¶

CLAESY computes the eigenvalues and eigenvectors of a 2-by-2 complex symmetric matrix.

Purpose:

```
CLAESY computes the eigendecomposition of a 2-by-2 symmetric matrix

( ( A, B );( B, C ) )

provided the norm of the matrix of eigenvectors is larger than

some threshold value.

RT1 is the eigenvalue of larger absolute value, and RT2 of

smaller absolute value.  If the eigenvectors are computed, then

on return ( CS1, SN1 ) is the unit eigenvector for RT1, hence

[  CS1     SN1   ] . [ A  B ] . [ CS1    -SN1   ] = [ RT1  0  ]

[ -SN1     CS1   ]   [ B  C ]   [ SN1     CS1   ]   [  0  RT2 ]```

Parameters

A

```
A is COMPLEX

The ( 1, 1 ) element of input matrix.```

B

```
B is COMPLEX

The ( 1, 2 ) element of input matrix.  The ( 2, 1 ) element

is also given by B, since the 2-by-2 matrix is symmetric.```

C

```
C is COMPLEX

The ( 2, 2 ) element of input matrix.```

RT1

```
RT1 is COMPLEX

The eigenvalue of larger modulus.```

RT2

```
RT2 is COMPLEX

The eigenvalue of smaller modulus.```

EVSCAL

```
EVSCAL is COMPLEX

The complex value by which the eigenvector matrix was scaled

to make it orthonormal.  If EVSCAL is zero, the eigenvectors

were not computed.  This means one of two things:  the 2-by-2

matrix could not be diagonalized, or the norm of the matrix

of eigenvectors before scaling was larger than the threshold

value THRESH (set below).```

CS1

```
CS1 is COMPLEX```

SN1

```
SN1 is COMPLEX

If EVSCAL .NE. 0,  ( CS1, SN1 ) is the unit right eigenvector

for RT1.```

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date

December 2016

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

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

Purpose:

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

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

complex symmetric matrix A.```

Returns

CLANSY

```
CLANSY = ( 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 CLANSY 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, CLANSY is

set to zero.```

A

```
A is COMPLEX 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

Univ. of Colorado Denver

NAG Ltd.

Date

December 2016

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

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

Purpose:

```
CLAQSY 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 COMPLEX 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

Univ. of Colorado Denver

NAG Ltd.

Date

December 2016

## subroutine csymv (character UPLO, integer N, complex ALPHA, complex, dimension( lda, * ) A, integer LDA, complex, dimension( * ) X, integer INCX, complex BETA, complex, dimension( * ) Y, integer INCY)¶

CSYMV computes a matrix-vector product for a complex symmetric matrix.

Purpose:

```
CSYMV  performs the matrix-vector  operation

y := alpha*A*x + beta*y,

where alpha and beta are scalars, x and y are n element vectors and

A is an n by n symmetric matrix.```

Parameters

UPLO

```
UPLO is CHARACTER*1

On entry, UPLO specifies whether the upper or lower

triangular part of the array A is to be referenced as

follows:

UPLO = 'U' or 'u'   Only the upper triangular part of A

is to be referenced.

UPLO = 'L' or 'l'   Only the lower triangular part of A

is to be referenced.

Unchanged on exit.```

N

```
N is INTEGER

On entry, N specifies the order of the matrix A.

N must be at least zero.

Unchanged on exit.```

ALPHA

```
ALPHA is COMPLEX

On entry, ALPHA specifies the scalar alpha.

Unchanged on exit.```

A

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

Before entry, with  UPLO = 'U' or 'u', the leading n by n

upper triangular part of the array A must contain the upper

triangular part of the symmetric matrix and the strictly

lower triangular part of A is not referenced.

Before entry, with UPLO = 'L' or 'l', the leading n by n

lower triangular part of the array A must contain the lower

triangular part of the symmetric matrix and the strictly

upper triangular part of A is not referenced.

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, N ).

Unchanged on exit.```

X

```
X is COMPLEX array, dimension at least

( 1 + ( N - 1 )*abs( INCX ) ).

Before entry, the incremented array X must contain the N-

element 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 COMPLEX

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 COMPLEX array, dimension at least

( 1 + ( N - 1 )*abs( INCY ) ).

Before entry, the incremented array Y must contain the n

element 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.```

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date

December 2016

## subroutine csyr (character UPLO, integer N, complex ALPHA, complex, dimension( * ) X, integer INCX, complex, dimension( lda, * ) A, integer LDA)¶

CSYR performs the symmetric rank-1 update of a complex symmetric matrix.

Purpose:

```
CSYR   performs the symmetric rank 1 operation

A := alpha*x*x**H + A,

where alpha is a complex scalar, x is an n element vector and A is an

n by n symmetric matrix.```

Parameters

UPLO

```
UPLO is CHARACTER*1

On entry, UPLO specifies whether the upper or lower

triangular part of the array A is to be referenced as

follows:

UPLO = 'U' or 'u'   Only the upper triangular part of A

is to be referenced.

UPLO = 'L' or 'l'   Only the lower triangular part of A

is to be referenced.

Unchanged on exit.```

N

```
N is INTEGER

On entry, N specifies the order of the matrix A.

N must be at least zero.

Unchanged on exit.```

ALPHA

```
ALPHA is COMPLEX

On entry, ALPHA specifies the scalar alpha.

Unchanged on exit.```

X

```
X is COMPLEX array, dimension at least

( 1 + ( N - 1 )*abs( INCX ) ).

Before entry, the incremented array X must contain the N-

element 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.```

A

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

Before entry, with  UPLO = 'U' or 'u', the leading n by n

upper triangular part of the array A must contain the upper

triangular part of the symmetric matrix and the strictly

lower triangular part of A is not referenced. On exit, the

upper triangular part of the array A is overwritten by the

upper triangular part of the updated matrix.

Before entry, with UPLO = 'L' or 'l', the leading n by n

lower triangular part of the array A must contain the lower

triangular part of the symmetric matrix and the strictly

upper triangular part of A is not referenced. On exit, the

lower triangular part of the array A is overwritten by the

lower triangular part of the updated matrix.```

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, N ).

Unchanged on exit.```

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date

December 2016

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

CSYSWAPR

Purpose:

```
CSYSWAPR 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 COMPLEX 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 CSYTRF.

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

Univ. of Colorado Denver

NAG Ltd.

Date

December 2016

## subroutine ctgsy2 (character TRANS, integer IJOB, integer M, integer N, complex, dimension( lda, * ) A, integer LDA, complex, dimension( ldb, * ) B, integer LDB, complex, dimension( ldc, * ) C, integer LDC, complex, dimension( ldd, * ) D, integer LDD, complex, dimension( lde, * ) E, integer LDE, complex, dimension( ldf, * ) F, integer LDF, real SCALE, real RDSUM, real RDSCAL, integer INFO)¶

CTGSY2 solves the generalized Sylvester equation (unblocked algorithm).

Purpose:

```
CTGSY2 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. A, B, D and E are upper triangular

(i.e., (A,D) and (B,E) in generalized Schur form).

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

Zx = scale * b, where Z is defined as

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

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

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

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

If TRANS = 'C', y in the conjugate transposed system Z**H*y = scale*b

is solved for, which is equivalent to solve for R and L in

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

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

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

= sigma_min(Z) using reverse communication with CLACON.

CTGSY2 also (IJOB >= 1) contributes to the computation in CTGSYL

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

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

CTGSYL.```

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 COMPLEX array, dimension (LDA, M)

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

LDA

```
LDA is INTEGER

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

B

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

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

LDB

```
LDB is INTEGER

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

C

```
C is COMPLEX 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 COMPLEX 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 COMPLEX 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 COMPLEX 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 CTGSYL, 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 CTGSY2 is called by

CTGSYL.```

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 CTGSY2 is called by

CTGSYL.```

INFO

```
INFO is INTEGER

On exit, if INFO is set to

=0: Successful exit

<0: If INFO = -i, input argument number i is illegal.

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

close eigenvalues.```

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

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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