complex16GEauxiliary(3) LAPACK complex16GEauxiliary(3)

# NAME¶

complex16GEauxiliary

# SYNOPSIS¶

## Functions¶

subroutine zgesc2 (N, A, LDA, RHS, IPIV, JPIV, SCALE)
ZGESC2 solves a system of linear equations using the LU factorization with complete pivoting computed by sgetc2. subroutine zgetc2 (N, A, LDA, IPIV, JPIV, INFO)
ZGETC2 computes the LU factorization with complete pivoting of the general n-by-n matrix. double precision function zlange (NORM, M, N, A, LDA, WORK)
ZLANGE returns the value of the 1-norm, Frobenius norm, infinity-norm, or the largest absolute value of any element of a general rectangular matrix. subroutine zlaqge (M, N, A, LDA, R, C, ROWCND, COLCND, AMAX, EQUED)
ZLAQGE scales a general rectangular matrix, using row and column scaling factors computed by sgeequ. subroutine ztgex2 (WANTQ, WANTZ, N, A, LDA, B, LDB, Q, LDQ, Z, LDZ, J1, INFO)
ZTGEX2 swaps adjacent diagonal blocks in an upper (quasi) triangular matrix pair by an unitary equivalence transformation.

# Detailed Description¶

This is the group of complex16 auxiliary functions for GE matrices

# Function Documentation¶

## subroutine zgesc2 (integer N, complex*16, dimension( lda, * ) A, integer LDA, complex*16, dimension( * ) RHS, integer, dimension( * ) IPIV, integer, dimension( * ) JPIV, double precision SCALE)¶

ZGESC2 solves a system of linear equations using the LU factorization with complete pivoting computed by sgetc2.

Purpose:

```
ZGESC2 solves a system of linear equations

A * X = scale* RHS

with a general N-by-N matrix A using the LU factorization with

complete pivoting computed by ZGETC2.```

Parameters

N

```
N is INTEGER

The number of columns of the matrix A.```

A

```
A is COMPLEX*16 array, dimension (LDA, N)

On entry, the  LU part of the factorization of the n-by-n

matrix A computed by ZGETC2:  A = P * L * U * Q```

LDA

```
LDA is INTEGER

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

RHS

```
RHS is COMPLEX*16 array, dimension N.

On entry, the right hand side vector b.

On exit, the solution vector X.```

IPIV

```
IPIV is INTEGER array, dimension (N).

The pivot indices; for 1 <= i <= N, row i of the

matrix has been interchanged with row IPIV(i).```

JPIV

```
JPIV is INTEGER array, dimension (N).

The pivot indices; for 1 <= j <= N, column j of the

matrix has been interchanged with column JPIV(j).```

SCALE

```
SCALE is DOUBLE PRECISION

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

0 <= SCALE <= 1 to prevent overflow in the solution.```

Author

Univ. of Tennessee

Univ. of California Berkeley

NAG Ltd.

Date

November 2017

Contributors:

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

## subroutine zgetc2 (integer N, complex*16, dimension( lda, * ) A, integer LDA, integer, dimension( * ) IPIV, integer, dimension( * ) JPIV, integer INFO)¶

ZGETC2 computes the LU factorization with complete pivoting of the general n-by-n matrix.

Purpose:

```
ZGETC2 computes an LU factorization, using complete pivoting, of the

n-by-n matrix A. The factorization has the form A = P * L * U * Q,

where P and Q are permutation matrices, L is lower triangular with

unit diagonal elements and U is upper triangular.

This is a level 1 BLAS version of the algorithm.```

Parameters

N

```
N is INTEGER

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

A

```
A is COMPLEX*16 array, dimension (LDA, N)

On entry, the n-by-n matrix to be factored.

On exit, the factors L and U from the factorization

A = P*L*U*Q; the unit diagonal elements of L are not stored.

If U(k, k) appears to be less than SMIN, U(k, k) is given the

value of SMIN, giving a nonsingular perturbed system.```

LDA

```
LDA is INTEGER

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

IPIV

```
IPIV is INTEGER array, dimension (N).

The pivot indices; for 1 <= i <= N, row i of the

matrix has been interchanged with row IPIV(i).```

JPIV

```
JPIV is INTEGER array, dimension (N).

The pivot indices; for 1 <= j <= N, column j of the

matrix has been interchanged with column JPIV(j).```

INFO

```
INFO is INTEGER

= 0: successful exit

> 0: if INFO = k, U(k, k) is likely to produce overflow if

one tries to solve for x in Ax = b. So U is perturbed

to avoid the overflow.```

Author

Univ. of Tennessee

Univ. of California Berkeley

NAG Ltd.

Date

June 2016

Contributors:

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

## double precision function zlange (character NORM, integer M, integer N, complex*16, dimension( lda, * ) A, integer LDA, double precision, dimension( * ) WORK)¶

ZLANGE returns the value of the 1-norm, Frobenius norm, infinity-norm, or the largest absolute value of any element of a general rectangular matrix.

Purpose:

```
ZLANGE  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 matrix A.```

Returns

ZLANGE

```
ZLANGE = ( 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 ZLANGE as described

above.```

M

```
M is INTEGER

The number of rows of the matrix A.  M >= 0.  When M = 0,

ZLANGE is set to zero.```

N

```
N is INTEGER

The number of columns of the matrix A.  N >= 0.  When N = 0,

ZLANGE is set to zero.```

A

```
A is COMPLEX*16 array, dimension (LDA,N)

The m by n matrix A.```

LDA

```
LDA is INTEGER

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

WORK

```
WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK)),

where LWORK >= M when NORM = 'I'; otherwise, WORK is not

referenced.```

Author

Univ. of Tennessee

Univ. of California Berkeley

NAG Ltd.

Date

December 2016

## subroutine zlaqge (integer M, integer N, complex*16, dimension( lda, * ) A, integer LDA, double precision, dimension( * ) R, double precision, dimension( * ) C, double precision ROWCND, double precision COLCND, double precision AMAX, character EQUED)¶

ZLAQGE scales a general rectangular matrix, using row and column scaling factors computed by sgeequ.

Purpose:

```
ZLAQGE equilibrates a general M by N matrix A using the row and

column 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.```

A

```
A is COMPLEX*16 array, dimension (LDA,N)

On entry, the M by N matrix A.

On exit, the equilibrated matrix.  See EQUED for the form of

the equilibrated matrix.```

LDA

```
LDA is INTEGER

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

R

```
R is DOUBLE PRECISION array, dimension (M)

The row scale factors for A.```

C

```
C is DOUBLE PRECISION array, dimension (N)

The column scale factors for A.```

ROWCND

```
ROWCND is DOUBLE PRECISION

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

COLCND

```
COLCND is DOUBLE PRECISION

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

AMAX

```
AMAX is DOUBLE PRECISION

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

NAG Ltd.

Date

December 2016

## subroutine ztgex2 (logical WANTQ, logical WANTZ, integer N, complex*16, dimension( lda, * ) A, integer LDA, complex*16, dimension( ldb, * ) B, integer LDB, complex*16, dimension( ldq, * ) Q, integer LDQ, complex*16, dimension( ldz, * ) Z, integer LDZ, integer J1, integer INFO)¶

ZTGEX2 swaps adjacent diagonal blocks in an upper (quasi) triangular matrix pair by an unitary equivalence transformation.

Purpose:

```
ZTGEX2 swaps adjacent diagonal 1 by 1 blocks (A11,B11) and (A22,B22)

in an upper triangular matrix pair (A, B) by an unitary equivalence

transformation.

(A, B) must be in generalized Schur canonical form, that is, A and

B are both upper triangular.

Optionally, the matrices Q and Z of generalized Schur vectors are

updated.

Q(in) * A(in) * Z(in)**H = Q(out) * A(out) * Z(out)**H

Q(in) * B(in) * Z(in)**H = Q(out) * B(out) * Z(out)**H```

Parameters

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.```

N

```
N is INTEGER

The order of the matrices A and B. N >= 0.```

A

```
A is COMPLEX*16 array, dimensions (LDA,N)

On entry, the matrix A in the pair (A, B).

On exit, the updated matrix A.```

LDA

```
LDA is INTEGER

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

B

```
B is COMPLEX*16 array, dimensions (LDB,N)

On entry, the matrix B in the pair (A, B).

On exit, the updated matrix B.```

LDB

```
LDB is INTEGER

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

Q

```
Q is COMPLEX*16 array, dimension (LDQ,N)

If WANTQ = .TRUE, on entry, the unitary matrix Q. On exit,

the updated matrix Q.

Not referenced if WANTQ = .FALSE..```

LDQ

```
LDQ is INTEGER

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

If WANTQ = .TRUE., LDQ >= N.```

Z

```
Z is COMPLEX*16 array, dimension (LDZ,N)

If WANTZ = .TRUE, on entry, the unitary matrix Z. On exit,

the updated matrix Z.

Not referenced if WANTZ = .FALSE..```

LDZ

```
LDZ is INTEGER

The leading dimension of the array Z. LDZ >= 1;

If WANTZ = .TRUE., LDZ >= N.```

J1

```
J1 is INTEGER

The index to the first block (A11, B11).```

INFO

```
INFO is INTEGER

=0:  Successful exit.

=1:  The transformed matrix pair (A, B) would be too far

from generalized Schur form; the problem is ill-

conditioned.```

Author

Univ. of Tennessee

Univ. of California Berkeley

NAG Ltd.

Date

June 2017

Further Details:

In the current code both weak and strong stability tests are performed. The user can omit the strong stability test by changing the internal logical parameter WANDS to .FALSE.. See ref.  for details.

Contributors:

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

References:

 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.
 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.

# Author¶

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