## NAME

complexGEauxiliary - complex## SYNOPSIS

### Functions

subroutine

**cgesc2**(N, A, LDA, RHS, IPIV, JPIV, SCALE)

**CGESC2**solves a system of linear equations using the LU factorization with complete pivoting computed by sgetc2. subroutine

**cgetc2**(N, A, LDA, IPIV, JPIV, INFO)

**CGETC2**computes the LU factorization with complete pivoting of the general n-by-n matrix. real function

**clange**(NORM, M, N, A, LDA, WORK)

**CLANGE**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

**claqge**(M, N, A, LDA, R, C, ROWCND, COLCND, AMAX, EQUED)

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

**ctgex2**(WANTQ, WANTZ, N, A, LDA, B, LDB, Q, LDQ, Z, LDZ, J1, INFO)

**CTGEX2**swaps adjacent diagonal blocks in an upper (quasi) triangular matrix pair by an unitary equivalence transformation.

## Detailed Description

This is the group of complex auxiliary functions for GE matrices## Function Documentation

### subroutine cgesc2 (integer N, complex, dimension( lda, * ) A, integer LDA, complex, dimension( * ) RHS, integer, dimension( * ) IPIV, integer, dimension( * ) JPIV, real SCALE)

**CGESC2**solves a system of linear equations using the LU factorization with complete pivoting computed by sgetc2.

**Purpose:**

CGESC2 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 CGETC2.

**Parameters**

*N*

N is INTEGER The number of columns of the matrix A.

*A*

A is COMPLEX array, dimension (LDA, N) On entry, the LU part of the factorization of the n-by-n matrix A computed by CGETC2: A = P * L * U * Q

*LDA*

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

*RHS*

RHS is COMPLEX 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 REAL 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
Univ. of Colorado Denver
NAG Ltd.

**Contributors:**

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

### subroutine cgetc2 (integer N, complex, dimension( lda, * ) A, integer LDA, integer, dimension( * ) IPIV, integer, dimension( * ) JPIV, integer INFO)

**CGETC2**computes the LU factorization with complete pivoting of the general n-by-n matrix.

**Purpose:**

CGETC2 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 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
Univ. of Colorado Denver
NAG Ltd.

**Contributors:**

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

### real function clange (character NORM, integer M, integer N, complex, dimension( lda, * ) A, integer LDA, real, dimension( * ) WORK)

**CLANGE**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:**

CLANGE 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**

CLANGE

CLANGE = ( 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 CLANGE as described above.

*M*

M is INTEGER The number of rows of the matrix A. M >= 0. When M = 0, CLANGE is set to zero.

*N*

N is INTEGER The number of columns of the matrix A. N >= 0. When N = 0, CLANGE is set to zero.

*A*

A is COMPLEX 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 REAL array, dimension (MAX(1,LWORK)), where LWORK >= M when NORM = 'I'; otherwise, WORK is not referenced.

**Author**

Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.

### subroutine claqge (integer M, integer N, complex, dimension( lda, * ) A, integer LDA, real, dimension( * ) R, real, dimension( * ) C, real ROWCND, real COLCND, real AMAX, character EQUED)

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

**Purpose:**

CLAQGE 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 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 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.

### subroutine ctgex2 (logical WANTQ, logical WANTZ, integer N, complex, dimension( lda, * ) A, integer LDA, complex, dimension( ldb, * ) B, integer LDB, complex, dimension( ldq, * ) Q, integer LDQ, complex, dimension( ldz, * ) Z, integer LDZ, integer J1, integer INFO)

**CTGEX2**swaps adjacent diagonal blocks in an upper (quasi) triangular matrix pair by an unitary equivalence transformation.

**Purpose:**

CTGEX2 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 array, dimension (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 array, dimension (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 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 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
Univ. of Colorado Denver
NAG Ltd.

**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. [2] for
details.

**Contributors:**

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

**References:**

[1] 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.

[2] 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.

[2] 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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