LAPACK  3.4.2
LAPACK: Linear Algebra PACKage
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complex16
Auxiliary routines
Collaboration diagram for complex16:

Functions/Subroutines

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/Subroutine 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.

Download ZGESC2 + dependencies [TGZ] [ZIP] [TXT]
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:
[in]N
          N is INTEGER
          The number of columns of the matrix A.
[in]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
[in]LDA
          LDA is INTEGER
          The leading dimension of the array A.  LDA >= max(1, N).
[in,out]RHS
          RHS is COMPLEX*16 array, dimension N.
          On entry, the right hand side vector b.
          On exit, the solution vector X.
[in]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).
[in]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).
[out]SCALE
          SCALE is DOUBLE PRECISION
           On exit, SCALE contains the scale factor. SCALE is chosen
           0 <= SCALE <= 1 to prevent owerflow in the solution.
Author:
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
September 2012
Contributors:
Bo Kagstrom and Peter Poromaa, Department of Computing Science, Umea University, S-901 87 Umea, Sweden.

Definition at line 116 of file zgesc2.f.

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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.

Download ZGETC2 + dependencies [TGZ] [ZIP] [TXT]
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:
[in]N
          N is INTEGER
          The order of the matrix A. N >= 0.
[in,out]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.
[in]LDA
          LDA is INTEGER
          The leading dimension of the array A.  LDA >= max(1, N).
[out]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).
[out]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).
[out]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.
Date:
September 2012
Contributors:
Bo Kagstrom and Peter Poromaa, Department of Computing Science, Umea University, S-901 87 Umea, Sweden.

Definition at line 112 of file zgetc2.f.

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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.

Download ZLANGE + dependencies [TGZ] [ZIP] [TXT]
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:
[in]NORM
          NORM is CHARACTER*1
          Specifies the value to be returned in ZLANGE as described
          above.
[in]M
          M is INTEGER
          The number of rows of the matrix A.  M >= 0.  When M = 0,
          ZLANGE is set to zero.
[in]N
          N is INTEGER
          The number of columns of the matrix A.  N >= 0.  When N = 0,
          ZLANGE is set to zero.
[in]A
          A is COMPLEX*16 array, dimension (LDA,N)
          The m by n matrix A.
[in]LDA
          LDA is INTEGER
          The leading dimension of the array A.  LDA >= max(M,1).
[out]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
Univ. of Colorado Denver
NAG Ltd.
Date:
September 2012

Definition at line 116 of file zlange.f.

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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.

Download ZLAQGE + dependencies [TGZ] [ZIP] [TXT]
Purpose: 
 ZLAQGE equilibrates a general M by N matrix A using the row and
 column scaling factors in the vectors R and C.
Parameters:
[in]M
          M is INTEGER
          The number of rows of the matrix A.  M >= 0.
[in]N
          N is INTEGER
          The number of columns of the matrix A.  N >= 0.
[in,out]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.
[in]LDA
          LDA is INTEGER
          The leading dimension of the array A.  LDA >= max(M,1).
[in]R
          R is DOUBLE PRECISION array, dimension (M)
          The row scale factors for A.
[in]C
          C is DOUBLE PRECISION array, dimension (N)
          The column scale factors for A.
[in]ROWCND
          ROWCND is DOUBLE PRECISION
          Ratio of the smallest R(i) to the largest R(i).
[in]COLCND
          COLCND is DOUBLE PRECISION
          Ratio of the smallest C(i) to the largest C(i).
[in]AMAX
          AMAX is DOUBLE PRECISION
          Absolute value of largest matrix entry.
[out]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.
Date:
September 2012

Definition at line 143 of file zlaqge.f.

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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.

Download ZTGEX2 + dependencies [TGZ] [ZIP] [TXT]
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:
[in]WANTQ
          WANTQ is LOGICAL
          .TRUE. : update the left transformation matrix Q;
          .FALSE.: do not update Q.
[in]WANTZ
          WANTZ is LOGICAL
          .TRUE. : update the right transformation matrix Z;
          .FALSE.: do not update Z.
[in]N
          N is INTEGER
          The order of the matrices A and B. N >= 0.
[in,out]A
          A is COMPLEX*16 arrays, dimensions (LDA,N)
          On entry, the matrix A in the pair (A, B).
          On exit, the updated matrix A.
[in]LDA
          LDA is INTEGER
          The leading dimension of the array A. LDA >= max(1,N).
[in,out]B
          B is COMPLEX*16 arrays, dimensions (LDB,N)
          On entry, the matrix B in the pair (A, B).
          On exit, the updated matrix B.
[in]LDB
          LDB is INTEGER
          The leading dimension of the array B. LDB >= max(1,N).
[in,out]Q
          Q is COMPLEX*16 array, dimension (LDZ,N)
          If WANTQ = .TRUE, on entry, the unitary matrix Q. On exit,
          the updated matrix Q.
          Not referenced if WANTQ = .FALSE..
[in]LDQ
          LDQ is INTEGER
          The leading dimension of the array Q. LDQ >= 1;
          If WANTQ = .TRUE., LDQ >= N.
[in,out]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..
[in]LDZ
          LDZ is INTEGER
          The leading dimension of the array Z. LDZ >= 1;
          If WANTZ = .TRUE., LDZ >= N.
[in]J1
          J1 is INTEGER
          The index to the first block (A11, B11).
[out]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.
Date:
September 2012
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.

Definition at line 190 of file ztgex2.f.

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