hecon_3(3)

NAME
hecon_3 - {he,sy}con_3: condition number estimate

SYNOPSIS

  Functions
subroutine checon_3 (uplo, n, a, lda, e, ipiv, anorm, rcond, work, info)
CHECON_3
subroutine csycon_3 (uplo, n, a, lda, e, ipiv, anorm, rcond, work, info)
CSYCON_3
subroutine dsycon_3 (uplo, n, a, lda, e, ipiv, anorm, rcond, work, iwork, info)
DSYCON_3
subroutine ssycon_3 (uplo, n, a, lda, e, ipiv, anorm, rcond, work, iwork, info)
SSYCON_3
subroutine zhecon_3 (uplo, n, a, lda, e, ipiv, anorm, rcond, work, info)
ZHECON_3
subroutine zsycon_3 (uplo, n, a, lda, e, ipiv, anorm, rcond, work, info)
ZSYCON_3

Detailed Description

Function Documentation

  subroutine checon_3 (character uplo, integer n, complex, dimension( lda, * ) a, integer lda, complex, dimension( * ) e, integer, dimension( * ) ipiv, real anorm, real rcond, complex, dimension( * ) work, integer info)

CHECON_3

Purpose:

     CHECON_3 estimates the reciprocal of the condition number (in the
     1-norm) of a complex Hermitian matrix A using the factorization
     computed by CHETRF_RK or CHETRF_BK:
        A = P*U*D*(U**H)*(P**T) or A = P*L*D*(L**H)*(P**T),
     where U (or L) is unit upper (or lower) triangular matrix,
     U**H (or L**H) is the conjugate of U (or L), P is a permutation
     matrix, P**T is the transpose of P, and D is Hermitian and block
     diagonal with 1-by-1 and 2-by-2 diagonal blocks.
     An estimate is obtained for norm(inv(A)), and the reciprocal of the
     condition number is computed as RCOND = 1 / (ANORM * norm(inv(A))).
     This routine uses BLAS3 solver CHETRS_3.

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 = P*U*D*(U**H)*(P**T);
              = 'L':  Lower triangular, form is A = P*L*D*(L**H)*(P**T).

    N

              N is INTEGER
              The order of the matrix A.  N >= 0.

    A

              A is COMPLEX array, dimension (LDA,N)
              Diagonal of the block diagonal matrix D and factors U or L
              as computed by CHETRF_RK and CHETRF_BK:
                a) ONLY diagonal elements of the Hermitian block diagonal
                   matrix D on the diagonal of A, i.e. D(k,k) = A(k,k);
                   (superdiagonal (or subdiagonal) elements of D
                    should be provided on entry in array E), and
                b) If UPLO = 'U': factor U in the superdiagonal part of A.
                   If UPLO = 'L': factor L in the subdiagonal part of A.

    LDA

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

    E

              E is COMPLEX array, dimension (N)
              On entry, contains the superdiagonal (or subdiagonal)
              elements of the Hermitian block diagonal matrix D
              with 1-by-1 or 2-by-2 diagonal blocks, where
              If UPLO = 'U': E(i) = D(i-1,i),i=2:N, E(1) not referenced;
              If UPLO = 'L': E(i) = D(i+1,i),i=1:N-1, E(N) not referenced.
              NOTE: For 1-by-1 diagonal block D(k), where
              1 <= k <= N, the element E(k) is not referenced in both
              UPLO = 'U' or UPLO = 'L' cases.

    IPIV

              IPIV is INTEGER array, dimension (N)
              Details of the interchanges and the block structure of D
              as determined by CHETRF_RK or CHETRF_BK.

    ANORM

              ANORM is REAL
              The 1-norm of the original matrix A.

    RCOND

              RCOND is REAL
              The reciprocal of the condition number of the matrix A,
              computed as RCOND = 1/(ANORM * AINVNM), where AINVNM is an
              estimate of the 1-norm of inv(A) computed in this routine.

    WORK

              WORK is COMPLEX array, dimension (2*N)

    INFO

              INFO is INTEGER
              = 0:  successful exit
              < 0:  if INFO = -i, the i-th argument had an illegal value

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Contributors:

      June 2017,  Igor Kozachenko,
                      Computer Science Division,
                      University of California, Berkeley
      September 2007, Sven Hammarling, Nicholas J. Higham, Craig Lucas,
                      School of Mathematics,
                      University of Manchester

Definition at line 164 of file checon_3.f.

  subroutine csycon_3 (character uplo, integer n, complex, dimension( lda, * ) a, integer lda, complex, dimension( * ) e, integer, dimension( * ) ipiv, real anorm, real rcond, complex, dimension( * ) work, integer info)

CSYCON_3

Purpose:

     CSYCON_3 estimates the reciprocal of the condition number (in the
     1-norm) of a complex symmetric matrix A using the factorization
     computed by CSYTRF_RK or CSYTRF_BK:
        A = P*U*D*(U**T)*(P**T) or A = P*L*D*(L**T)*(P**T),
     where U (or L) is unit upper (or lower) triangular matrix,
     U**T (or L**T) is the transpose of U (or L), P is a permutation
     matrix, P**T is the transpose of P, and D is symmetric and block
     diagonal with 1-by-1 and 2-by-2 diagonal blocks.
     An estimate is obtained for norm(inv(A)), and the reciprocal of the
     condition number is computed as RCOND = 1 / (ANORM * norm(inv(A))).
     This routine uses BLAS3 solver CSYTRS_3.

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 = P*U*D*(U**T)*(P**T);
              = 'L':  Lower triangular, form is A = P*L*D*(L**T)*(P**T).

    N

              N is INTEGER
              The order of the matrix A.  N >= 0.

    A

              A is COMPLEX array, dimension (LDA,N)
              Diagonal of the block diagonal matrix D and factors U or L
              as computed by CSYTRF_RK and CSYTRF_BK:
                a) ONLY diagonal elements of the symmetric block diagonal
                   matrix D on the diagonal of A, i.e. D(k,k) = A(k,k);
                   (superdiagonal (or subdiagonal) elements of D
                    should be provided on entry in array E), and
                b) If UPLO = 'U': factor U in the superdiagonal part of A.
                   If UPLO = 'L': factor L in the subdiagonal part of A.

    LDA

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

    E

              E is COMPLEX array, dimension (N)
              On entry, contains the superdiagonal (or subdiagonal)
              elements of the symmetric block diagonal matrix D
              with 1-by-1 or 2-by-2 diagonal blocks, where
              If UPLO = 'U': E(i) = D(i-1,i),i=2:N, E(1) not referenced;
              If UPLO = 'L': E(i) = D(i+1,i),i=1:N-1, E(N) not referenced.
              NOTE: For 1-by-1 diagonal block D(k), where
              1 <= k <= N, the element E(k) is not referenced in both
              UPLO = 'U' or UPLO = 'L' cases.

    IPIV

              IPIV is INTEGER array, dimension (N)
              Details of the interchanges and the block structure of D
              as determined by CSYTRF_RK or CSYTRF_BK.

    ANORM

              ANORM is REAL
              The 1-norm of the original matrix A.

    RCOND

              RCOND is REAL
              The reciprocal of the condition number of the matrix A,
              computed as RCOND = 1/(ANORM * AINVNM), where AINVNM is an
              estimate of the 1-norm of inv(A) computed in this routine.

    WORK

              WORK is COMPLEX array, dimension (2*N)

    INFO

              INFO is INTEGER
              = 0:  successful exit
              < 0:  if INFO = -i, the i-th argument had an illegal value

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Contributors:

      June 2017,  Igor Kozachenko,
                      Computer Science Division,
                      University of California, Berkeley
      September 2007, Sven Hammarling, Nicholas J. Higham, Craig Lucas,
                      School of Mathematics,
                      University of Manchester

Definition at line 164 of file csycon_3.f.

  subroutine dsycon_3 (character uplo, integer n, double precision, dimension( lda, * ) a, integer lda, double precision, dimension( * ) e, integer, dimension( * ) ipiv, double precision anorm, double precision rcond, double precision, dimension( * ) work, integer, dimension( * ) iwork, integer info)

DSYCON_3

Purpose:

     DSYCON_3 estimates the reciprocal of the condition number (in the
     1-norm) of a real symmetric matrix A using the factorization
     computed by DSYTRF_RK or DSYTRF_BK:
        A = P*U*D*(U**T)*(P**T) or A = P*L*D*(L**T)*(P**T),
     where U (or L) is unit upper (or lower) triangular matrix,
     U**T (or L**T) is the transpose of U (or L), P is a permutation
     matrix, P**T is the transpose of P, and D is symmetric and block
     diagonal with 1-by-1 and 2-by-2 diagonal blocks.
     An estimate is obtained for norm(inv(A)), and the reciprocal of the
     condition number is computed as RCOND = 1 / (ANORM * norm(inv(A))).
     This routine uses BLAS3 solver DSYTRS_3.

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 = P*U*D*(U**T)*(P**T);
              = 'L':  Lower triangular, form is A = P*L*D*(L**T)*(P**T).

    N

              N is INTEGER
              The order of the matrix A.  N >= 0.

    A

              A is DOUBLE PRECISION array, dimension (LDA,N)
              Diagonal of the block diagonal matrix D and factors U or L
              as computed by DSYTRF_RK and DSYTRF_BK:
                a) ONLY diagonal elements of the symmetric block diagonal
                   matrix D on the diagonal of A, i.e. D(k,k) = A(k,k);
                   (superdiagonal (or subdiagonal) elements of D
                    should be provided on entry in array E), and
                b) If UPLO = 'U': factor U in the superdiagonal part of A.
                   If UPLO = 'L': factor L in the subdiagonal part of A.

    LDA

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

    E

              E is DOUBLE PRECISION array, dimension (N)
              On entry, contains the superdiagonal (or subdiagonal)
              elements of the symmetric block diagonal matrix D
              with 1-by-1 or 2-by-2 diagonal blocks, where
              If UPLO = 'U': E(i) = D(i-1,i),i=2:N, E(1) not referenced;
              If UPLO = 'L': E(i) = D(i+1,i),i=1:N-1, E(N) not referenced.
              NOTE: For 1-by-1 diagonal block D(k), where
              1 <= k <= N, the element E(k) is not referenced in both
              UPLO = 'U' or UPLO = 'L' cases.

    IPIV

              IPIV is INTEGER array, dimension (N)
              Details of the interchanges and the block structure of D
              as determined by DSYTRF_RK or DSYTRF_BK.

    ANORM

              ANORM is DOUBLE PRECISION
              The 1-norm of the original matrix A.

    RCOND

              RCOND is DOUBLE PRECISION
              The reciprocal of the condition number of the matrix A,
              computed as RCOND = 1/(ANORM * AINVNM), where AINVNM is an
              estimate of the 1-norm of inv(A) computed in this routine.

    WORK

              WORK is DOUBLE PRECISION array, dimension (2*N)

    IWORK

              IWORK is INTEGER array, dimension (N)

    INFO

              INFO is INTEGER
              = 0:  successful exit
              < 0:  if INFO = -i, the i-th argument had an illegal value

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Contributors:

      June 2017,  Igor Kozachenko,
                      Computer Science Division,
                      University of California, Berkeley
      September 2007, Sven Hammarling, Nicholas J. Higham, Craig Lucas,
                      School of Mathematics,
                      University of Manchester

Definition at line 169 of file dsycon_3.f.

  subroutine ssycon_3 (character uplo, integer n, real, dimension( lda, * ) a, integer lda, real, dimension( * ) e, integer, dimension( * ) ipiv, real anorm, real rcond, real, dimension( * ) work, integer, dimension( * ) iwork, integer info)

SSYCON_3

Purpose:

     SSYCON_3 estimates the reciprocal of the condition number (in the
     1-norm) of a real symmetric matrix A using the factorization
     computed by DSYTRF_RK or DSYTRF_BK:
        A = P*U*D*(U**T)*(P**T) or A = P*L*D*(L**T)*(P**T),
     where U (or L) is unit upper (or lower) triangular matrix,
     U**T (or L**T) is the transpose of U (or L), P is a permutation
     matrix, P**T is the transpose of P, and D is symmetric and block
     diagonal with 1-by-1 and 2-by-2 diagonal blocks.
     An estimate is obtained for norm(inv(A)), and the reciprocal of the
     condition number is computed as RCOND = 1 / (ANORM * norm(inv(A))).
     This routine uses BLAS3 solver SSYTRS_3.

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 = P*U*D*(U**T)*(P**T);
              = 'L':  Lower triangular, form is A = P*L*D*(L**T)*(P**T).

    N

              N is INTEGER
              The order of the matrix A.  N >= 0.

    A

              A is REAL array, dimension (LDA,N)
              Diagonal of the block diagonal matrix D and factors U or L
              as computed by SSYTRF_RK and SSYTRF_BK:
                a) ONLY diagonal elements of the symmetric block diagonal
                   matrix D on the diagonal of A, i.e. D(k,k) = A(k,k);
                   (superdiagonal (or subdiagonal) elements of D
                    should be provided on entry in array E), and
                b) If UPLO = 'U': factor U in the superdiagonal part of A.
                   If UPLO = 'L': factor L in the subdiagonal part of A.

    LDA

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

    E

              E is REAL array, dimension (N)
              On entry, contains the superdiagonal (or subdiagonal)
              elements of the symmetric block diagonal matrix D
              with 1-by-1 or 2-by-2 diagonal blocks, where
              If UPLO = 'U': E(i) = D(i-1,i),i=2:N, E(1) not referenced;
              If UPLO = 'L': E(i) = D(i+1,i),i=1:N-1, E(N) not referenced.
              NOTE: For 1-by-1 diagonal block D(k), where
              1 <= k <= N, the element E(k) is not referenced in both
              UPLO = 'U' or UPLO = 'L' cases.

    IPIV

              IPIV is INTEGER array, dimension (N)
              Details of the interchanges and the block structure of D
              as determined by SSYTRF_RK or SSYTRF_BK.

    ANORM

              ANORM is REAL
              The 1-norm of the original matrix A.

    RCOND

              RCOND is REAL
              The reciprocal of the condition number of the matrix A,
              computed as RCOND = 1/(ANORM * AINVNM), where AINVNM is an
              estimate of the 1-norm of inv(A) computed in this routine.

    WORK

              WORK is REAL array, dimension (2*N)

    IWORK

              IWORK is INTEGER array, dimension (N)

    INFO

              INFO is INTEGER
              = 0:  successful exit
              < 0:  if INFO = -i, the i-th argument had an illegal value

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Contributors:

      June 2017,  Igor Kozachenko,
                      Computer Science Division,
                      University of California, Berkeley
      September 2007, Sven Hammarling, Nicholas J. Higham, Craig Lucas,
                      School of Mathematics,
                      University of Manchester

Definition at line 169 of file ssycon_3.f.

  subroutine zhecon_3 (character uplo, integer n, complex*16, dimension( lda, * ) a, integer lda, complex*16, dimension( * ) e, integer, dimension( * ) ipiv, double precision anorm, double precision rcond, complex*16, dimension( * ) work, integer info)

ZHECON_3

Purpose:

     ZHECON_3 estimates the reciprocal of the condition number (in the
     1-norm) of a complex Hermitian matrix A using the factorization
     computed by ZHETRF_RK or ZHETRF_BK:
        A = P*U*D*(U**H)*(P**T) or A = P*L*D*(L**H)*(P**T),
     where U (or L) is unit upper (or lower) triangular matrix,
     U**H (or L**H) is the conjugate of U (or L), P is a permutation
     matrix, P**T is the transpose of P, and D is Hermitian and block
     diagonal with 1-by-1 and 2-by-2 diagonal blocks.
     An estimate is obtained for norm(inv(A)), and the reciprocal of the
     condition number is computed as RCOND = 1 / (ANORM * norm(inv(A))).
     This routine uses BLAS3 solver ZHETRS_3.

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 = P*U*D*(U**H)*(P**T);
              = 'L':  Lower triangular, form is A = P*L*D*(L**H)*(P**T).

    N

              N is INTEGER
              The order of the matrix A.  N >= 0.

    A

              A is COMPLEX*16 array, dimension (LDA,N)
              Diagonal of the block diagonal matrix D and factors U or L
              as computed by ZHETRF_RK and ZHETRF_BK:
                a) ONLY diagonal elements of the Hermitian block diagonal
                   matrix D on the diagonal of A, i.e. D(k,k) = A(k,k);
                   (superdiagonal (or subdiagonal) elements of D
                    should be provided on entry in array E), and
                b) If UPLO = 'U': factor U in the superdiagonal part of A.
                   If UPLO = 'L': factor L in the subdiagonal part of A.

    LDA

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

    E

              E is COMPLEX*16 array, dimension (N)
              On entry, contains the superdiagonal (or subdiagonal)
              elements of the Hermitian block diagonal matrix D
              with 1-by-1 or 2-by-2 diagonal blocks, where
              If UPLO = 'U': E(i) = D(i-1,i),i=2:N, E(1) not referenced;
              If UPLO = 'L': E(i) = D(i+1,i),i=1:N-1, E(N) not referenced.
              NOTE: For 1-by-1 diagonal block D(k), where
              1 <= k <= N, the element E(k) is not referenced in both
              UPLO = 'U' or UPLO = 'L' cases.

    IPIV

              IPIV is INTEGER array, dimension (N)
              Details of the interchanges and the block structure of D
              as determined by ZHETRF_RK or ZHETRF_BK.

    ANORM

              ANORM is DOUBLE PRECISION
              The 1-norm of the original matrix A.

    RCOND

              RCOND is DOUBLE PRECISION
              The reciprocal of the condition number of the matrix A,
              computed as RCOND = 1/(ANORM * AINVNM), where AINVNM is an
              estimate of the 1-norm of inv(A) computed in this routine.

    WORK

              WORK is COMPLEX*16 array, dimension (2*N)

    INFO

              INFO is INTEGER
              = 0:  successful exit
              < 0:  if INFO = -i, the i-th argument had an illegal value

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Contributors:

      June 2017,  Igor Kozachenko,
                      Computer Science Division,
                      University of California, Berkeley
      September 2007, Sven Hammarling, Nicholas J. Higham, Craig Lucas,
                      School of Mathematics,
                      University of Manchester

Definition at line 164 of file zhecon_3.f.

  subroutine zsycon_3 (character uplo, integer n, complex*16, dimension( lda, * ) a, integer lda, complex*16, dimension( * ) e, integer, dimension( * ) ipiv, double precision anorm, double precision rcond, complex*16, dimension( * ) work, integer info)

ZSYCON_3

Purpose:

     ZSYCON_3 estimates the reciprocal of the condition number (in the
     1-norm) of a complex symmetric matrix A using the factorization
     computed by ZSYTRF_RK or ZSYTRF_BK:
        A = P*U*D*(U**T)*(P**T) or A = P*L*D*(L**T)*(P**T),
     where U (or L) is unit upper (or lower) triangular matrix,
     U**T (or L**T) is the transpose of U (or L), P is a permutation
     matrix, P**T is the transpose of P, and D is symmetric and block
     diagonal with 1-by-1 and 2-by-2 diagonal blocks.
     An estimate is obtained for norm(inv(A)), and the reciprocal of the
     condition number is computed as RCOND = 1 / (ANORM * norm(inv(A))).
     This routine uses BLAS3 solver ZSYTRS_3.

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 = P*U*D*(U**T)*(P**T);
              = 'L':  Lower triangular, form is A = P*L*D*(L**T)*(P**T).

    N

              N is INTEGER
              The order of the matrix A.  N >= 0.

    A

              A is COMPLEX*16 array, dimension (LDA,N)
              Diagonal of the block diagonal matrix D and factors U or L
              as computed by ZSYTRF_RK and ZSYTRF_BK:
                a) ONLY diagonal elements of the symmetric block diagonal
                   matrix D on the diagonal of A, i.e. D(k,k) = A(k,k);
                   (superdiagonal (or subdiagonal) elements of D
                    should be provided on entry in array E), and
                b) If UPLO = 'U': factor U in the superdiagonal part of A.
                   If UPLO = 'L': factor L in the subdiagonal part of A.

    LDA

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

    E

              E is COMPLEX*16 array, dimension (N)
              On entry, contains the superdiagonal (or subdiagonal)
              elements of the symmetric block diagonal matrix D
              with 1-by-1 or 2-by-2 diagonal blocks, where
              If UPLO = 'U': E(i) = D(i-1,i),i=2:N, E(1) not referenced;
              If UPLO = 'L': E(i) = D(i+1,i),i=1:N-1, E(N) not referenced.
              NOTE: For 1-by-1 diagonal block D(k), where
              1 <= k <= N, the element E(k) is not referenced in both
              UPLO = 'U' or UPLO = 'L' cases.

    IPIV

              IPIV is INTEGER array, dimension (N)
              Details of the interchanges and the block structure of D
              as determined by ZSYTRF_RK or ZSYTRF_BK.

    ANORM

              ANORM is DOUBLE PRECISION
              The 1-norm of the original matrix A.

    RCOND

              RCOND is DOUBLE PRECISION
              The reciprocal of the condition number of the matrix A,
              computed as RCOND = 1/(ANORM * AINVNM), where AINVNM is an
              estimate of the 1-norm of inv(A) computed in this routine.

    WORK

              WORK is COMPLEX*16 array, dimension (2*N)

    INFO

              INFO is INTEGER
              = 0:  successful exit
              < 0:  if INFO = -i, the i-th argument had an illegal value

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Contributors:

      June 2017,  Igor Kozachenko,
                      Computer Science Division,
                      University of California, Berkeley
      September 2007, Sven Hammarling, Nicholas J. Higham, Craig Lucas,
                      School of Mathematics,
                      University of Manchester

Definition at line 164 of file zsycon_3.f.

Author

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