Provided by: liblapack-doc_3.12.0-3build1.1_all bug

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

   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

   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

   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

   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

   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

Author

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