cgeru

complex_blas_level2(3)              LAPACK              complex_blas_level2(3)



NAME
       complex_blas_level2

SYNOPSIS
   Functions
       subroutine cgbmv (TRANS, M, N, KL, KU, ALPHA, A, LDA, X, INCX, BETA, Y,
           INCY)
           CGBMV
       subroutine cgemv (TRANS, M, N, ALPHA, A, LDA, X, INCX, BETA, Y, INCY)
           CGEMV
       subroutine cgerc (M, N, ALPHA, X, INCX, Y, INCY, A, LDA)
           CGERC
       subroutine cgeru (M, N, ALPHA, X, INCX, Y, INCY, A, LDA)
           CGERU
       subroutine chbmv (UPLO, N, K, ALPHA, A, LDA, X, INCX, BETA, Y, INCY)
           CHBMV
       subroutine chemv (UPLO, N, ALPHA, A, LDA, X, INCX, BETA, Y, INCY)
           CHEMV
       subroutine cher (UPLO, N, ALPHA, X, INCX, A, LDA)
           CHER
       subroutine cher2 (UPLO, N, ALPHA, X, INCX, Y, INCY, A, LDA)
           CHER2
       subroutine chpmv (UPLO, N, ALPHA, AP, X, INCX, BETA, Y, INCY)
           CHPMV
       subroutine chpr (UPLO, N, ALPHA, X, INCX, AP)
           CHPR
       subroutine chpr2 (UPLO, N, ALPHA, X, INCX, Y, INCY, AP)
           CHPR2
       subroutine ctbmv (UPLO, TRANS, DIAG, N, K, A, LDA, X, INCX)
           CTBMV
       subroutine ctbsv (UPLO, TRANS, DIAG, N, K, A, LDA, X, INCX)
           CTBSV
       subroutine ctpmv (UPLO, TRANS, DIAG, N, AP, X, INCX)
           CTPMV
       subroutine ctpsv (UPLO, TRANS, DIAG, N, AP, X, INCX)
           CTPSV
       subroutine ctrmv (UPLO, TRANS, DIAG, N, A, LDA, X, INCX)
           CTRMV
       subroutine ctrsv (UPLO, TRANS, DIAG, N, A, LDA, X, INCX)
           CTRSV

Detailed Description
       This is the group of complex LEVEL 2 BLAS routines.

Function Documentation
   subroutine cgbmv (character TRANS, integer M, integer N, integer KL,
       integer KU, complex ALPHA, complex, dimension(lda,*) A, integer LDA,
       complex, dimension(*) X, integer INCX, complex BETA, complex,
       dimension(*) Y, integer INCY)
       CGBMV

       Purpose:


            CGBMV  performs one of the matrix-vector operations

               y := alpha*A*x + beta*y,   or   y := alpha*A**T*x + beta*y,   or

               y := alpha*A**H*x + beta*y,

            where alpha and beta are scalars, x and y are vectors and A is an
            m by n band matrix, with kl sub-diagonals and ku super-diagonals.

       Parameters:
           TRANS

                     TRANS is CHARACTER*1
                      On entry, TRANS specifies the operation to be performed as
                      follows:

                         TRANS = 'N' or 'n'   y := alpha*A*x + beta*y.

                         TRANS = 'T' or 't'   y := alpha*A**T*x + beta*y.

                         TRANS = 'C' or 'c'   y := alpha*A**H*x + beta*y.

           M

                     M is INTEGER
                      On entry, M specifies the number of rows of the matrix A.
                      M must be at least zero.

           N

                     N is INTEGER
                      On entry, N specifies the number of columns of the matrix A.
                      N must be at least zero.

           KL

                     KL is INTEGER
                      On entry, KL specifies the number of sub-diagonals of the
                      matrix A. KL must satisfy  0 .le. KL.

           KU

                     KU is INTEGER
                      On entry, KU specifies the number of super-diagonals of the
                      matrix A. KU must satisfy  0 .le. KU.

           ALPHA

                     ALPHA is COMPLEX
                      On entry, ALPHA specifies the scalar alpha.

           A

                     A is COMPLEX array, dimension ( LDA, N )
                      Before entry, the leading ( kl + ku + 1 ) by n part of the
                      array A must contain the matrix of coefficients, supplied
                      column by column, with the leading diagonal of the matrix in
                      row ( ku + 1 ) of the array, the first super-diagonal
                      starting at position 2 in row ku, the first sub-diagonal
                      starting at position 1 in row ( ku + 2 ), and so on.
                      Elements in the array A that do not correspond to elements
                      in the band matrix (such as the top left ku by ku triangle)
                      are not referenced.
                      The following program segment will transfer a band matrix
                      from conventional full matrix storage to band storage:

                            DO 20, J = 1, N
                               K = KU + 1 - J
                               DO 10, I = MAX( 1, J - KU ), MIN( M, J + KL )
                                  A( K + I, J ) = matrix( I, J )
                         10    CONTINUE
                         20 CONTINUE

           LDA

                     LDA is INTEGER
                      On entry, LDA specifies the first dimension of A as declared
                      in the calling (sub) program. LDA must be at least
                      ( kl + ku + 1 ).

           X

                     X is COMPLEX array, dimension at least
                      ( 1 + ( n - 1 )*abs( INCX ) ) when TRANS = 'N' or 'n'
                      and at least
                      ( 1 + ( m - 1 )*abs( INCX ) ) otherwise.
                      Before entry, the incremented array X must contain the
                      vector x.

           INCX

                     INCX is INTEGER
                      On entry, INCX specifies the increment for the elements of
                      X. INCX must not be zero.

           BETA

                     BETA is COMPLEX
                      On entry, BETA specifies the scalar beta. When BETA is
                      supplied as zero then Y need not be set on input.

           Y

                     Y is COMPLEX array, dimension at least
                      ( 1 + ( m - 1 )*abs( INCY ) ) when TRANS = 'N' or 'n'
                      and at least
                      ( 1 + ( n - 1 )*abs( INCY ) ) otherwise.
                      Before entry, the incremented array Y must contain the
                      vector y. On exit, Y is overwritten by the updated vector y.

           INCY

                     INCY is INTEGER
                      On entry, INCY specifies the increment for the elements of
                      Y. INCY must not be zero.

       Author:
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Date:
           December 2016

       Further Details:


             Level 2 Blas routine.
             The vector and matrix arguments are not referenced when N = 0, or M = 0

             -- Written on 22-October-1986.
                Jack Dongarra, Argonne National Lab.
                Jeremy Du Croz, Nag Central Office.
                Sven Hammarling, Nag Central Office.
                Richard Hanson, Sandia National Labs.

       Definition at line 189 of file cgbmv.f.

   subroutine cgemv (character TRANS, integer M, integer N, complex ALPHA,
       complex, dimension(lda,*) A, integer LDA, complex, dimension(*) X,
       integer INCX, complex BETA, complex, dimension(*) Y, integer INCY)
       CGEMV

       Purpose:


            CGEMV performs one of the matrix-vector operations

               y := alpha*A*x + beta*y,   or   y := alpha*A**T*x + beta*y,   or

               y := alpha*A**H*x + beta*y,

            where alpha and beta are scalars, x and y are vectors and A is an
            m by n matrix.

       Parameters:
           TRANS

                     TRANS is CHARACTER*1
                      On entry, TRANS specifies the operation to be performed as
                      follows:

                         TRANS = 'N' or 'n'   y := alpha*A*x + beta*y.

                         TRANS = 'T' or 't'   y := alpha*A**T*x + beta*y.

                         TRANS = 'C' or 'c'   y := alpha*A**H*x + beta*y.

           M

                     M is INTEGER
                      On entry, M specifies the number of rows of the matrix A.
                      M must be at least zero.

           N

                     N is INTEGER
                      On entry, N specifies the number of columns of the matrix A.
                      N must be at least zero.

           ALPHA

                     ALPHA is COMPLEX
                      On entry, ALPHA specifies the scalar alpha.

           A

                     A is COMPLEX array, dimension ( LDA, N )
                      Before entry, the leading m by n part of the array A must
                      contain the matrix of coefficients.

           LDA

                     LDA is INTEGER
                      On entry, LDA specifies the first dimension of A as declared
                      in the calling (sub) program. LDA must be at least
                      max( 1, m ).

           X

                     X is COMPLEX array, dimension at least
                      ( 1 + ( n - 1 )*abs( INCX ) ) when TRANS = 'N' or 'n'
                      and at least
                      ( 1 + ( m - 1 )*abs( INCX ) ) otherwise.
                      Before entry, the incremented array X must contain the
                      vector x.

           INCX

                     INCX is INTEGER
                      On entry, INCX specifies the increment for the elements of
                      X. INCX must not be zero.

           BETA

                     BETA is COMPLEX
                      On entry, BETA specifies the scalar beta. When BETA is
                      supplied as zero then Y need not be set on input.

           Y

                     Y is COMPLEX array, dimension at least
                      ( 1 + ( m - 1 )*abs( INCY ) ) when TRANS = 'N' or 'n'
                      and at least
                      ( 1 + ( n - 1 )*abs( INCY ) ) otherwise.
                      Before entry with BETA non-zero, the incremented array Y
                      must contain the vector y. On exit, Y is overwritten by the
                      updated vector y.

           INCY

                     INCY is INTEGER
                      On entry, INCY specifies the increment for the elements of
                      Y. INCY must not be zero.

       Author:
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Date:
           December 2016

       Further Details:


             Level 2 Blas routine.
             The vector and matrix arguments are not referenced when N = 0, or M = 0

             -- Written on 22-October-1986.
                Jack Dongarra, Argonne National Lab.
                Jeremy Du Croz, Nag Central Office.
                Sven Hammarling, Nag Central Office.
                Richard Hanson, Sandia National Labs.

       Definition at line 160 of file cgemv.f.

   subroutine cgerc (integer M, integer N, complex ALPHA, complex,
       dimension(*) X, integer INCX, complex, dimension(*) Y, integer INCY,
       complex, dimension(lda,*) A, integer LDA)
       CGERC

       Purpose:


            CGERC  performs the rank 1 operation

               A := alpha*x*y**H + A,

            where alpha is a scalar, x is an m element vector, y is an n element
            vector and A is an m by n matrix.

       Parameters:
           M

                     M is INTEGER
                      On entry, M specifies the number of rows of the matrix A.
                      M must be at least zero.

           N

                     N is INTEGER
                      On entry, N specifies the number of columns of the matrix A.
                      N must be at least zero.

           ALPHA

                     ALPHA is COMPLEX
                      On entry, ALPHA specifies the scalar alpha.

           X

                     X is COMPLEX array, dimension at least
                      ( 1 + ( m - 1 )*abs( INCX ) ).
                      Before entry, the incremented array X must contain the m
                      element vector x.

           INCX

                     INCX is INTEGER
                      On entry, INCX specifies the increment for the elements of
                      X. INCX must not be zero.

           Y

                     Y is COMPLEX array, dimension at least
                      ( 1 + ( n - 1 )*abs( INCY ) ).
                      Before entry, the incremented array Y must contain the n
                      element vector y.

           INCY

                     INCY is INTEGER
                      On entry, INCY specifies the increment for the elements of
                      Y. INCY must not be zero.

           A

                     A is COMPLEX array, dimension ( LDA, N )
                      Before entry, the leading m by n part of the array A must
                      contain the matrix of coefficients. On exit, A is
                      overwritten by the updated matrix.

           LDA

                     LDA is INTEGER
                      On entry, LDA specifies the first dimension of A as declared
                      in the calling (sub) program. LDA must be at least
                      max( 1, m ).

       Author:
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Date:
           December 2016

       Further Details:


             Level 2 Blas routine.

             -- Written on 22-October-1986.
                Jack Dongarra, Argonne National Lab.
                Jeremy Du Croz, Nag Central Office.
                Sven Hammarling, Nag Central Office.
                Richard Hanson, Sandia National Labs.

       Definition at line 132 of file cgerc.f.

   subroutine cgeru (integer M, integer N, complex ALPHA, complex,
       dimension(*) X, integer INCX, complex, dimension(*) Y, integer INCY,
       complex, dimension(lda,*) A, integer LDA)
       CGERU

       Purpose:


            CGERU  performs the rank 1 operation

               A := alpha*x*y**T + A,

            where alpha is a scalar, x is an m element vector, y is an n element
            vector and A is an m by n matrix.

       Parameters:
           M

                     M is INTEGER
                      On entry, M specifies the number of rows of the matrix A.
                      M must be at least zero.

           N

                     N is INTEGER
                      On entry, N specifies the number of columns of the matrix A.
                      N must be at least zero.

           ALPHA

                     ALPHA is COMPLEX
                      On entry, ALPHA specifies the scalar alpha.

           X

                     X is COMPLEX array, dimension at least
                      ( 1 + ( m - 1 )*abs( INCX ) ).
                      Before entry, the incremented array X must contain the m
                      element vector x.

           INCX

                     INCX is INTEGER
                      On entry, INCX specifies the increment for the elements of
                      X. INCX must not be zero.

           Y

                     Y is COMPLEX array, dimension at least
                      ( 1 + ( n - 1 )*abs( INCY ) ).
                      Before entry, the incremented array Y must contain the n
                      element vector y.

           INCY

                     INCY is INTEGER
                      On entry, INCY specifies the increment for the elements of
                      Y. INCY must not be zero.

           A

                     A is COMPLEX array, dimension ( LDA, N )
                      Before entry, the leading m by n part of the array A must
                      contain the matrix of coefficients. On exit, A is
                      overwritten by the updated matrix.

           LDA

                     LDA is INTEGER
                      On entry, LDA specifies the first dimension of A as declared
                      in the calling (sub) program. LDA must be at least
                      max( 1, m ).

       Author:
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Date:
           December 2016

       Further Details:


             Level 2 Blas routine.

             -- Written on 22-October-1986.
                Jack Dongarra, Argonne National Lab.
                Jeremy Du Croz, Nag Central Office.
                Sven Hammarling, Nag Central Office.
                Richard Hanson, Sandia National Labs.

       Definition at line 132 of file cgeru.f.

   subroutine chbmv (character UPLO, integer N, integer K, complex ALPHA,
       complex, dimension(lda,*) A, integer LDA, complex, dimension(*) X,
       integer INCX, complex BETA, complex, dimension(*) Y, integer INCY)
       CHBMV

       Purpose:


            CHBMV  performs the matrix-vector  operation

               y := alpha*A*x + beta*y,

            where alpha and beta are scalars, x and y are n element vectors and
            A is an n by n hermitian band matrix, with k super-diagonals.

       Parameters:
           UPLO

                     UPLO is CHARACTER*1
                      On entry, UPLO specifies whether the upper or lower
                      triangular part of the band matrix A is being supplied as
                      follows:

                         UPLO = 'U' or 'u'   The upper triangular part of A is
                                             being supplied.

                         UPLO = 'L' or 'l'   The lower triangular part of A is
                                             being supplied.

           N

                     N is INTEGER
                      On entry, N specifies the order of the matrix A.
                      N must be at least zero.

           K

                     K is INTEGER
                      On entry, K specifies the number of super-diagonals of the
                      matrix A. K must satisfy  0 .le. K.

           ALPHA

                     ALPHA is COMPLEX
                      On entry, ALPHA specifies the scalar alpha.

           A

                     A is COMPLEX array, dimension ( LDA, N )
                      Before entry with UPLO = 'U' or 'u', the leading ( k + 1 )
                      by n part of the array A must contain the upper triangular
                      band part of the hermitian matrix, supplied column by
                      column, with the leading diagonal of the matrix in row
                      ( k + 1 ) of the array, the first super-diagonal starting at
                      position 2 in row k, and so on. The top left k by k triangle
                      of the array A is not referenced.
                      The following program segment will transfer the upper
                      triangular part of a hermitian band matrix from conventional
                      full matrix storage to band storage:

                            DO 20, J = 1, N
                               M = K + 1 - J
                               DO 10, I = MAX( 1, J - K ), J
                                  A( M + I, J ) = matrix( I, J )
                         10    CONTINUE
                         20 CONTINUE

                      Before entry with UPLO = 'L' or 'l', the leading ( k + 1 )
                      by n part of the array A must contain the lower triangular
                      band part of the hermitian matrix, supplied column by
                      column, with the leading diagonal of the matrix in row 1 of
                      the array, the first sub-diagonal starting at position 1 in
                      row 2, and so on. The bottom right k by k triangle of the
                      array A is not referenced.
                      The following program segment will transfer the lower
                      triangular part of a hermitian band matrix from conventional
                      full matrix storage to band storage:

                            DO 20, J = 1, N
                               M = 1 - J
                               DO 10, I = J, MIN( N, J + K )
                                  A( M + I, J ) = matrix( I, J )
                         10    CONTINUE
                         20 CONTINUE

                      Note that the imaginary parts of the diagonal elements need
                      not be set and are assumed to be zero.

           LDA

                     LDA is INTEGER
                      On entry, LDA specifies the first dimension of A as declared
                      in the calling (sub) program. LDA must be at least
                      ( k + 1 ).

           X

                     X is COMPLEX array, dimension at least
                      ( 1 + ( n - 1 )*abs( INCX ) ).
                      Before entry, the incremented array X must contain the
                      vector x.

           INCX

                     INCX is INTEGER
                      On entry, INCX specifies the increment for the elements of
                      X. INCX must not be zero.

           BETA

                     BETA is COMPLEX
                      On entry, BETA specifies the scalar beta.

           Y

                     Y is COMPLEX array, dimension at least
                      ( 1 + ( n - 1 )*abs( INCY ) ).
                      Before entry, the incremented array Y must contain the
                      vector y. On exit, Y is overwritten by the updated vector y.

           INCY

                     INCY is INTEGER
                      On entry, INCY specifies the increment for the elements of
                      Y. INCY must not be zero.

       Author:
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Date:
           December 2016

       Further Details:


             Level 2 Blas routine.
             The vector and matrix arguments are not referenced when N = 0, or M = 0

             -- Written on 22-October-1986.
                Jack Dongarra, Argonne National Lab.
                Jeremy Du Croz, Nag Central Office.
                Sven Hammarling, Nag Central Office.
                Richard Hanson, Sandia National Labs.

       Definition at line 189 of file chbmv.f.

   subroutine chemv (character UPLO, integer N, complex ALPHA, complex,
       dimension(lda,*) A, integer LDA, complex, dimension(*) X, integer INCX,
       complex BETA, complex, dimension(*) Y, integer INCY)
       CHEMV

       Purpose:


            CHEMV  performs the matrix-vector  operation

               y := alpha*A*x + beta*y,

            where alpha and beta are scalars, x and y are n element vectors and
            A is an n by n hermitian matrix.

       Parameters:
           UPLO

                     UPLO is CHARACTER*1
                      On entry, UPLO specifies whether the upper or lower
                      triangular part of the array A is to be referenced as
                      follows:

                         UPLO = 'U' or 'u'   Only the upper triangular part of A
                                             is to be referenced.

                         UPLO = 'L' or 'l'   Only the lower triangular part of A
                                             is to be referenced.

           N

                     N is INTEGER
                      On entry, N specifies the order of the matrix A.
                      N must be at least zero.

           ALPHA

                     ALPHA is COMPLEX
                      On entry, ALPHA specifies the scalar alpha.

           A

                     A is COMPLEX array, dimension ( LDA, N )
                      Before entry with  UPLO = 'U' or 'u', the leading n by n
                      upper triangular part of the array A must contain the upper
                      triangular part of the hermitian matrix and the strictly
                      lower triangular part of A is not referenced.
                      Before entry with UPLO = 'L' or 'l', the leading n by n
                      lower triangular part of the array A must contain the lower
                      triangular part of the hermitian matrix and the strictly
                      upper triangular part of A is not referenced.
                      Note that the imaginary parts of the diagonal elements need
                      not be set and are assumed to be zero.

           LDA

                     LDA is INTEGER
                      On entry, LDA specifies the first dimension of A as declared
                      in the calling (sub) program. LDA must be at least
                      max( 1, n ).

           X

                     X is COMPLEX array, dimension at least
                      ( 1 + ( n - 1 )*abs( INCX ) ).
                      Before entry, the incremented array X must contain the n
                      element vector x.

           INCX

                     INCX is INTEGER
                      On entry, INCX specifies the increment for the elements of
                      X. INCX must not be zero.

           BETA

                     BETA is COMPLEX
                      On entry, BETA specifies the scalar beta. When BETA is
                      supplied as zero then Y need not be set on input.

           Y

                     Y is COMPLEX array, dimension at least
                      ( 1 + ( n - 1 )*abs( INCY ) ).
                      Before entry, the incremented array Y must contain the n
                      element vector y. On exit, Y is overwritten by the updated
                      vector y.

           INCY

                     INCY is INTEGER
                      On entry, INCY specifies the increment for the elements of
                      Y. INCY must not be zero.

       Author:
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Date:
           December 2016

       Further Details:


             Level 2 Blas routine.
             The vector and matrix arguments are not referenced when N = 0, or M = 0

             -- Written on 22-October-1986.
                Jack Dongarra, Argonne National Lab.
                Jeremy Du Croz, Nag Central Office.
                Sven Hammarling, Nag Central Office.
                Richard Hanson, Sandia National Labs.

       Definition at line 156 of file chemv.f.

   subroutine cher (character UPLO, integer N, real ALPHA, complex,
       dimension(*) X, integer INCX, complex, dimension(lda,*) A, integer LDA)
       CHER

       Purpose:


            CHER   performs the hermitian rank 1 operation

               A := alpha*x*x**H + A,

            where alpha is a real scalar, x is an n element vector and A is an
            n by n hermitian matrix.

       Parameters:
           UPLO

                     UPLO is CHARACTER*1
                      On entry, UPLO specifies whether the upper or lower
                      triangular part of the array A is to be referenced as
                      follows:

                         UPLO = 'U' or 'u'   Only the upper triangular part of A
                                             is to be referenced.

                         UPLO = 'L' or 'l'   Only the lower triangular part of A
                                             is to be referenced.

           N

                     N is INTEGER
                      On entry, N specifies the order of the matrix A.
                      N must be at least zero.

           ALPHA

                     ALPHA is REAL
                      On entry, ALPHA specifies the scalar alpha.

           X

                     X is COMPLEX array, dimension at least
                      ( 1 + ( n - 1 )*abs( INCX ) ).
                      Before entry, the incremented array X must contain the n
                      element vector x.

           INCX

                     INCX is INTEGER
                      On entry, INCX specifies the increment for the elements of
                      X. INCX must not be zero.

           A

                     A is COMPLEX array, dimension ( LDA, N )
                      Before entry with  UPLO = 'U' or 'u', the leading n by n
                      upper triangular part of the array A must contain the upper
                      triangular part of the hermitian matrix and the strictly
                      lower triangular part of A is not referenced. On exit, the
                      upper triangular part of the array A is overwritten by the
                      upper triangular part of the updated matrix.
                      Before entry with UPLO = 'L' or 'l', the leading n by n
                      lower triangular part of the array A must contain the lower
                      triangular part of the hermitian matrix and the strictly
                      upper triangular part of A is not referenced. On exit, the
                      lower triangular part of the array A is overwritten by the
                      lower triangular part of the updated matrix.
                      Note that the imaginary parts of the diagonal elements need
                      not be set, they are assumed to be zero, and on exit they
                      are set to zero.

           LDA

                     LDA is INTEGER
                      On entry, LDA specifies the first dimension of A as declared
                      in the calling (sub) program. LDA must be at least
                      max( 1, n ).

       Author:
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Date:
           December 2016

       Further Details:


             Level 2 Blas routine.

             -- Written on 22-October-1986.
                Jack Dongarra, Argonne National Lab.
                Jeremy Du Croz, Nag Central Office.
                Sven Hammarling, Nag Central Office.
                Richard Hanson, Sandia National Labs.

       Definition at line 137 of file cher.f.

   subroutine cher2 (character UPLO, integer N, complex ALPHA, complex,
       dimension(*) X, integer INCX, complex, dimension(*) Y, integer INCY,
       complex, dimension(lda,*) A, integer LDA)
       CHER2

       Purpose:


            CHER2  performs the hermitian rank 2 operation

               A := alpha*x*y**H + conjg( alpha )*y*x**H + A,

            where alpha is a scalar, x and y are n element vectors and A is an n
            by n hermitian matrix.

       Parameters:
           UPLO

                     UPLO is CHARACTER*1
                      On entry, UPLO specifies whether the upper or lower
                      triangular part of the array A is to be referenced as
                      follows:

                         UPLO = 'U' or 'u'   Only the upper triangular part of A
                                             is to be referenced.

                         UPLO = 'L' or 'l'   Only the lower triangular part of A
                                             is to be referenced.

           N

                     N is INTEGER
                      On entry, N specifies the order of the matrix A.
                      N must be at least zero.

           ALPHA

                     ALPHA is COMPLEX
                      On entry, ALPHA specifies the scalar alpha.

           X

                     X is COMPLEX array, dimension at least
                      ( 1 + ( n - 1 )*abs( INCX ) ).
                      Before entry, the incremented array X must contain the n
                      element vector x.

           INCX

                     INCX is INTEGER
                      On entry, INCX specifies the increment for the elements of
                      X. INCX must not be zero.

           Y

                     Y is COMPLEX array, dimension at least
                      ( 1 + ( n - 1 )*abs( INCY ) ).
                      Before entry, the incremented array Y must contain the n
                      element vector y.

           INCY

                     INCY is INTEGER
                      On entry, INCY specifies the increment for the elements of
                      Y. INCY must not be zero.

           A

                     A is COMPLEX array, dimension ( LDA, N )
                      Before entry with  UPLO = 'U' or 'u', the leading n by n
                      upper triangular part of the array A must contain the upper
                      triangular part of the hermitian matrix and the strictly
                      lower triangular part of A is not referenced. On exit, the
                      upper triangular part of the array A is overwritten by the
                      upper triangular part of the updated matrix.
                      Before entry with UPLO = 'L' or 'l', the leading n by n
                      lower triangular part of the array A must contain the lower
                      triangular part of the hermitian matrix and the strictly
                      upper triangular part of A is not referenced. On exit, the
                      lower triangular part of the array A is overwritten by the
                      lower triangular part of the updated matrix.
                      Note that the imaginary parts of the diagonal elements need
                      not be set, they are assumed to be zero, and on exit they
                      are set to zero.

           LDA

                     LDA is INTEGER
                      On entry, LDA specifies the first dimension of A as declared
                      in the calling (sub) program. LDA must be at least
                      max( 1, n ).

       Author:
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Date:
           December 2016

       Further Details:


             Level 2 Blas routine.

             -- Written on 22-October-1986.
                Jack Dongarra, Argonne National Lab.
                Jeremy Du Croz, Nag Central Office.
                Sven Hammarling, Nag Central Office.
                Richard Hanson, Sandia National Labs.

       Definition at line 152 of file cher2.f.

   subroutine chpmv (character UPLO, integer N, complex ALPHA, complex,
       dimension(*) AP, complex, dimension(*) X, integer INCX, complex BETA,
       complex, dimension(*) Y, integer INCY)
       CHPMV

       Purpose:


            CHPMV  performs the matrix-vector operation

               y := alpha*A*x + beta*y,

            where alpha and beta are scalars, x and y are n element vectors and
            A is an n by n hermitian matrix, supplied in packed form.

       Parameters:
           UPLO

                     UPLO is CHARACTER*1
                      On entry, UPLO specifies whether the upper or lower
                      triangular part of the matrix A is supplied in the packed
                      array AP as follows:

                         UPLO = 'U' or 'u'   The upper triangular part of A is
                                             supplied in AP.

                         UPLO = 'L' or 'l'   The lower triangular part of A is
                                             supplied in AP.

           N

                     N is INTEGER
                      On entry, N specifies the order of the matrix A.
                      N must be at least zero.

           ALPHA

                     ALPHA is COMPLEX
                      On entry, ALPHA specifies the scalar alpha.

           AP

                     AP is COMPLEX array, dimension at least
                      ( ( n*( n + 1 ) )/2 ).
                      Before entry with UPLO = 'U' or 'u', the array AP must
                      contain the upper triangular part of the hermitian matrix
                      packed sequentially, column by column, so that AP( 1 )
                      contains a( 1, 1 ), AP( 2 ) and AP( 3 ) contain a( 1, 2 )
                      and a( 2, 2 ) respectively, and so on.
                      Before entry with UPLO = 'L' or 'l', the array AP must
                      contain the lower triangular part of the hermitian matrix
                      packed sequentially, column by column, so that AP( 1 )
                      contains a( 1, 1 ), AP( 2 ) and AP( 3 ) contain a( 2, 1 )
                      and a( 3, 1 ) respectively, and so on.
                      Note that the imaginary parts of the diagonal elements need
                      not be set and are assumed to be zero.

           X

                     X is COMPLEX array, dimension at least
                      ( 1 + ( n - 1 )*abs( INCX ) ).
                      Before entry, the incremented array X must contain the n
                      element vector x.

           INCX

                     INCX is INTEGER
                      On entry, INCX specifies the increment for the elements of
                      X. INCX must not be zero.

           BETA

                     BETA is COMPLEX
                      On entry, BETA specifies the scalar beta. When BETA is
                      supplied as zero then Y need not be set on input.

           Y

                     Y is COMPLEX array, dimension at least
                      ( 1 + ( n - 1 )*abs( INCY ) ).
                      Before entry, the incremented array Y must contain the n
                      element vector y. On exit, Y is overwritten by the updated
                      vector y.

           INCY

                     INCY is INTEGER
                      On entry, INCY specifies the increment for the elements of
                      Y. INCY must not be zero.

       Author:
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Date:
           December 2016

       Further Details:


             Level 2 Blas routine.
             The vector and matrix arguments are not referenced when N = 0, or M = 0

             -- Written on 22-October-1986.
                Jack Dongarra, Argonne National Lab.
                Jeremy Du Croz, Nag Central Office.
                Sven Hammarling, Nag Central Office.
                Richard Hanson, Sandia National Labs.

       Definition at line 151 of file chpmv.f.

   subroutine chpr (character UPLO, integer N, real ALPHA, complex,
       dimension(*) X, integer INCX, complex, dimension(*) AP)
       CHPR

       Purpose:


            CHPR    performs the hermitian rank 1 operation

               A := alpha*x*x**H + A,

            where alpha is a real scalar, x is an n element vector and A is an
            n by n hermitian matrix, supplied in packed form.

       Parameters:
           UPLO

                     UPLO is CHARACTER*1
                      On entry, UPLO specifies whether the upper or lower
                      triangular part of the matrix A is supplied in the packed
                      array AP as follows:

                         UPLO = 'U' or 'u'   The upper triangular part of A is
                                             supplied in AP.

                         UPLO = 'L' or 'l'   The lower triangular part of A is
                                             supplied in AP.

           N

                     N is INTEGER
                      On entry, N specifies the order of the matrix A.
                      N must be at least zero.

           ALPHA

                     ALPHA is REAL
                      On entry, ALPHA specifies the scalar alpha.

           X

                     X is COMPLEX array, dimension at least
                      ( 1 + ( n - 1 )*abs( INCX ) ).
                      Before entry, the incremented array X must contain the n
                      element vector x.

           INCX

                     INCX is INTEGER
                      On entry, INCX specifies the increment for the elements of
                      X. INCX must not be zero.

           AP

                     AP is COMPLEX array, dimension at least
                      ( ( n*( n + 1 ) )/2 ).
                      Before entry with  UPLO = 'U' or 'u', the array AP must
                      contain the upper triangular part of the hermitian matrix
                      packed sequentially, column by column, so that AP( 1 )
                      contains a( 1, 1 ), AP( 2 ) and AP( 3 ) contain a( 1, 2 )
                      and a( 2, 2 ) respectively, and so on. On exit, the array
                      AP is overwritten by the upper triangular part of the
                      updated matrix.
                      Before entry with UPLO = 'L' or 'l', the array AP must
                      contain the lower triangular part of the hermitian matrix
                      packed sequentially, column by column, so that AP( 1 )
                      contains a( 1, 1 ), AP( 2 ) and AP( 3 ) contain a( 2, 1 )
                      and a( 3, 1 ) respectively, and so on. On exit, the array
                      AP is overwritten by the lower triangular part of the
                      updated matrix.
                      Note that the imaginary parts of the diagonal elements need
                      not be set, they are assumed to be zero, and on exit they
                      are set to zero.

       Author:
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Date:
           December 2016

       Further Details:


             Level 2 Blas routine.

             -- Written on 22-October-1986.
                Jack Dongarra, Argonne National Lab.
                Jeremy Du Croz, Nag Central Office.
                Sven Hammarling, Nag Central Office.
                Richard Hanson, Sandia National Labs.

       Definition at line 132 of file chpr.f.

   subroutine chpr2 (character UPLO, integer N, complex ALPHA, complex,
       dimension(*) X, integer INCX, complex, dimension(*) Y, integer INCY,
       complex, dimension(*) AP)
       CHPR2

       Purpose:


            CHPR2  performs the hermitian rank 2 operation

               A := alpha*x*y**H + conjg( alpha )*y*x**H + A,

            where alpha is a scalar, x and y are n element vectors and A is an
            n by n hermitian matrix, supplied in packed form.

       Parameters:
           UPLO

                     UPLO is CHARACTER*1
                      On entry, UPLO specifies whether the upper or lower
                      triangular part of the matrix A is supplied in the packed
                      array AP as follows:

                         UPLO = 'U' or 'u'   The upper triangular part of A is
                                             supplied in AP.

                         UPLO = 'L' or 'l'   The lower triangular part of A is
                                             supplied in AP.

           N

                     N is INTEGER
                      On entry, N specifies the order of the matrix A.
                      N must be at least zero.

           ALPHA

                     ALPHA is COMPLEX
                      On entry, ALPHA specifies the scalar alpha.

           X

                     X is COMPLEX array, dimension at least
                      ( 1 + ( n - 1 )*abs( INCX ) ).
                      Before entry, the incremented array X must contain the n
                      element vector x.

           INCX

                     INCX is INTEGER
                      On entry, INCX specifies the increment for the elements of
                      X. INCX must not be zero.

           Y

                     Y is COMPLEX array, dimension at least
                      ( 1 + ( n - 1 )*abs( INCY ) ).
                      Before entry, the incremented array Y must contain the n
                      element vector y.

           INCY

                     INCY is INTEGER
                      On entry, INCY specifies the increment for the elements of
                      Y. INCY must not be zero.

           AP

                     AP is COMPLEX array, dimension at least
                      ( ( n*( n + 1 ) )/2 ).
                      Before entry with  UPLO = 'U' or 'u', the array AP must
                      contain the upper triangular part of the hermitian matrix
                      packed sequentially, column by column, so that AP( 1 )
                      contains a( 1, 1 ), AP( 2 ) and AP( 3 ) contain a( 1, 2 )
                      and a( 2, 2 ) respectively, and so on. On exit, the array
                      AP is overwritten by the upper triangular part of the
                      updated matrix.
                      Before entry with UPLO = 'L' or 'l', the array AP must
                      contain the lower triangular part of the hermitian matrix
                      packed sequentially, column by column, so that AP( 1 )
                      contains a( 1, 1 ), AP( 2 ) and AP( 3 ) contain a( 2, 1 )
                      and a( 3, 1 ) respectively, and so on. On exit, the array
                      AP is overwritten by the lower triangular part of the
                      updated matrix.
                      Note that the imaginary parts of the diagonal elements need
                      not be set, they are assumed to be zero, and on exit they
                      are set to zero.

       Author:
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Date:
           December 2016

       Further Details:


             Level 2 Blas routine.

             -- Written on 22-October-1986.
                Jack Dongarra, Argonne National Lab.
                Jeremy Du Croz, Nag Central Office.
                Sven Hammarling, Nag Central Office.
                Richard Hanson, Sandia National Labs.

       Definition at line 147 of file chpr2.f.

   subroutine ctbmv (character UPLO, character TRANS, character DIAG, integer
       N, integer K, complex, dimension(lda,*) A, integer LDA, complex,
       dimension(*) X, integer INCX)
       CTBMV

       Purpose:


            CTBMV  performs one of the matrix-vector operations

               x := A*x,   or   x := A**T*x,   or   x := A**H*x,

            where x is an n element vector and  A is an n by n unit, or non-unit,
            upper or lower triangular band matrix, with ( k + 1 ) diagonals.

       Parameters:
           UPLO

                     UPLO is CHARACTER*1
                      On entry, UPLO specifies whether the matrix is an upper or
                      lower triangular matrix as follows:

                         UPLO = 'U' or 'u'   A is an upper triangular matrix.

                         UPLO = 'L' or 'l'   A is a lower triangular matrix.

           TRANS

                     TRANS is CHARACTER*1
                      On entry, TRANS specifies the operation to be performed as
                      follows:

                         TRANS = 'N' or 'n'   x := A*x.

                         TRANS = 'T' or 't'   x := A**T*x.

                         TRANS = 'C' or 'c'   x := A**H*x.

           DIAG

                     DIAG is CHARACTER*1
                      On entry, DIAG specifies whether or not A is unit
                      triangular as follows:

                         DIAG = 'U' or 'u'   A is assumed to be unit triangular.

                         DIAG = 'N' or 'n'   A is not assumed to be unit
                                             triangular.

           N

                     N is INTEGER
                      On entry, N specifies the order of the matrix A.
                      N must be at least zero.

           K

                     K is INTEGER
                      On entry with UPLO = 'U' or 'u', K specifies the number of
                      super-diagonals of the matrix A.
                      On entry with UPLO = 'L' or 'l', K specifies the number of
                      sub-diagonals of the matrix A.
                      K must satisfy  0 .le. K.

           A

                     A is COMPLEX array, dimension ( LDA, N ).
                      Before entry with UPLO = 'U' or 'u', the leading ( k + 1 )
                      by n part of the array A must contain the upper triangular
                      band part of the matrix of coefficients, supplied column by
                      column, with the leading diagonal of the matrix in row
                      ( k + 1 ) of the array, the first super-diagonal starting at
                      position 2 in row k, and so on. The top left k by k triangle
                      of the array A is not referenced.
                      The following program segment will transfer an upper
                      triangular band matrix from conventional full matrix storage
                      to band storage:

                            DO 20, J = 1, N
                               M = K + 1 - J
                               DO 10, I = MAX( 1, J - K ), J
                                  A( M + I, J ) = matrix( I, J )
                         10    CONTINUE
                         20 CONTINUE

                      Before entry with UPLO = 'L' or 'l', the leading ( k + 1 )
                      by n part of the array A must contain the lower triangular
                      band part of the matrix of coefficients, supplied column by
                      column, with the leading diagonal of the matrix in row 1 of
                      the array, the first sub-diagonal starting at position 1 in
                      row 2, and so on. The bottom right k by k triangle of the
                      array A is not referenced.
                      The following program segment will transfer a lower
                      triangular band matrix from conventional full matrix storage
                      to band storage:

                            DO 20, J = 1, N
                               M = 1 - J
                               DO 10, I = J, MIN( N, J + K )
                                  A( M + I, J ) = matrix( I, J )
                         10    CONTINUE
                         20 CONTINUE

                      Note that when DIAG = 'U' or 'u' the elements of the array A
                      corresponding to the diagonal elements of the matrix are not
                      referenced, but are assumed to be unity.

           LDA

                     LDA is INTEGER
                      On entry, LDA specifies the first dimension of A as declared
                      in the calling (sub) program. LDA must be at least
                      ( k + 1 ).

           X

                     X is COMPLEX array, dimension at least
                      ( 1 + ( n - 1 )*abs( INCX ) ).
                      Before entry, the incremented array X must contain the n
                      element vector x. On exit, X is overwritten with the
                      transformed vector x.

           INCX

                     INCX is INTEGER
                      On entry, INCX specifies the increment for the elements of
                      X. INCX must not be zero.

       Author:
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Date:
           December 2016

       Further Details:


             Level 2 Blas routine.
             The vector and matrix arguments are not referenced when N = 0, or M = 0

             -- Written on 22-October-1986.
                Jack Dongarra, Argonne National Lab.
                Jeremy Du Croz, Nag Central Office.
                Sven Hammarling, Nag Central Office.
                Richard Hanson, Sandia National Labs.

       Definition at line 188 of file ctbmv.f.

   subroutine ctbsv (character UPLO, character TRANS, character DIAG, integer
       N, integer K, complex, dimension(lda,*) A, integer LDA, complex,
       dimension(*) X, integer INCX)
       CTBSV

       Purpose:


            CTBSV  solves one of the systems of equations

               A*x = b,   or   A**T*x = b,   or   A**H*x = b,

            where b and x are n element vectors and A is an n by n unit, or
            non-unit, upper or lower triangular band matrix, with ( k + 1 )
            diagonals.

            No test for singularity or near-singularity is included in this
            routine. Such tests must be performed before calling this routine.

       Parameters:
           UPLO

                     UPLO is CHARACTER*1
                      On entry, UPLO specifies whether the matrix is an upper or
                      lower triangular matrix as follows:

                         UPLO = 'U' or 'u'   A is an upper triangular matrix.

                         UPLO = 'L' or 'l'   A is a lower triangular matrix.

           TRANS

                     TRANS is CHARACTER*1
                      On entry, TRANS specifies the equations to be solved as
                      follows:

                         TRANS = 'N' or 'n'   A*x = b.

                         TRANS = 'T' or 't'   A**T*x = b.

                         TRANS = 'C' or 'c'   A**H*x = b.

           DIAG

                     DIAG is CHARACTER*1
                      On entry, DIAG specifies whether or not A is unit
                      triangular as follows:

                         DIAG = 'U' or 'u'   A is assumed to be unit triangular.

                         DIAG = 'N' or 'n'   A is not assumed to be unit
                                             triangular.

           N

                     N is INTEGER
                      On entry, N specifies the order of the matrix A.
                      N must be at least zero.

           K

                     K is INTEGER
                      On entry with UPLO = 'U' or 'u', K specifies the number of
                      super-diagonals of the matrix A.
                      On entry with UPLO = 'L' or 'l', K specifies the number of
                      sub-diagonals of the matrix A.
                      K must satisfy  0 .le. K.

           A

                     A is COMPLEX array, dimension ( LDA, N )
                      Before entry with UPLO = 'U' or 'u', the leading ( k + 1 )
                      by n part of the array A must contain the upper triangular
                      band part of the matrix of coefficients, supplied column by
                      column, with the leading diagonal of the matrix in row
                      ( k + 1 ) of the array, the first super-diagonal starting at
                      position 2 in row k, and so on. The top left k by k triangle
                      of the array A is not referenced.
                      The following program segment will transfer an upper
                      triangular band matrix from conventional full matrix storage
                      to band storage:

                            DO 20, J = 1, N
                               M = K + 1 - J
                               DO 10, I = MAX( 1, J - K ), J
                                  A( M + I, J ) = matrix( I, J )
                         10    CONTINUE
                         20 CONTINUE

                      Before entry with UPLO = 'L' or 'l', the leading ( k + 1 )
                      by n part of the array A must contain the lower triangular
                      band part of the matrix of coefficients, supplied column by
                      column, with the leading diagonal of the matrix in row 1 of
                      the array, the first sub-diagonal starting at position 1 in
                      row 2, and so on. The bottom right k by k triangle of the
                      array A is not referenced.
                      The following program segment will transfer a lower
                      triangular band matrix from conventional full matrix storage
                      to band storage:

                            DO 20, J = 1, N
                               M = 1 - J
                               DO 10, I = J, MIN( N, J + K )
                                  A( M + I, J ) = matrix( I, J )
                         10    CONTINUE
                         20 CONTINUE

                      Note that when DIAG = 'U' or 'u' the elements of the array A
                      corresponding to the diagonal elements of the matrix are not
                      referenced, but are assumed to be unity.

           LDA

                     LDA is INTEGER
                      On entry, LDA specifies the first dimension of A as declared
                      in the calling (sub) program. LDA must be at least
                      ( k + 1 ).

           X

                     X is COMPLEX array, dimension at least
                      ( 1 + ( n - 1 )*abs( INCX ) ).
                      Before entry, the incremented array X must contain the n
                      element right-hand side vector b. On exit, X is overwritten
                      with the solution vector x.

           INCX

                     INCX is INTEGER
                      On entry, INCX specifies the increment for the elements of
                      X. INCX must not be zero.

       Author:
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Date:
           December 2016

       Further Details:


             Level 2 Blas routine.

             -- Written on 22-October-1986.
                Jack Dongarra, Argonne National Lab.
                Jeremy Du Croz, Nag Central Office.
                Sven Hammarling, Nag Central Office.
                Richard Hanson, Sandia National Labs.

       Definition at line 191 of file ctbsv.f.

   subroutine ctpmv (character UPLO, character TRANS, character DIAG, integer
       N, complex, dimension(*) AP, complex, dimension(*) X, integer INCX)
       CTPMV

       Purpose:


            CTPMV  performs one of the matrix-vector operations

               x := A*x,   or   x := A**T*x,   or   x := A**H*x,

            where x is an n element vector and  A is an n by n unit, or non-unit,
            upper or lower triangular matrix, supplied in packed form.

       Parameters:
           UPLO

                     UPLO is CHARACTER*1
                      On entry, UPLO specifies whether the matrix is an upper or
                      lower triangular matrix as follows:

                         UPLO = 'U' or 'u'   A is an upper triangular matrix.

                         UPLO = 'L' or 'l'   A is a lower triangular matrix.

           TRANS

                     TRANS is CHARACTER*1
                      On entry, TRANS specifies the operation to be performed as
                      follows:

                         TRANS = 'N' or 'n'   x := A*x.

                         TRANS = 'T' or 't'   x := A**T*x.

                         TRANS = 'C' or 'c'   x := A**H*x.

           DIAG

                     DIAG is CHARACTER*1
                      On entry, DIAG specifies whether or not A is unit
                      triangular as follows:

                         DIAG = 'U' or 'u'   A is assumed to be unit triangular.

                         DIAG = 'N' or 'n'   A is not assumed to be unit
                                             triangular.

           N

                     N is INTEGER
                      On entry, N specifies the order of the matrix A.
                      N must be at least zero.

           AP

                     AP is COMPLEX array, dimension at least
                      ( ( n*( n + 1 ) )/2 ).
                      Before entry with  UPLO = 'U' or 'u', the array AP must
                      contain the upper triangular matrix packed sequentially,
                      column by column, so that AP( 1 ) contains a( 1, 1 ),
                      AP( 2 ) and AP( 3 ) contain a( 1, 2 ) and a( 2, 2 )
                      respectively, and so on.
                      Before entry with UPLO = 'L' or 'l', the array AP must
                      contain the lower triangular matrix packed sequentially,
                      column by column, so that AP( 1 ) contains a( 1, 1 ),
                      AP( 2 ) and AP( 3 ) contain a( 2, 1 ) and a( 3, 1 )
                      respectively, and so on.
                      Note that when  DIAG = 'U' or 'u', the diagonal elements of
                      A are not referenced, but are assumed to be unity.

           X

                     X is COMPLEX array, dimension at least
                      ( 1 + ( n - 1 )*abs( INCX ) ).
                      Before entry, the incremented array X must contain the n
                      element vector x. On exit, X is overwritten with the
                      transformed vector x.

           INCX

                     INCX is INTEGER
                      On entry, INCX specifies the increment for the elements of
                      X. INCX must not be zero.

       Author:
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Date:
           December 2016

       Further Details:


             Level 2 Blas routine.
             The vector and matrix arguments are not referenced when N = 0, or M = 0

             -- Written on 22-October-1986.
                Jack Dongarra, Argonne National Lab.
                Jeremy Du Croz, Nag Central Office.
                Sven Hammarling, Nag Central Office.
                Richard Hanson, Sandia National Labs.

       Definition at line 144 of file ctpmv.f.

   subroutine ctpsv (character UPLO, character TRANS, character DIAG, integer
       N, complex, dimension(*) AP, complex, dimension(*) X, integer INCX)
       CTPSV

       Purpose:


            CTPSV  solves one of the systems of equations

               A*x = b,   or   A**T*x = b,   or   A**H*x = b,

            where b and x are n element vectors and A is an n by n unit, or
            non-unit, upper or lower triangular matrix, supplied in packed form.

            No test for singularity or near-singularity is included in this
            routine. Such tests must be performed before calling this routine.

       Parameters:
           UPLO

                     UPLO is CHARACTER*1
                      On entry, UPLO specifies whether the matrix is an upper or
                      lower triangular matrix as follows:

                         UPLO = 'U' or 'u'   A is an upper triangular matrix.

                         UPLO = 'L' or 'l'   A is a lower triangular matrix.

           TRANS

                     TRANS is CHARACTER*1
                      On entry, TRANS specifies the equations to be solved as
                      follows:

                         TRANS = 'N' or 'n'   A*x = b.

                         TRANS = 'T' or 't'   A**T*x = b.

                         TRANS = 'C' or 'c'   A**H*x = b.

           DIAG

                     DIAG is CHARACTER*1
                      On entry, DIAG specifies whether or not A is unit
                      triangular as follows:

                         DIAG = 'U' or 'u'   A is assumed to be unit triangular.

                         DIAG = 'N' or 'n'   A is not assumed to be unit
                                             triangular.

           N

                     N is INTEGER
                      On entry, N specifies the order of the matrix A.
                      N must be at least zero.

           AP

                     AP is COMPLEX array, dimension at least
                      ( ( n*( n + 1 ) )/2 ).
                      Before entry with  UPLO = 'U' or 'u', the array AP must
                      contain the upper triangular matrix packed sequentially,
                      column by column, so that AP( 1 ) contains a( 1, 1 ),
                      AP( 2 ) and AP( 3 ) contain a( 1, 2 ) and a( 2, 2 )
                      respectively, and so on.
                      Before entry with UPLO = 'L' or 'l', the array AP must
                      contain the lower triangular matrix packed sequentially,
                      column by column, so that AP( 1 ) contains a( 1, 1 ),
                      AP( 2 ) and AP( 3 ) contain a( 2, 1 ) and a( 3, 1 )
                      respectively, and so on.
                      Note that when  DIAG = 'U' or 'u', the diagonal elements of
                      A are not referenced, but are assumed to be unity.

           X

                     X is COMPLEX array, dimension at least
                      ( 1 + ( n - 1 )*abs( INCX ) ).
                      Before entry, the incremented array X must contain the n
                      element right-hand side vector b. On exit, X is overwritten
                      with the solution vector x.

           INCX

                     INCX is INTEGER
                      On entry, INCX specifies the increment for the elements of
                      X. INCX must not be zero.

       Author:
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Date:
           December 2016

       Further Details:


             Level 2 Blas routine.

             -- Written on 22-October-1986.
                Jack Dongarra, Argonne National Lab.
                Jeremy Du Croz, Nag Central Office.
                Sven Hammarling, Nag Central Office.
                Richard Hanson, Sandia National Labs.

       Definition at line 146 of file ctpsv.f.

   subroutine ctrmv (character UPLO, character TRANS, character DIAG, integer
       N, complex, dimension(lda,*) A, integer LDA, complex, dimension(*) X,
       integer INCX)
       CTRMV

       Purpose:


            CTRMV  performs one of the matrix-vector operations

               x := A*x,   or   x := A**T*x,   or   x := A**H*x,

            where x is an n element vector and  A is an n by n unit, or non-unit,
            upper or lower triangular matrix.

       Parameters:
           UPLO

                     UPLO is CHARACTER*1
                      On entry, UPLO specifies whether the matrix is an upper or
                      lower triangular matrix as follows:

                         UPLO = 'U' or 'u'   A is an upper triangular matrix.

                         UPLO = 'L' or 'l'   A is a lower triangular matrix.

           TRANS

                     TRANS is CHARACTER*1
                      On entry, TRANS specifies the operation to be performed as
                      follows:

                         TRANS = 'N' or 'n'   x := A*x.

                         TRANS = 'T' or 't'   x := A**T*x.

                         TRANS = 'C' or 'c'   x := A**H*x.

           DIAG

                     DIAG is CHARACTER*1
                      On entry, DIAG specifies whether or not A is unit
                      triangular as follows:

                         DIAG = 'U' or 'u'   A is assumed to be unit triangular.

                         DIAG = 'N' or 'n'   A is not assumed to be unit
                                             triangular.

           N

                     N is INTEGER
                      On entry, N specifies the order of the matrix A.
                      N must be at least zero.

           A

                     A is COMPLEX array, dimension ( LDA, N ).
                      Before entry with  UPLO = 'U' or 'u', the leading n by n
                      upper triangular part of the array A must contain the upper
                      triangular matrix and the strictly lower triangular part of
                      A is not referenced.
                      Before entry with UPLO = 'L' or 'l', the leading n by n
                      lower triangular part of the array A must contain the lower
                      triangular matrix and the strictly upper triangular part of
                      A is not referenced.
                      Note that when  DIAG = 'U' or 'u', the diagonal elements of
                      A are not referenced either, but are assumed to be unity.

           LDA

                     LDA is INTEGER
                      On entry, LDA specifies the first dimension of A as declared
                      in the calling (sub) program. LDA must be at least
                      max( 1, n ).

           X

                     X is COMPLEX array, dimension at least
                      ( 1 + ( n - 1 )*abs( INCX ) ).
                      Before entry, the incremented array X must contain the n
                      element vector x. On exit, X is overwritten with the
                      transformed vector x.

           INCX

                     INCX is INTEGER
                      On entry, INCX specifies the increment for the elements of
                      X. INCX must not be zero.

       Author:
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Date:
           December 2016

       Further Details:


             Level 2 Blas routine.
             The vector and matrix arguments are not referenced when N = 0, or M = 0

             -- Written on 22-October-1986.
                Jack Dongarra, Argonne National Lab.
                Jeremy Du Croz, Nag Central Office.
                Sven Hammarling, Nag Central Office.
                Richard Hanson, Sandia National Labs.

       Definition at line 149 of file ctrmv.f.

   subroutine ctrsv (character UPLO, character TRANS, character DIAG, integer
       N, complex, dimension(lda,*) A, integer LDA, complex, dimension(*) X,
       integer INCX)
       CTRSV

       Purpose:


            CTRSV  solves one of the systems of equations

               A*x = b,   or   A**T*x = b,   or   A**H*x = b,

            where b and x are n element vectors and A is an n by n unit, or
            non-unit, upper or lower triangular matrix.

            No test for singularity or near-singularity is included in this
            routine. Such tests must be performed before calling this routine.

       Parameters:
           UPLO

                     UPLO is CHARACTER*1
                      On entry, UPLO specifies whether the matrix is an upper or
                      lower triangular matrix as follows:

                         UPLO = 'U' or 'u'   A is an upper triangular matrix.

                         UPLO = 'L' or 'l'   A is a lower triangular matrix.

           TRANS

                     TRANS is CHARACTER*1
                      On entry, TRANS specifies the equations to be solved as
                      follows:

                         TRANS = 'N' or 'n'   A*x = b.

                         TRANS = 'T' or 't'   A**T*x = b.

                         TRANS = 'C' or 'c'   A**H*x = b.

           DIAG

                     DIAG is CHARACTER*1
                      On entry, DIAG specifies whether or not A is unit
                      triangular as follows:

                         DIAG = 'U' or 'u'   A is assumed to be unit triangular.

                         DIAG = 'N' or 'n'   A is not assumed to be unit
                                             triangular.

           N

                     N is INTEGER
                      On entry, N specifies the order of the matrix A.
                      N must be at least zero.

           A

                     A is COMPLEX array, dimension ( LDA, N )
                      Before entry with  UPLO = 'U' or 'u', the leading n by n
                      upper triangular part of the array A must contain the upper
                      triangular matrix and the strictly lower triangular part of
                      A is not referenced.
                      Before entry with UPLO = 'L' or 'l', the leading n by n
                      lower triangular part of the array A must contain the lower
                      triangular matrix and the strictly upper triangular part of
                      A is not referenced.
                      Note that when  DIAG = 'U' or 'u', the diagonal elements of
                      A are not referenced either, but are assumed to be unity.

           LDA

                     LDA is INTEGER
                      On entry, LDA specifies the first dimension of A as declared
                      in the calling (sub) program. LDA must be at least
                      max( 1, n ).

           X

                     X is COMPLEX array, dimension at least
                      ( 1 + ( n - 1 )*abs( INCX ) ).
                      Before entry, the incremented array X must contain the n
                      element right-hand side vector b. On exit, X is overwritten
                      with the solution vector x.

           INCX

                     INCX is INTEGER
                      On entry, INCX specifies the increment for the elements of
                      X. INCX must not be zero.

       Author:
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Date:
           December 2016

       Further Details:


             Level 2 Blas routine.

             -- Written on 22-October-1986.
                Jack Dongarra, Argonne National Lab.
                Jeremy Du Croz, Nag Central Office.
                Sven Hammarling, Nag Central Office.
                Richard Hanson, Sandia National Labs.

       Definition at line 151 of file ctrsv.f.

Author
       Generated automatically by Doxygen for LAPACK from the source code.




Version 3.8.0                   Wed May 2 2018          complex_blas_level2(3)