LAPACK  3.4.2
LAPACK: Linear Algebra PACKage
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Functions/Subroutines
double
Level2
Collaboration diagram for double:

Functions/Subroutines

subroutine dgbmv (TRANS, M, N, KL, KU, ALPHA, A, LDA, X, INCX, BETA, Y, INCY)
 DGBMV
subroutine dgemv (TRANS, M, N, ALPHA, A, LDA, X, INCX, BETA, Y, INCY)
 DGEMV
subroutine dger (M, N, ALPHA, X, INCX, Y, INCY, A, LDA)
 DGER
subroutine dsbmv (UPLO, N, K, ALPHA, A, LDA, X, INCX, BETA, Y, INCY)
 DSBMV
subroutine dspmv (UPLO, N, ALPHA, AP, X, INCX, BETA, Y, INCY)
 DSPMV
subroutine dspr (UPLO, N, ALPHA, X, INCX, AP)
 DSPR
subroutine dspr2 (UPLO, N, ALPHA, X, INCX, Y, INCY, AP)
 DSPR2
subroutine dsymv (UPLO, N, ALPHA, A, LDA, X, INCX, BETA, Y, INCY)
 DSYMV
subroutine dsyr (UPLO, N, ALPHA, X, INCX, A, LDA)
 DSYR
subroutine dsyr2 (UPLO, N, ALPHA, X, INCX, Y, INCY, A, LDA)
 DSYR2
subroutine dtbmv (UPLO, TRANS, DIAG, N, K, A, LDA, X, INCX)
 DTBMV
subroutine dtbsv (UPLO, TRANS, DIAG, N, K, A, LDA, X, INCX)
 DTBSV
subroutine dtpmv (UPLO, TRANS, DIAG, N, AP, X, INCX)
 DTPMV
subroutine dtpsv (UPLO, TRANS, DIAG, N, AP, X, INCX)
 DTPSV
subroutine dtrmv (UPLO, TRANS, DIAG, N, A, LDA, X, INCX)
 DTRMV

Detailed Description

This is the group of double LEVEL 2 BLAS routines.

Function/Subroutine Documentation

subroutine dgbmv ( character  TRANS,
integer  M,
integer  N,
integer  KL,
integer  KU,
double precision  ALPHA,
double precision, dimension(lda,*)  A,
integer  LDA,
double precision, dimension(*)  X,
integer  INCX,
double precision  BETA,
double precision, dimension(*)  Y,
integer  INCY 
)

DGBMV

Purpose: 
 DGBMV  performs one of the matrix-vector operations

    y := alpha*A*x + beta*y,   or   y := alpha*A**T*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:
[in]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**T*x + beta*y.
[in]M
          M is INTEGER
           On entry, M specifies the number of rows of the matrix A.
           M must be at least zero.
[in]N
          N is INTEGER
           On entry, N specifies the number of columns of the matrix A.
           N must be at least zero.
[in]KL
          KL is INTEGER
           On entry, KL specifies the number of sub-diagonals of the
           matrix A. KL must satisfy  0 .le. KL.
[in]KU
          KU is INTEGER
           On entry, KU specifies the number of super-diagonals of the
           matrix A. KU must satisfy  0 .le. KU.
[in]ALPHA
          ALPHA is DOUBLE PRECISION.
           On entry, ALPHA specifies the scalar alpha.
[in]A
          A is DOUBLE PRECISION array of 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
[in]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 ).
[in]X
          X is DOUBLE PRECISION array of 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.
[in]INCX
          INCX is INTEGER
           On entry, INCX specifies the increment for the elements of
           X. INCX must not be zero.
[in]BETA
          BETA is DOUBLE PRECISION.
           On entry, BETA specifies the scalar beta. When BETA is
           supplied as zero then Y need not be set on input.
[in,out]Y
          Y is DOUBLE PRECISION array of 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.
[in]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:
November 2011
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 186 of file dgbmv.f.

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subroutine dgemv ( character  TRANS,
integer  M,
integer  N,
double precision  ALPHA,
double precision, dimension(lda,*)  A,
integer  LDA,
double precision, dimension(*)  X,
integer  INCX,
double precision  BETA,
double precision, dimension(*)  Y,
integer  INCY 
)

DGEMV

Purpose: 
 DGEMV  performs one of the matrix-vector operations

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

 where alpha and beta are scalars, x and y are vectors and A is an
 m by n matrix.
Parameters:
[in]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**T*x + beta*y.
[in]M
          M is INTEGER
           On entry, M specifies the number of rows of the matrix A.
           M must be at least zero.
[in]N
          N is INTEGER
           On entry, N specifies the number of columns of the matrix A.
           N must be at least zero.
[in]ALPHA
          ALPHA is DOUBLE PRECISION.
           On entry, ALPHA specifies the scalar alpha.
[in]A
          A is DOUBLE PRECISION array of DIMENSION ( LDA, n ).
           Before entry, the leading m by n part of the array A must
           contain the matrix of coefficients.
[in]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 ).
[in]X
          X is DOUBLE PRECISION array of 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.
[in]INCX
          INCX is INTEGER
           On entry, INCX specifies the increment for the elements of
           X. INCX must not be zero.
[in]BETA
          BETA is DOUBLE PRECISION.
           On entry, BETA specifies the scalar beta. When BETA is
           supplied as zero then Y need not be set on input.
[in,out]Y
          Y is DOUBLE PRECISION array of 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.
[in]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:
November 2011
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 157 of file dgemv.f.

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subroutine dger ( integer  M,
integer  N,
double precision  ALPHA,
double precision, dimension(*)  X,
integer  INCX,
double precision, dimension(*)  Y,
integer  INCY,
double precision, dimension(lda,*)  A,
integer  LDA 
)

DGER

Purpose: 
 DGER   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:
[in]M
          M is INTEGER
           On entry, M specifies the number of rows of the matrix A.
           M must be at least zero.
[in]N
          N is INTEGER
           On entry, N specifies the number of columns of the matrix A.
           N must be at least zero.
[in]ALPHA
          ALPHA is DOUBLE PRECISION.
           On entry, ALPHA specifies the scalar alpha.
[in]X
          X is DOUBLE PRECISION array of dimension at least
           ( 1 + ( m - 1 )*abs( INCX ) ).
           Before entry, the incremented array X must contain the m
           element vector x.
[in]INCX
          INCX is INTEGER
           On entry, INCX specifies the increment for the elements of
           X. INCX must not be zero.
[in]Y
          Y is DOUBLE PRECISION array of dimension at least
           ( 1 + ( n - 1 )*abs( INCY ) ).
           Before entry, the incremented array Y must contain the n
           element vector y.
[in]INCY
          INCY is INTEGER
           On entry, INCY specifies the increment for the elements of
           Y. INCY must not be zero.
[in,out]A
          A is DOUBLE PRECISION array of 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.
[in]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:
November 2011
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 131 of file dger.f.

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subroutine dsbmv ( character  UPLO,
integer  N,
integer  K,
double precision  ALPHA,
double precision, dimension(lda,*)  A,
integer  LDA,
double precision, dimension(*)  X,
integer  INCX,
double precision  BETA,
double precision, dimension(*)  Y,
integer  INCY 
)

DSBMV

Purpose: 
 DSBMV  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 symmetric band matrix, with k super-diagonals.
Parameters:
[in]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.
[in]N
          N is INTEGER
           On entry, N specifies the order of the matrix A.
           N must be at least zero.
[in]K
          K is INTEGER
           On entry, K specifies the number of super-diagonals of the
           matrix A. K must satisfy  0 .le. K.
[in]ALPHA
          ALPHA is DOUBLE PRECISION.
           On entry, ALPHA specifies the scalar alpha.
[in]A
          A is DOUBLE PRECISION array of 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 symmetric 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 symmetric 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 symmetric 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 symmetric 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
[in]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 ).
[in]X
          X is DOUBLE PRECISION array of DIMENSION at least
           ( 1 + ( n - 1 )*abs( INCX ) ).
           Before entry, the incremented array X must contain the
           vector x.
[in]INCX
          INCX is INTEGER
           On entry, INCX specifies the increment for the elements of
           X. INCX must not be zero.
[in]BETA
          BETA is DOUBLE PRECISION.
           On entry, BETA specifies the scalar beta.
[in,out]Y
          Y is DOUBLE PRECISION array of 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.
[in]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:
November 2011
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 185 of file dsbmv.f.

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subroutine dspmv ( character  UPLO,
integer  N,
double precision  ALPHA,
double precision, dimension(*)  AP,
double precision, dimension(*)  X,
integer  INCX,
double precision  BETA,
double precision, dimension(*)  Y,
integer  INCY 
)

DSPMV

Purpose: 
 DSPMV  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 symmetric matrix, supplied in packed form.
Parameters:
[in]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.
[in]N
          N is INTEGER
           On entry, N specifies the order of the matrix A.
           N must be at least zero.
[in]ALPHA
          ALPHA is DOUBLE PRECISION.
           On entry, ALPHA specifies the scalar alpha.
[in]AP
          AP is DOUBLE PRECISION array of 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 symmetric 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 symmetric 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.
[in]X
          X is DOUBLE PRECISION array of dimension at least
           ( 1 + ( n - 1 )*abs( INCX ) ).
           Before entry, the incremented array X must contain the n
           element vector x.
[in]INCX
          INCX is INTEGER
           On entry, INCX specifies the increment for the elements of
           X. INCX must not be zero.
[in]BETA
          BETA is DOUBLE PRECISION.
           On entry, BETA specifies the scalar beta. When BETA is
           supplied as zero then Y need not be set on input.
[in,out]Y
          Y is DOUBLE PRECISION array of 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.
[in]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:
November 2011
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 148 of file dspmv.f.

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subroutine dspr ( character  UPLO,
integer  N,
double precision  ALPHA,
double precision, dimension(*)  X,
integer  INCX,
double precision, dimension(*)  AP 
)

DSPR

Purpose: 
 DSPR    performs the symmetric rank 1 operation

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

 where alpha is a real scalar, x is an n element vector and A is an
 n by n symmetric matrix, supplied in packed form.
Parameters:
[in]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.
[in]N
          N is INTEGER
           On entry, N specifies the order of the matrix A.
           N must be at least zero.
[in]ALPHA
          ALPHA is DOUBLE PRECISION.
           On entry, ALPHA specifies the scalar alpha.
[in]X
          X is DOUBLE PRECISION array of dimension at least
           ( 1 + ( n - 1 )*abs( INCX ) ).
           Before entry, the incremented array X must contain the n
           element vector x.
[in]INCX
          INCX is INTEGER
           On entry, INCX specifies the increment for the elements of
           X. INCX must not be zero.
[in,out]AP
          AP is DOUBLE PRECISION array of 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 symmetric 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 symmetric 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.
Author:
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
November 2011
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 128 of file dspr.f.

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subroutine dspr2 ( character  UPLO,
integer  N,
double precision  ALPHA,
double precision, dimension(*)  X,
integer  INCX,
double precision, dimension(*)  Y,
integer  INCY,
double precision, dimension(*)  AP 
)

DSPR2

Purpose: 
 DSPR2  performs the symmetric rank 2 operation

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

 where alpha is a scalar, x and y are n element vectors and A is an
 n by n symmetric matrix, supplied in packed form.
Parameters:
[in]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.
[in]N
          N is INTEGER
           On entry, N specifies the order of the matrix A.
           N must be at least zero.
[in]ALPHA
          ALPHA is DOUBLE PRECISION.
           On entry, ALPHA specifies the scalar alpha.
[in]X
          X is DOUBLE PRECISION array of dimension at least
           ( 1 + ( n - 1 )*abs( INCX ) ).
           Before entry, the incremented array X must contain the n
           element vector x.
[in]INCX
          INCX is INTEGER
           On entry, INCX specifies the increment for the elements of
           X. INCX must not be zero.
[in]Y
          Y is DOUBLE PRECISION array of dimension at least
           ( 1 + ( n - 1 )*abs( INCY ) ).
           Before entry, the incremented array Y must contain the n
           element vector y.
[in]INCY
          INCY is INTEGER
           On entry, INCY specifies the increment for the elements of
           Y. INCY must not be zero.
[in,out]AP
          AP is DOUBLE PRECISION array of 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 symmetric 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 symmetric 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.
Author:
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
November 2011
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 143 of file dspr2.f.

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subroutine dsymv ( character  UPLO,
integer  N,
double precision  ALPHA,
double precision, dimension(lda,*)  A,
integer  LDA,
double precision, dimension(*)  X,
integer  INCX,
double precision  BETA,
double precision, dimension(*)  Y,
integer  INCY 
)

DSYMV

Purpose: 
 DSYMV  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 symmetric matrix.
Parameters:
[in]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.
[in]N
          N is INTEGER
           On entry, N specifies the order of the matrix A.
           N must be at least zero.
[in]ALPHA
          ALPHA is DOUBLE PRECISION.
           On entry, ALPHA specifies the scalar alpha.
[in]A
          A is DOUBLE PRECISION array of 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 symmetric 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 symmetric matrix and the strictly
           upper triangular part of A is not referenced.
[in]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 ).
[in]X
          X is DOUBLE PRECISION array of dimension at least
           ( 1 + ( n - 1 )*abs( INCX ) ).
           Before entry, the incremented array X must contain the n
           element vector x.
[in]INCX
          INCX is INTEGER
           On entry, INCX specifies the increment for the elements of
           X. INCX must not be zero.
[in]BETA
          BETA is DOUBLE PRECISION.
           On entry, BETA specifies the scalar beta. When BETA is
           supplied as zero then Y need not be set on input.
[in,out]Y
          Y is DOUBLE PRECISION array of 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.
[in]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:
November 2011
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 153 of file dsymv.f.

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subroutine dsyr ( character  UPLO,
integer  N,
double precision  ALPHA,
double precision, dimension(*)  X,
integer  INCX,
double precision, dimension(lda,*)  A,
integer  LDA 
)

DSYR

Purpose: 
 DSYR   performs the symmetric rank 1 operation

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

 where alpha is a real scalar, x is an n element vector and A is an
 n by n symmetric matrix.
Parameters:
[in]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.
[in]N
          N is INTEGER
           On entry, N specifies the order of the matrix A.
           N must be at least zero.
[in]ALPHA
          ALPHA is DOUBLE PRECISION.
           On entry, ALPHA specifies the scalar alpha.
[in]X
          X is DOUBLE PRECISION array of dimension at least
           ( 1 + ( n - 1 )*abs( INCX ) ).
           Before entry, the incremented array X must contain the n
           element vector x.
[in]INCX
          INCX is INTEGER
           On entry, INCX specifies the increment for the elements of
           X. INCX must not be zero.
[in,out]A
          A is DOUBLE PRECISION array of 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 symmetric 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 symmetric 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.
[in]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:
November 2011
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 133 of file dsyr.f.

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subroutine dsyr2 ( character  UPLO,
integer  N,
double precision  ALPHA,
double precision, dimension(*)  X,
integer  INCX,
double precision, dimension(*)  Y,
integer  INCY,
double precision, dimension(lda,*)  A,
integer  LDA 
)

DSYR2

Purpose: 
 DSYR2  performs the symmetric rank 2 operation

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

 where alpha is a scalar, x and y are n element vectors and A is an n
 by n symmetric matrix.
Parameters:
[in]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.
[in]N
          N is INTEGER
           On entry, N specifies the order of the matrix A.
           N must be at least zero.
[in]ALPHA
          ALPHA is DOUBLE PRECISION.
           On entry, ALPHA specifies the scalar alpha.
[in]X
          X is DOUBLE PRECISION array of dimension at least
           ( 1 + ( n - 1 )*abs( INCX ) ).
           Before entry, the incremented array X must contain the n
           element vector x.
[in]INCX
          INCX is INTEGER
           On entry, INCX specifies the increment for the elements of
           X. INCX must not be zero.
[in]Y
          Y is DOUBLE PRECISION array of dimension at least
           ( 1 + ( n - 1 )*abs( INCY ) ).
           Before entry, the incremented array Y must contain the n
           element vector y.
[in]INCY
          INCY is INTEGER
           On entry, INCY specifies the increment for the elements of
           Y. INCY must not be zero.
[in,out]A
          A is DOUBLE PRECISION array of 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 symmetric 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 symmetric 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.
[in]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:
November 2011
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 148 of file dsyr2.f.

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subroutine dtbmv ( character  UPLO,
character  TRANS,
character  DIAG,
integer  N,
integer  K,
double precision, dimension(lda,*)  A,
integer  LDA,
double precision, dimension(*)  X,
integer  INCX 
)

DTBMV

Purpose: 
 DTBMV  performs one of the matrix-vector operations

    x := A*x,   or   x := A**T*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:
[in]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.
[in]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**T*x.
[in]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.
[in]N
          N is INTEGER
           On entry, N specifies the order of the matrix A.
           N must be at least zero.
[in]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.
[in]A
          A is DOUBLE PRECISION array of 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.
[in]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 ).
[in,out]X
          X is DOUBLE PRECISION array of 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
           tranformed vector x.
[in]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:
November 2011
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 187 of file dtbmv.f.

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subroutine dtbsv ( character  UPLO,
character  TRANS,
character  DIAG,
integer  N,
integer  K,
double precision, dimension(lda,*)  A,
integer  LDA,
double precision, dimension(*)  X,
integer  INCX 
)

DTBSV

Purpose: 
 DTBSV  solves one of the systems of equations

    A*x = b,   or   A**T*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:
[in]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.
[in]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**T*x = b.
[in]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.
[in]N
          N is INTEGER
           On entry, N specifies the order of the matrix A.
           N must be at least zero.
[in]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.
[in]A
          A is DOUBLE PRECISION array of 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.
[in]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 ).
[in,out]X
          X is DOUBLE PRECISION array of 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.
[in]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:
November 2011
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 190 of file dtbsv.f.

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subroutine dtpmv ( character  UPLO,
character  TRANS,
character  DIAG,
integer  N,
double precision, dimension(*)  AP,
double precision, dimension(*)  X,
integer  INCX 
)

DTPMV

Purpose: 
 DTPMV  performs one of the matrix-vector operations

    x := A*x,   or   x := A**T*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:
[in]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.
[in]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**T*x.
[in]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.
[in]N
          N is INTEGER
           On entry, N specifies the order of the matrix A.
           N must be at least zero.
[in]AP
          AP is DOUBLE PRECISION array of 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.
[in,out]X
          X is DOUBLE PRECISION array of 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
           tranformed vector x.
[in]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:
November 2011
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 143 of file dtpmv.f.

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subroutine dtpsv ( character  UPLO,
character  TRANS,
character  DIAG,
integer  N,
double precision, dimension(*)  AP,
double precision, dimension(*)  X,
integer  INCX 
)

DTPSV

Purpose: 
 DTPSV  solves one of the systems of equations

    A*x = b,   or   A**T*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:
[in]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.
[in]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**T*x = b.
[in]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.
[in]N
          N is INTEGER
           On entry, N specifies the order of the matrix A.
           N must be at least zero.
[in]AP
          AP is DOUBLE PRECISION array of 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.
[in,out]X
          X is DOUBLE PRECISION array of 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.
[in]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:
November 2011
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 145 of file dtpsv.f.

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subroutine dtrmv ( character  UPLO,
character  TRANS,
character  DIAG,
integer  N,
double precision, dimension(lda,*)  A,
integer  LDA,
double precision, dimension(*)  X,
integer  INCX 
)

DTRMV

Purpose: 
 DTRMV  performs one of the matrix-vector operations

    x := A*x,   or   x := A**T*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:
[in]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.
[in]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**T*x.
[in]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.
[in]N
          N is INTEGER
           On entry, N specifies the order of the matrix A.
           N must be at least zero.
[in]A
          A is DOUBLE PRECISION array of 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.
[in]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 ).
[in,out]X
          X is DOUBLE PRECISION array of 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
           tranformed vector x.
[in]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:
November 2011
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 148 of file dtrmv.f.

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