subroutinecgbmv(characterTRANS,integerM,integerN,integerKL,integerKU,complexALPHA,complex,dimension(lda,*)A,integerLDA,complex,dimension(*)X,integerINCX,complexBETA,complex,dimension(*)Y,integerINCY)CGBMVPurpose:
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.
ParametersTRANS
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.
FurtherDetails:
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.
subroutinecgemv(characterTRANS,integerM,integerN,complexALPHA,complex,dimension(lda,*)A,integerLDA,complex,dimension(*)X,integerINCX,complexBETA,complex,dimension(*)Y,integerINCY)CGEMVPurpose:
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.
ParametersTRANS
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.
FurtherDetails:
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.
subroutinecgerc(integerM,integerN,complexALPHA,complex,dimension(*)X,integerINCX,complex,dimension(*)Y,integerINCY,complex,dimension(lda,*)A,integerLDA)CGERCPurpose:
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.
ParametersM
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.
FurtherDetails:
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.
subroutinecgeru(integerM,integerN,complexALPHA,complex,dimension(*)X,integerINCX,complex,dimension(*)Y,integerINCY,complex,dimension(lda,*)A,integerLDA)CGERUPurpose:
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.
ParametersM
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.
FurtherDetails:
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.
subroutinechbmv(characterUPLO,integerN,integerK,complexALPHA,complex,dimension(lda,*)A,integerLDA,complex,dimension(*)X,integerINCX,complexBETA,complex,dimension(*)Y,integerINCY)CHBMVPurpose:
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.
ParametersUPLO
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.
FurtherDetails:
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.
subroutinechemv(characterUPLO,integerN,complexALPHA,complex,dimension(lda,*)A,integerLDA,complex,dimension(*)X,integerINCX,complexBETA,complex,dimension(*)Y,integerINCY)CHEMVPurpose:
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.
ParametersUPLO
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.
FurtherDetails:
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.
subroutinecher(characterUPLO,integerN,realALPHA,complex,dimension(*)X,integerINCX,complex,dimension(lda,*)A,integerLDA)CHERPurpose:
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.
ParametersUPLO
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.
FurtherDetails:
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.
subroutinecher2(characterUPLO,integerN,complexALPHA,complex,dimension(*)X,integerINCX,complex,dimension(*)Y,integerINCY,complex,dimension(lda,*)A,integerLDA)CHER2Purpose:
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.
ParametersUPLO
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.
FurtherDetails:
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.
subroutinechpmv(characterUPLO,integerN,complexALPHA,complex,dimension(*)AP,complex,dimension(*)X,integerINCX,complexBETA,complex,dimension(*)Y,integerINCY)CHPMVPurpose:
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.
ParametersUPLO
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.
FurtherDetails:
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.
subroutinechpr(characterUPLO,integerN,realALPHA,complex,dimension(*)X,integerINCX,complex,dimension(*)AP)CHPRPurpose:
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.
ParametersUPLO
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.
FurtherDetails:
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.
subroutinechpr2(characterUPLO,integerN,complexALPHA,complex,dimension(*)X,integerINCX,complex,dimension(*)Y,integerINCY,complex,dimension(*)AP)CHPR2Purpose:
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.
ParametersUPLO
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.
FurtherDetails:
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.
subroutinectbmv(characterUPLO,characterTRANS,characterDIAG,integerN,integerK,complex,dimension(lda,*)A,integerLDA,complex,dimension(*)X,integerINCX)CTBMVPurpose:
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.
ParametersUPLO
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.
FurtherDetails:
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.
subroutinectbsv(characterUPLO,characterTRANS,characterDIAG,integerN,integerK,complex,dimension(lda,*)A,integerLDA,complex,dimension(*)X,integerINCX)CTBSVPurpose:
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.
ParametersUPLO
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.
FurtherDetails:
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.
subroutinectpmv(characterUPLO,characterTRANS,characterDIAG,integerN,complex,dimension(*)AP,complex,dimension(*)X,integerINCX)CTPMVPurpose:
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.
ParametersUPLO
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.
FurtherDetails:
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.
subroutinectpsv(characterUPLO,characterTRANS,characterDIAG,integerN,complex,dimension(*)AP,complex,dimension(*)X,integerINCX)CTPSVPurpose:
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.
ParametersUPLO
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.
FurtherDetails:
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.
subroutinectrmv(characterUPLO,characterTRANS,characterDIAG,integerN,complex,dimension(lda,*)A,integerLDA,complex,dimension(*)X,integerINCX)CTRMVPurpose:
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.
ParametersUPLO
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.
FurtherDetails:
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.
subroutinectrsv(characterUPLO,characterTRANS,characterDIAG,integerN,complex,dimension(lda,*)A,integerLDA,complex,dimension(*)X,integerINCX)CTRSVPurpose:
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.
ParametersUPLO
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.
FurtherDetails:
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.
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