.TH "complex16_blas_level2" 3 "Sun Nov 27 2022" "Version 3.11.0" "LAPACK" \" -*- nroff -*-
.ad l
.nh
.SH NAME
complex16_blas_level2 \- complex16
.SH SYNOPSIS
.br
.PP
.SS "Functions"
.in +1c
.ti -1c
.RI "subroutine \fBzgbmv\fP (TRANS, M, N, KL, KU, ALPHA, A, LDA, X, INCX, BETA, Y, INCY)"
.br
.RI "\fBZGBMV\fP "
.ti -1c
.RI "subroutine \fBzgemv\fP (TRANS, M, N, ALPHA, A, LDA, X, INCX, BETA, Y, INCY)"
.br
.RI "\fBZGEMV\fP "
.ti -1c
.RI "subroutine \fBzgerc\fP (M, N, ALPHA, X, INCX, Y, INCY, A, LDA)"
.br
.RI "\fBZGERC\fP "
.ti -1c
.RI "subroutine \fBzgeru\fP (M, N, ALPHA, X, INCX, Y, INCY, A, LDA)"
.br
.RI "\fBZGERU\fP "
.ti -1c
.RI "subroutine \fBzhbmv\fP (UPLO, N, K, ALPHA, A, LDA, X, INCX, BETA, Y, INCY)"
.br
.RI "\fBZHBMV\fP "
.ti -1c
.RI "subroutine \fBzhemv\fP (UPLO, N, ALPHA, A, LDA, X, INCX, BETA, Y, INCY)"
.br
.RI "\fBZHEMV\fP "
.ti -1c
.RI "subroutine \fBzher\fP (UPLO, N, ALPHA, X, INCX, A, LDA)"
.br
.RI "\fBZHER\fP "
.ti -1c
.RI "subroutine \fBzher2\fP (UPLO, N, ALPHA, X, INCX, Y, INCY, A, LDA)"
.br
.RI "\fBZHER2\fP "
.ti -1c
.RI "subroutine \fBzhpmv\fP (UPLO, N, ALPHA, AP, X, INCX, BETA, Y, INCY)"
.br
.RI "\fBZHPMV\fP "
.ti -1c
.RI "subroutine \fBzhpr\fP (UPLO, N, ALPHA, X, INCX, AP)"
.br
.RI "\fBZHPR\fP "
.ti -1c
.RI "subroutine \fBzhpr2\fP (UPLO, N, ALPHA, X, INCX, Y, INCY, AP)"
.br
.RI "\fBZHPR2\fP "
.ti -1c
.RI "subroutine \fBztbmv\fP (UPLO, TRANS, DIAG, N, K, A, LDA, X, INCX)"
.br
.RI "\fBZTBMV\fP "
.ti -1c
.RI "subroutine \fBztbsv\fP (UPLO, TRANS, DIAG, N, K, A, LDA, X, INCX)"
.br
.RI "\fBZTBSV\fP "
.ti -1c
.RI "subroutine \fBztpmv\fP (UPLO, TRANS, DIAG, N, AP, X, INCX)"
.br
.RI "\fBZTPMV\fP "
.ti -1c
.RI "subroutine \fBztpsv\fP (UPLO, TRANS, DIAG, N, AP, X, INCX)"
.br
.RI "\fBZTPSV\fP "
.ti -1c
.RI "subroutine \fBztrmv\fP (UPLO, TRANS, DIAG, N, A, LDA, X, INCX)"
.br
.RI "\fBZTRMV\fP "
.ti -1c
.RI "subroutine \fBztrsv\fP (UPLO, TRANS, DIAG, N, A, LDA, X, INCX)"
.br
.RI "\fBZTRSV\fP "
.in -1c
.SH "Detailed Description"
.PP
This is the group of complex16 LEVEL 2 BLAS routines\&.
.SH "Function Documentation"
.PP
.SS "subroutine zgbmv (character TRANS, integer M, integer N, integer KL, integer KU, complex*16 ALPHA, complex*16, dimension(lda,*) A, integer LDA, complex*16, dimension(*) X, integer INCX, complex*16 BETA, complex*16, dimension(*) Y, integer INCY)"
.PP
\fBZGBMV\fP
.PP
\fBPurpose:\fP
.RS 4
.PP
.nf
ZGBMV 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\&.
.fi
.PP
.RE
.PP
\fBParameters\fP
.RS 4
\fITRANS\fP
.PP
.nf
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\&.
.fi
.PP
.br
\fIM\fP
.PP
.nf
M is INTEGER
On entry, M specifies the number of rows of the matrix A\&.
M must be at least zero\&.
.fi
.PP
.br
\fIN\fP
.PP
.nf
N is INTEGER
On entry, N specifies the number of columns of the matrix A\&.
N must be at least zero\&.
.fi
.PP
.br
\fIKL\fP
.PP
.nf
KL is INTEGER
On entry, KL specifies the number of sub-diagonals of the
matrix A\&. KL must satisfy 0 \&.le\&. KL\&.
.fi
.PP
.br
\fIKU\fP
.PP
.nf
KU is INTEGER
On entry, KU specifies the number of super-diagonals of the
matrix A\&. KU must satisfy 0 \&.le\&. KU\&.
.fi
.PP
.br
\fIALPHA\fP
.PP
.nf
ALPHA is COMPLEX*16
On entry, ALPHA specifies the scalar alpha\&.
.fi
.PP
.br
\fIA\fP
.PP
.nf
A is COMPLEX*16 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
.fi
.PP
.br
\fILDA\fP
.PP
.nf
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 )\&.
.fi
.PP
.br
\fIX\fP
.PP
.nf
X is COMPLEX*16 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\&.
.fi
.PP
.br
\fIINCX\fP
.PP
.nf
INCX is INTEGER
On entry, INCX specifies the increment for the elements of
X\&. INCX must not be zero\&.
.fi
.PP
.br
\fIBETA\fP
.PP
.nf
BETA is COMPLEX*16
On entry, BETA specifies the scalar beta\&. When BETA is
supplied as zero then Y need not be set on input\&.
.fi
.PP
.br
\fIY\fP
.PP
.nf
Y is COMPLEX*16 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\&.
.fi
.PP
.br
\fIINCY\fP
.PP
.nf
INCY is INTEGER
On entry, INCY specifies the increment for the elements of
Y\&. INCY must not be zero\&.
.fi
.PP
.RE
.PP
\fBAuthor\fP
.RS 4
Univ\&. of Tennessee
.PP
Univ\&. of California Berkeley
.PP
Univ\&. of Colorado Denver
.PP
NAG Ltd\&.
.RE
.PP
\fBFurther Details:\fP
.RS 4
.PP
.nf
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\&.
.fi
.PP
.RE
.PP
.SS "subroutine zgemv (character TRANS, integer M, integer N, complex*16 ALPHA, complex*16, dimension(lda,*) A, integer LDA, complex*16, dimension(*) X, integer INCX, complex*16 BETA, complex*16, dimension(*) Y, integer INCY)"
.PP
\fBZGEMV\fP
.PP
\fBPurpose:\fP
.RS 4
.PP
.nf
ZGEMV 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\&.
.fi
.PP
.RE
.PP
\fBParameters\fP
.RS 4
\fITRANS\fP
.PP
.nf
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\&.
.fi
.PP
.br
\fIM\fP
.PP
.nf
M is INTEGER
On entry, M specifies the number of rows of the matrix A\&.
M must be at least zero\&.
.fi
.PP
.br
\fIN\fP
.PP
.nf
N is INTEGER
On entry, N specifies the number of columns of the matrix A\&.
N must be at least zero\&.
.fi
.PP
.br
\fIALPHA\fP
.PP
.nf
ALPHA is COMPLEX*16
On entry, ALPHA specifies the scalar alpha\&.
.fi
.PP
.br
\fIA\fP
.PP
.nf
A is COMPLEX*16 array, dimension ( LDA, N )
Before entry, the leading m by n part of the array A must
contain the matrix of coefficients\&.
.fi
.PP
.br
\fILDA\fP
.PP
.nf
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 )\&.
.fi
.PP
.br
\fIX\fP
.PP
.nf
X is COMPLEX*16 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\&.
.fi
.PP
.br
\fIINCX\fP
.PP
.nf
INCX is INTEGER
On entry, INCX specifies the increment for the elements of
X\&. INCX must not be zero\&.
.fi
.PP
.br
\fIBETA\fP
.PP
.nf
BETA is COMPLEX*16
On entry, BETA specifies the scalar beta\&. When BETA is
supplied as zero then Y need not be set on input\&.
.fi
.PP
.br
\fIY\fP
.PP
.nf
Y is COMPLEX*16 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\&.
.fi
.PP
.br
\fIINCY\fP
.PP
.nf
INCY is INTEGER
On entry, INCY specifies the increment for the elements of
Y\&. INCY must not be zero\&.
.fi
.PP
.RE
.PP
\fBAuthor\fP
.RS 4
Univ\&. of Tennessee
.PP
Univ\&. of California Berkeley
.PP
Univ\&. of Colorado Denver
.PP
NAG Ltd\&.
.RE
.PP
\fBFurther Details:\fP
.RS 4
.PP
.nf
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\&.
.fi
.PP
.RE
.PP
.SS "subroutine zgerc (integer M, integer N, complex*16 ALPHA, complex*16, dimension(*) X, integer INCX, complex*16, dimension(*) Y, integer INCY, complex*16, dimension(lda,*) A, integer LDA)"
.PP
\fBZGERC\fP
.PP
\fBPurpose:\fP
.RS 4
.PP
.nf
ZGERC 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\&.
.fi
.PP
.RE
.PP
\fBParameters\fP
.RS 4
\fIM\fP
.PP
.nf
M is INTEGER
On entry, M specifies the number of rows of the matrix A\&.
M must be at least zero\&.
.fi
.PP
.br
\fIN\fP
.PP
.nf
N is INTEGER
On entry, N specifies the number of columns of the matrix A\&.
N must be at least zero\&.
.fi
.PP
.br
\fIALPHA\fP
.PP
.nf
ALPHA is COMPLEX*16
On entry, ALPHA specifies the scalar alpha\&.
.fi
.PP
.br
\fIX\fP
.PP
.nf
X is COMPLEX*16 array, dimension at least
( 1 + ( m - 1 )*abs( INCX ) )\&.
Before entry, the incremented array X must contain the m
element vector x\&.
.fi
.PP
.br
\fIINCX\fP
.PP
.nf
INCX is INTEGER
On entry, INCX specifies the increment for the elements of
X\&. INCX must not be zero\&.
.fi
.PP
.br
\fIY\fP
.PP
.nf
Y is COMPLEX*16 array, dimension at least
( 1 + ( n - 1 )*abs( INCY ) )\&.
Before entry, the incremented array Y must contain the n
element vector y\&.
.fi
.PP
.br
\fIINCY\fP
.PP
.nf
INCY is INTEGER
On entry, INCY specifies the increment for the elements of
Y\&. INCY must not be zero\&.
.fi
.PP
.br
\fIA\fP
.PP
.nf
A is COMPLEX*16 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\&.
.fi
.PP
.br
\fILDA\fP
.PP
.nf
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 )\&.
.fi
.PP
.RE
.PP
\fBAuthor\fP
.RS 4
Univ\&. of Tennessee
.PP
Univ\&. of California Berkeley
.PP
Univ\&. of Colorado Denver
.PP
NAG Ltd\&.
.RE
.PP
\fBFurther Details:\fP
.RS 4
.PP
.nf
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\&.
.fi
.PP
.RE
.PP
.SS "subroutine zgeru (integer M, integer N, complex*16 ALPHA, complex*16, dimension(*) X, integer INCX, complex*16, dimension(*) Y, integer INCY, complex*16, dimension(lda,*) A, integer LDA)"
.PP
\fBZGERU\fP
.PP
\fBPurpose:\fP
.RS 4
.PP
.nf
ZGERU 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\&.
.fi
.PP
.RE
.PP
\fBParameters\fP
.RS 4
\fIM\fP
.PP
.nf
M is INTEGER
On entry, M specifies the number of rows of the matrix A\&.
M must be at least zero\&.
.fi
.PP
.br
\fIN\fP
.PP
.nf
N is INTEGER
On entry, N specifies the number of columns of the matrix A\&.
N must be at least zero\&.
.fi
.PP
.br
\fIALPHA\fP
.PP
.nf
ALPHA is COMPLEX*16
On entry, ALPHA specifies the scalar alpha\&.
.fi
.PP
.br
\fIX\fP
.PP
.nf
X is COMPLEX*16 array, dimension at least
( 1 + ( m - 1 )*abs( INCX ) )\&.
Before entry, the incremented array X must contain the m
element vector x\&.
.fi
.PP
.br
\fIINCX\fP
.PP
.nf
INCX is INTEGER
On entry, INCX specifies the increment for the elements of
X\&. INCX must not be zero\&.
.fi
.PP
.br
\fIY\fP
.PP
.nf
Y is COMPLEX*16 array, dimension at least
( 1 + ( n - 1 )*abs( INCY ) )\&.
Before entry, the incremented array Y must contain the n
element vector y\&.
.fi
.PP
.br
\fIINCY\fP
.PP
.nf
INCY is INTEGER
On entry, INCY specifies the increment for the elements of
Y\&. INCY must not be zero\&.
.fi
.PP
.br
\fIA\fP
.PP
.nf
A is COMPLEX*16 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\&.
.fi
.PP
.br
\fILDA\fP
.PP
.nf
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 )\&.
.fi
.PP
.RE
.PP
\fBAuthor\fP
.RS 4
Univ\&. of Tennessee
.PP
Univ\&. of California Berkeley
.PP
Univ\&. of Colorado Denver
.PP
NAG Ltd\&.
.RE
.PP
\fBFurther Details:\fP
.RS 4
.PP
.nf
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\&.
.fi
.PP
.RE
.PP
.SS "subroutine zhbmv (character UPLO, integer N, integer K, complex*16 ALPHA, complex*16, dimension(lda,*) A, integer LDA, complex*16, dimension(*) X, integer INCX, complex*16 BETA, complex*16, dimension(*) Y, integer INCY)"
.PP
\fBZHBMV\fP
.PP
\fBPurpose:\fP
.RS 4
.PP
.nf
ZHBMV 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\&.
.fi
.PP
.RE
.PP
\fBParameters\fP
.RS 4
\fIUPLO\fP
.PP
.nf
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\&.
.fi
.PP
.br
\fIN\fP
.PP
.nf
N is INTEGER
On entry, N specifies the order of the matrix A\&.
N must be at least zero\&.
.fi
.PP
.br
\fIK\fP
.PP
.nf
K is INTEGER
On entry, K specifies the number of super-diagonals of the
matrix A\&. K must satisfy 0 \&.le\&. K\&.
.fi
.PP
.br
\fIALPHA\fP
.PP
.nf
ALPHA is COMPLEX*16
On entry, ALPHA specifies the scalar alpha\&.
.fi
.PP
.br
\fIA\fP
.PP
.nf
A is COMPLEX*16 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\&.
.fi
.PP
.br
\fILDA\fP
.PP
.nf
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 )\&.
.fi
.PP
.br
\fIX\fP
.PP
.nf
X is COMPLEX*16 array, dimension at least
( 1 + ( n - 1 )*abs( INCX ) )\&.
Before entry, the incremented array X must contain the
vector x\&.
.fi
.PP
.br
\fIINCX\fP
.PP
.nf
INCX is INTEGER
On entry, INCX specifies the increment for the elements of
X\&. INCX must not be zero\&.
.fi
.PP
.br
\fIBETA\fP
.PP
.nf
BETA is COMPLEX*16
On entry, BETA specifies the scalar beta\&.
.fi
.PP
.br
\fIY\fP
.PP
.nf
Y is COMPLEX*16 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\&.
.fi
.PP
.br
\fIINCY\fP
.PP
.nf
INCY is INTEGER
On entry, INCY specifies the increment for the elements of
Y\&. INCY must not be zero\&.
.fi
.PP
.RE
.PP
\fBAuthor\fP
.RS 4
Univ\&. of Tennessee
.PP
Univ\&. of California Berkeley
.PP
Univ\&. of Colorado Denver
.PP
NAG Ltd\&.
.RE
.PP
\fBFurther Details:\fP
.RS 4
.PP
.nf
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\&.
.fi
.PP
.RE
.PP
.SS "subroutine zhemv (character UPLO, integer N, complex*16 ALPHA, complex*16, dimension(lda,*) A, integer LDA, complex*16, dimension(*) X, integer INCX, complex*16 BETA, complex*16, dimension(*) Y, integer INCY)"
.PP
\fBZHEMV\fP
.PP
\fBPurpose:\fP
.RS 4
.PP
.nf
ZHEMV 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\&.
.fi
.PP
.RE
.PP
\fBParameters\fP
.RS 4
\fIUPLO\fP
.PP
.nf
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\&.
.fi
.PP
.br
\fIN\fP
.PP
.nf
N is INTEGER
On entry, N specifies the order of the matrix A\&.
N must be at least zero\&.
.fi
.PP
.br
\fIALPHA\fP
.PP
.nf
ALPHA is COMPLEX*16
On entry, ALPHA specifies the scalar alpha\&.
.fi
.PP
.br
\fIA\fP
.PP
.nf
A is COMPLEX*16 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\&.
.fi
.PP
.br
\fILDA\fP
.PP
.nf
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 )\&.
.fi
.PP
.br
\fIX\fP
.PP
.nf
X is COMPLEX*16 array, dimension at least
( 1 + ( n - 1 )*abs( INCX ) )\&.
Before entry, the incremented array X must contain the n
element vector x\&.
.fi
.PP
.br
\fIINCX\fP
.PP
.nf
INCX is INTEGER
On entry, INCX specifies the increment for the elements of
X\&. INCX must not be zero\&.
.fi
.PP
.br
\fIBETA\fP
.PP
.nf
BETA is COMPLEX*16
On entry, BETA specifies the scalar beta\&. When BETA is
supplied as zero then Y need not be set on input\&.
.fi
.PP
.br
\fIY\fP
.PP
.nf
Y is COMPLEX*16 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\&.
.fi
.PP
.br
\fIINCY\fP
.PP
.nf
INCY is INTEGER
On entry, INCY specifies the increment for the elements of
Y\&. INCY must not be zero\&.
.fi
.PP
.RE
.PP
\fBAuthor\fP
.RS 4
Univ\&. of Tennessee
.PP
Univ\&. of California Berkeley
.PP
Univ\&. of Colorado Denver
.PP
NAG Ltd\&.
.RE
.PP
\fBFurther Details:\fP
.RS 4
.PP
.nf
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\&.
.fi
.PP
.RE
.PP
.SS "subroutine zher (character UPLO, integer N, double precision ALPHA, complex*16, dimension(*) X, integer INCX, complex*16, dimension(lda,*) A, integer LDA)"
.PP
\fBZHER\fP
.PP
\fBPurpose:\fP
.RS 4
.PP
.nf
ZHER 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\&.
.fi
.PP
.RE
.PP
\fBParameters\fP
.RS 4
\fIUPLO\fP
.PP
.nf
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\&.
.fi
.PP
.br
\fIN\fP
.PP
.nf
N is INTEGER
On entry, N specifies the order of the matrix A\&.
N must be at least zero\&.
.fi
.PP
.br
\fIALPHA\fP
.PP
.nf
ALPHA is DOUBLE PRECISION\&.
On entry, ALPHA specifies the scalar alpha\&.
.fi
.PP
.br
\fIX\fP
.PP
.nf
X is COMPLEX*16 array, dimension at least
( 1 + ( n - 1 )*abs( INCX ) )\&.
Before entry, the incremented array X must contain the n
element vector x\&.
.fi
.PP
.br
\fIINCX\fP
.PP
.nf
INCX is INTEGER
On entry, INCX specifies the increment for the elements of
X\&. INCX must not be zero\&.
.fi
.PP
.br
\fIA\fP
.PP
.nf
A is COMPLEX*16 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\&.
.fi
.PP
.br
\fILDA\fP
.PP
.nf
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 )\&.
.fi
.PP
.RE
.PP
\fBAuthor\fP
.RS 4
Univ\&. of Tennessee
.PP
Univ\&. of California Berkeley
.PP
Univ\&. of Colorado Denver
.PP
NAG Ltd\&.
.RE
.PP
\fBFurther Details:\fP
.RS 4
.PP
.nf
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\&.
.fi
.PP
.RE
.PP
.SS "subroutine zher2 (character UPLO, integer N, complex*16 ALPHA, complex*16, dimension(*) X, integer INCX, complex*16, dimension(*) Y, integer INCY, complex*16, dimension(lda,*) A, integer LDA)"
.PP
\fBZHER2\fP
.PP
\fBPurpose:\fP
.RS 4
.PP
.nf
ZHER2 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\&.
.fi
.PP
.RE
.PP
\fBParameters\fP
.RS 4
\fIUPLO\fP
.PP
.nf
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\&.
.fi
.PP
.br
\fIN\fP
.PP
.nf
N is INTEGER
On entry, N specifies the order of the matrix A\&.
N must be at least zero\&.
.fi
.PP
.br
\fIALPHA\fP
.PP
.nf
ALPHA is COMPLEX*16
On entry, ALPHA specifies the scalar alpha\&.
.fi
.PP
.br
\fIX\fP
.PP
.nf
X is COMPLEX*16 array, dimension at least
( 1 + ( n - 1 )*abs( INCX ) )\&.
Before entry, the incremented array X must contain the n
element vector x\&.
.fi
.PP
.br
\fIINCX\fP
.PP
.nf
INCX is INTEGER
On entry, INCX specifies the increment for the elements of
X\&. INCX must not be zero\&.
.fi
.PP
.br
\fIY\fP
.PP
.nf
Y is COMPLEX*16 array, dimension at least
( 1 + ( n - 1 )*abs( INCY ) )\&.
Before entry, the incremented array Y must contain the n
element vector y\&.
.fi
.PP
.br
\fIINCY\fP
.PP
.nf
INCY is INTEGER
On entry, INCY specifies the increment for the elements of
Y\&. INCY must not be zero\&.
.fi
.PP
.br
\fIA\fP
.PP
.nf
A is COMPLEX*16 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\&.
.fi
.PP
.br
\fILDA\fP
.PP
.nf
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 )\&.
.fi
.PP
.RE
.PP
\fBAuthor\fP
.RS 4
Univ\&. of Tennessee
.PP
Univ\&. of California Berkeley
.PP
Univ\&. of Colorado Denver
.PP
NAG Ltd\&.
.RE
.PP
\fBFurther Details:\fP
.RS 4
.PP
.nf
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\&.
.fi
.PP
.RE
.PP
.SS "subroutine zhpmv (character UPLO, integer N, complex*16 ALPHA, complex*16, dimension(*) AP, complex*16, dimension(*) X, integer INCX, complex*16 BETA, complex*16, dimension(*) Y, integer INCY)"
.PP
\fBZHPMV\fP
.PP
\fBPurpose:\fP
.RS 4
.PP
.nf
ZHPMV 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\&.
.fi
.PP
.RE
.PP
\fBParameters\fP
.RS 4
\fIUPLO\fP
.PP
.nf
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\&.
.fi
.PP
.br
\fIN\fP
.PP
.nf
N is INTEGER
On entry, N specifies the order of the matrix A\&.
N must be at least zero\&.
.fi
.PP
.br
\fIALPHA\fP
.PP
.nf
ALPHA is COMPLEX*16
On entry, ALPHA specifies the scalar alpha\&.
.fi
.PP
.br
\fIAP\fP
.PP
.nf
AP is COMPLEX*16 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\&.
.fi
.PP
.br
\fIX\fP
.PP
.nf
X is COMPLEX*16 array, dimension at least
( 1 + ( n - 1 )*abs( INCX ) )\&.
Before entry, the incremented array X must contain the n
element vector x\&.
.fi
.PP
.br
\fIINCX\fP
.PP
.nf
INCX is INTEGER
On entry, INCX specifies the increment for the elements of
X\&. INCX must not be zero\&.
.fi
.PP
.br
\fIBETA\fP
.PP
.nf
BETA is COMPLEX*16
On entry, BETA specifies the scalar beta\&. When BETA is
supplied as zero then Y need not be set on input\&.
.fi
.PP
.br
\fIY\fP
.PP
.nf
Y is COMPLEX*16 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\&.
.fi
.PP
.br
\fIINCY\fP
.PP
.nf
INCY is INTEGER
On entry, INCY specifies the increment for the elements of
Y\&. INCY must not be zero\&.
.fi
.PP
.RE
.PP
\fBAuthor\fP
.RS 4
Univ\&. of Tennessee
.PP
Univ\&. of California Berkeley
.PP
Univ\&. of Colorado Denver
.PP
NAG Ltd\&.
.RE
.PP
\fBFurther Details:\fP
.RS 4
.PP
.nf
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\&.
.fi
.PP
.RE
.PP
.SS "subroutine zhpr (character UPLO, integer N, double precision ALPHA, complex*16, dimension(*) X, integer INCX, complex*16, dimension(*) AP)"
.PP
\fBZHPR\fP
.PP
\fBPurpose:\fP
.RS 4
.PP
.nf
ZHPR 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\&.
.fi
.PP
.RE
.PP
\fBParameters\fP
.RS 4
\fIUPLO\fP
.PP
.nf
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\&.
.fi
.PP
.br
\fIN\fP
.PP
.nf
N is INTEGER
On entry, N specifies the order of the matrix A\&.
N must be at least zero\&.
.fi
.PP
.br
\fIALPHA\fP
.PP
.nf
ALPHA is DOUBLE PRECISION\&.
On entry, ALPHA specifies the scalar alpha\&.
.fi
.PP
.br
\fIX\fP
.PP
.nf
X is COMPLEX*16 array, dimension at least
( 1 + ( n - 1 )*abs( INCX ) )\&.
Before entry, the incremented array X must contain the n
element vector x\&.
.fi
.PP
.br
\fIINCX\fP
.PP
.nf
INCX is INTEGER
On entry, INCX specifies the increment for the elements of
X\&. INCX must not be zero\&.
.fi
.PP
.br
\fIAP\fP
.PP
.nf
AP is COMPLEX*16 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\&.
.fi
.PP
.RE
.PP
\fBAuthor\fP
.RS 4
Univ\&. of Tennessee
.PP
Univ\&. of California Berkeley
.PP
Univ\&. of Colorado Denver
.PP
NAG Ltd\&.
.RE
.PP
\fBFurther Details:\fP
.RS 4
.PP
.nf
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\&.
.fi
.PP
.RE
.PP
.SS "subroutine zhpr2 (character UPLO, integer N, complex*16 ALPHA, complex*16, dimension(*) X, integer INCX, complex*16, dimension(*) Y, integer INCY, complex*16, dimension(*) AP)"
.PP
\fBZHPR2\fP
.PP
\fBPurpose:\fP
.RS 4
.PP
.nf
ZHPR2 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\&.
.fi
.PP
.RE
.PP
\fBParameters\fP
.RS 4
\fIUPLO\fP
.PP
.nf
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\&.
.fi
.PP
.br
\fIN\fP
.PP
.nf
N is INTEGER
On entry, N specifies the order of the matrix A\&.
N must be at least zero\&.
.fi
.PP
.br
\fIALPHA\fP
.PP
.nf
ALPHA is COMPLEX*16
On entry, ALPHA specifies the scalar alpha\&.
.fi
.PP
.br
\fIX\fP
.PP
.nf
X is COMPLEX*16 array, dimension at least
( 1 + ( n - 1 )*abs( INCX ) )\&.
Before entry, the incremented array X must contain the n
element vector x\&.
.fi
.PP
.br
\fIINCX\fP
.PP
.nf
INCX is INTEGER
On entry, INCX specifies the increment for the elements of
X\&. INCX must not be zero\&.
.fi
.PP
.br
\fIY\fP
.PP
.nf
Y is COMPLEX*16 array, dimension at least
( 1 + ( n - 1 )*abs( INCY ) )\&.
Before entry, the incremented array Y must contain the n
element vector y\&.
.fi
.PP
.br
\fIINCY\fP
.PP
.nf
INCY is INTEGER
On entry, INCY specifies the increment for the elements of
Y\&. INCY must not be zero\&.
.fi
.PP
.br
\fIAP\fP
.PP
.nf
AP is COMPLEX*16 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\&.
.fi
.PP
.RE
.PP
\fBAuthor\fP
.RS 4
Univ\&. of Tennessee
.PP
Univ\&. of California Berkeley
.PP
Univ\&. of Colorado Denver
.PP
NAG Ltd\&.
.RE
.PP
\fBFurther Details:\fP
.RS 4
.PP
.nf
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\&.
.fi
.PP
.RE
.PP
.SS "subroutine ztbmv (character UPLO, character TRANS, character DIAG, integer N, integer K, complex*16, dimension(lda,*) A, integer LDA, complex*16, dimension(*) X, integer INCX)"
.PP
\fBZTBMV\fP
.PP
\fBPurpose:\fP
.RS 4
.PP
.nf
ZTBMV 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\&.
.fi
.PP
.RE
.PP
\fBParameters\fP
.RS 4
\fIUPLO\fP
.PP
.nf
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\&.
.fi
.PP
.br
\fITRANS\fP
.PP
.nf
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\&.
.fi
.PP
.br
\fIDIAG\fP
.PP
.nf
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\&.
.fi
.PP
.br
\fIN\fP
.PP
.nf
N is INTEGER
On entry, N specifies the order of the matrix A\&.
N must be at least zero\&.
.fi
.PP
.br
\fIK\fP
.PP
.nf
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\&.
.fi
.PP
.br
\fIA\fP
.PP
.nf
A is COMPLEX*16 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\&.
.fi
.PP
.br
\fILDA\fP
.PP
.nf
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 )\&.
.fi
.PP
.br
\fIX\fP
.PP
.nf
X is COMPLEX*16 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\&.
.fi
.PP
.br
\fIINCX\fP
.PP
.nf
INCX is INTEGER
On entry, INCX specifies the increment for the elements of
X\&. INCX must not be zero\&.
.fi
.PP
.RE
.PP
\fBAuthor\fP
.RS 4
Univ\&. of Tennessee
.PP
Univ\&. of California Berkeley
.PP
Univ\&. of Colorado Denver
.PP
NAG Ltd\&.
.RE
.PP
\fBFurther Details:\fP
.RS 4
.PP
.nf
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\&.
.fi
.PP
.RE
.PP
.SS "subroutine ztbsv (character UPLO, character TRANS, character DIAG, integer N, integer K, complex*16, dimension(lda,*) A, integer LDA, complex*16, dimension(*) X, integer INCX)"
.PP
\fBZTBSV\fP
.PP
\fBPurpose:\fP
.RS 4
.PP
.nf
ZTBSV 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\&.
.fi
.PP
.RE
.PP
\fBParameters\fP
.RS 4
\fIUPLO\fP
.PP
.nf
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\&.
.fi
.PP
.br
\fITRANS\fP
.PP
.nf
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\&.
.fi
.PP
.br
\fIDIAG\fP
.PP
.nf
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\&.
.fi
.PP
.br
\fIN\fP
.PP
.nf
N is INTEGER
On entry, N specifies the order of the matrix A\&.
N must be at least zero\&.
.fi
.PP
.br
\fIK\fP
.PP
.nf
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\&.
.fi
.PP
.br
\fIA\fP
.PP
.nf
A is COMPLEX*16 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\&.
.fi
.PP
.br
\fILDA\fP
.PP
.nf
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 )\&.
.fi
.PP
.br
\fIX\fP
.PP
.nf
X is COMPLEX*16 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\&.
.fi
.PP
.br
\fIINCX\fP
.PP
.nf
INCX is INTEGER
On entry, INCX specifies the increment for the elements of
X\&. INCX must not be zero\&.
.fi
.PP
.RE
.PP
\fBAuthor\fP
.RS 4
Univ\&. of Tennessee
.PP
Univ\&. of California Berkeley
.PP
Univ\&. of Colorado Denver
.PP
NAG Ltd\&.
.RE
.PP
\fBFurther Details:\fP
.RS 4
.PP
.nf
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\&.
.fi
.PP
.RE
.PP
.SS "subroutine ztpmv (character UPLO, character TRANS, character DIAG, integer N, complex*16, dimension(*) AP, complex*16, dimension(*) X, integer INCX)"
.PP
\fBZTPMV\fP
.PP
\fBPurpose:\fP
.RS 4
.PP
.nf
ZTPMV 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\&.
.fi
.PP
.RE
.PP
\fBParameters\fP
.RS 4
\fIUPLO\fP
.PP
.nf
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\&.
.fi
.PP
.br
\fITRANS\fP
.PP
.nf
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\&.
.fi
.PP
.br
\fIDIAG\fP
.PP
.nf
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\&.
.fi
.PP
.br
\fIN\fP
.PP
.nf
N is INTEGER
On entry, N specifies the order of the matrix A\&.
N must be at least zero\&.
.fi
.PP
.br
\fIAP\fP
.PP
.nf
AP is COMPLEX*16 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\&.
.fi
.PP
.br
\fIX\fP
.PP
.nf
X is COMPLEX*16 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\&.
.fi
.PP
.br
\fIINCX\fP
.PP
.nf
INCX is INTEGER
On entry, INCX specifies the increment for the elements of
X\&. INCX must not be zero\&.
.fi
.PP
.RE
.PP
\fBAuthor\fP
.RS 4
Univ\&. of Tennessee
.PP
Univ\&. of California Berkeley
.PP
Univ\&. of Colorado Denver
.PP
NAG Ltd\&.
.RE
.PP
\fBFurther Details:\fP
.RS 4
.PP
.nf
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\&.
.fi
.PP
.RE
.PP
.SS "subroutine ztpsv (character UPLO, character TRANS, character DIAG, integer N, complex*16, dimension(*) AP, complex*16, dimension(*) X, integer INCX)"
.PP
\fBZTPSV\fP
.PP
\fBPurpose:\fP
.RS 4
.PP
.nf
ZTPSV 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\&.
.fi
.PP
.RE
.PP
\fBParameters\fP
.RS 4
\fIUPLO\fP
.PP
.nf
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\&.
.fi
.PP
.br
\fITRANS\fP
.PP
.nf
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\&.
.fi
.PP
.br
\fIDIAG\fP
.PP
.nf
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\&.
.fi
.PP
.br
\fIN\fP
.PP
.nf
N is INTEGER
On entry, N specifies the order of the matrix A\&.
N must be at least zero\&.
.fi
.PP
.br
\fIAP\fP
.PP
.nf
AP is COMPLEX*16 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\&.
.fi
.PP
.br
\fIX\fP
.PP
.nf
X is COMPLEX*16 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\&.
.fi
.PP
.br
\fIINCX\fP
.PP
.nf
INCX is INTEGER
On entry, INCX specifies the increment for the elements of
X\&. INCX must not be zero\&.
.fi
.PP
.RE
.PP
\fBAuthor\fP
.RS 4
Univ\&. of Tennessee
.PP
Univ\&. of California Berkeley
.PP
Univ\&. of Colorado Denver
.PP
NAG Ltd\&.
.RE
.PP
\fBFurther Details:\fP
.RS 4
.PP
.nf
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\&.
.fi
.PP
.RE
.PP
.SS "subroutine ztrmv (character UPLO, character TRANS, character DIAG, integer N, complex*16, dimension(lda,*) A, integer LDA, complex*16, dimension(*) X, integer INCX)"
.PP
\fBZTRMV\fP
.PP
\fBPurpose:\fP
.RS 4
.PP
.nf
ZTRMV 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\&.
.fi
.PP
.RE
.PP
\fBParameters\fP
.RS 4
\fIUPLO\fP
.PP
.nf
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\&.
.fi
.PP
.br
\fITRANS\fP
.PP
.nf
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\&.
.fi
.PP
.br
\fIDIAG\fP
.PP
.nf
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\&.
.fi
.PP
.br
\fIN\fP
.PP
.nf
N is INTEGER
On entry, N specifies the order of the matrix A\&.
N must be at least zero\&.
.fi
.PP
.br
\fIA\fP
.PP
.nf
A is COMPLEX*16 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\&.
.fi
.PP
.br
\fILDA\fP
.PP
.nf
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 )\&.
.fi
.PP
.br
\fIX\fP
.PP
.nf
X is COMPLEX*16 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\&.
.fi
.PP
.br
\fIINCX\fP
.PP
.nf
INCX is INTEGER
On entry, INCX specifies the increment for the elements of
X\&. INCX must not be zero\&.
.fi
.PP
.RE
.PP
\fBAuthor\fP
.RS 4
Univ\&. of Tennessee
.PP
Univ\&. of California Berkeley
.PP
Univ\&. of Colorado Denver
.PP
NAG Ltd\&.
.RE
.PP
\fBFurther Details:\fP
.RS 4
.PP
.nf
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\&.
.fi
.PP
.RE
.PP
.SS "subroutine ztrsv (character UPLO, character TRANS, character DIAG, integer N, complex*16, dimension(lda,*) A, integer LDA, complex*16, dimension(*) X, integer INCX)"
.PP
\fBZTRSV\fP
.PP
\fBPurpose:\fP
.RS 4
.PP
.nf
ZTRSV 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\&.
.fi
.PP
.RE
.PP
\fBParameters\fP
.RS 4
\fIUPLO\fP
.PP
.nf
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\&.
.fi
.PP
.br
\fITRANS\fP
.PP
.nf
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\&.
.fi
.PP
.br
\fIDIAG\fP
.PP
.nf
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\&.
.fi
.PP
.br
\fIN\fP
.PP
.nf
N is INTEGER
On entry, N specifies the order of the matrix A\&.
N must be at least zero\&.
.fi
.PP
.br
\fIA\fP
.PP
.nf
A is COMPLEX*16 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\&.
.fi
.PP
.br
\fILDA\fP
.PP
.nf
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 )\&.
.fi
.PP
.br
\fIX\fP
.PP
.nf
X is COMPLEX*16 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\&.
.fi
.PP
.br
\fIINCX\fP
.PP
.nf
INCX is INTEGER
On entry, INCX specifies the increment for the elements of
X\&. INCX must not be zero\&.
.fi
.PP
.RE
.PP
\fBAuthor\fP
.RS 4
Univ\&. of Tennessee
.PP
Univ\&. of California Berkeley
.PP
Univ\&. of Colorado Denver
.PP
NAG Ltd\&.
.RE
.PP
\fBFurther Details:\fP
.RS 4
.PP
.nf
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\&.
.fi
.PP
.RE
.PP
.SH "Author"
.PP
Generated automatically by Doxygen for LAPACK from the source code\&.