## table of contents

complex16_blas_level2(3) | LAPACK | complex16_blas_level2(3) |

# NAME¶

complex16_blas_level2

# SYNOPSIS¶

## Functions¶

subroutine **zgbmv** (TRANS, M, N, KL, KU, ALPHA, A, LDA, X,
INCX, BETA, Y, INCY)

**ZGBMV** subroutine **zgemv** (TRANS, M, N, ALPHA, A, LDA, X, INCX,
BETA, Y, INCY)

**ZGEMV** subroutine **zgerc** (M, N, ALPHA, X, INCX, Y, INCY, A, LDA)

**ZGERC** subroutine **zgeru** (M, N, ALPHA, X, INCX, Y, INCY, A, LDA)

**ZGERU** subroutine **zhbmv** (UPLO, N, K, ALPHA, A, LDA, X, INCX,
BETA, Y, INCY)

**ZHBMV** subroutine **zhemv** (UPLO, N, ALPHA, A, LDA, X, INCX, BETA,
Y, INCY)

**ZHEMV** subroutine **zher** (UPLO, N, ALPHA, X, INCX, A, LDA)

**ZHER** subroutine **zher2** (UPLO, N, ALPHA, X, INCX, Y, INCY, A, LDA)

**ZHER2** subroutine **zhpmv** (UPLO, N, ALPHA, AP, X, INCX, BETA, Y,
INCY)

**ZHPMV** subroutine **zhpr** (UPLO, N, ALPHA, X, INCX, AP)

**ZHPR** subroutine **zhpr2** (UPLO, N, ALPHA, X, INCX, Y, INCY, AP)

**ZHPR2** subroutine **ztbmv** (UPLO, TRANS, DIAG, N, K, A, LDA, X,
INCX)

**ZTBMV** subroutine **ztbsv** (UPLO, TRANS, DIAG, N, K, A, LDA, X,
INCX)

**ZTBSV** subroutine **ztpmv** (UPLO, TRANS, DIAG, N, AP, X, INCX)

**ZTPMV** subroutine **ztpsv** (UPLO, TRANS, DIAG, N, AP, X, INCX)

**ZTPSV** subroutine **ztrmv** (UPLO, TRANS, DIAG, N, A, LDA, X, INCX)

**ZTRMV** subroutine **ztrsv** (UPLO, TRANS, DIAG, N, A, LDA, X, INCX)

**ZTRSV**

# Detailed Description¶

This is the group of complex16 LEVEL 2 BLAS routines.

# Function Documentation¶

## 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)¶

**ZGBMV**

**Purpose:**

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.

**Parameters**

*TRANS*

TRANS is CHARACTER*1

On entry, TRANS specifies the operation to be performed as

follows:

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

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

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

*M*

M is INTEGER

On entry, M specifies the number of rows of the matrix A.

M must be at least zero.

*N*

N is INTEGER

On entry, N specifies the number of columns of the matrix A.

N must be at least zero.

*KL*

KL is INTEGER

On entry, KL specifies the number of sub-diagonals of the

matrix A. KL must satisfy 0 .le. KL.

*KU*

KU is INTEGER

On entry, KU specifies the number of super-diagonals of the

matrix A. KU must satisfy 0 .le. KU.

*ALPHA*

ALPHA is COMPLEX*16

On entry, ALPHA specifies the scalar alpha.

*A*

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

*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*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.

*INCX*

INCX is INTEGER

On entry, INCX specifies the increment for the elements of

X. INCX must not be zero.

*BETA*

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.

*Y*

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.

*INCY*

INCY is INTEGER

On entry, INCY specifies the increment for the elements of

Y. INCY must not be zero.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

**Further Details:**

Level 2 Blas routine.

The vector and matrix arguments are not referenced when N = 0, or M = 0

-- Written on 22-October-1986.

Jack Dongarra, Argonne National Lab.

Jeremy Du Croz, Nag Central Office.

Sven Hammarling, Nag Central Office.

Richard Hanson, Sandia National Labs.

## 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)¶

**ZGEMV**

**Purpose:**

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.

**Parameters**

*TRANS*

TRANS is CHARACTER*1

On entry, TRANS specifies the operation to be performed as

follows:

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

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

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

*M*

M is INTEGER

On entry, M specifies the number of rows of the matrix A.

M must be at least zero.

*N*

N is INTEGER

On entry, N specifies the number of columns of the matrix A.

N must be at least zero.

*ALPHA*

ALPHA is COMPLEX*16

On entry, ALPHA specifies the scalar alpha.

*A*

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.

*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*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.

*INCX*

INCX is INTEGER

On entry, INCX specifies the increment for the elements of

X. INCX must not be zero.

*BETA*

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.

*Y*

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.

*INCY*

INCY is INTEGER

On entry, INCY specifies the increment for the elements of

Y. INCY must not be zero.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

**Further Details:**

Level 2 Blas routine.

The vector and matrix arguments are not referenced when N = 0, or M = 0

-- Written on 22-October-1986.

Jack Dongarra, Argonne National Lab.

Jeremy Du Croz, Nag Central Office.

Sven Hammarling, Nag Central Office.

Richard Hanson, Sandia National Labs.

## 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)¶

**ZGERC**

**Purpose:**

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.

**Parameters**

*M*

M is INTEGER

On entry, M specifies the number of rows of the matrix A.

M must be at least zero.

*N*

N is INTEGER

On entry, N specifies the number of columns of the matrix A.

N must be at least zero.

*ALPHA*

ALPHA is COMPLEX*16

On entry, ALPHA specifies the scalar alpha.

*X*

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.

*INCX*

INCX is INTEGER

On entry, INCX specifies the increment for the elements of

X. INCX must not be zero.

*Y*

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.

*INCY*

INCY is INTEGER

On entry, INCY specifies the increment for the elements of

Y. INCY must not be zero.

*A*

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.

*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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

**Further Details:**

Level 2 Blas routine.

-- Written on 22-October-1986.

Jack Dongarra, Argonne National Lab.

Jeremy Du Croz, Nag Central Office.

Sven Hammarling, Nag Central Office.

Richard Hanson, Sandia National Labs.

## 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)¶

**ZGERU**

**Purpose:**

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.

**Parameters**

*M*

M is INTEGER

On entry, M specifies the number of rows of the matrix A.

M must be at least zero.

*N*

N is INTEGER

On entry, N specifies the number of columns of the matrix A.

N must be at least zero.

*ALPHA*

ALPHA is COMPLEX*16

On entry, ALPHA specifies the scalar alpha.

*X*

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.

*INCX*

INCX is INTEGER

On entry, INCX specifies the increment for the elements of

X. INCX must not be zero.

*Y*

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.

*INCY*

INCY is INTEGER

On entry, INCY specifies the increment for the elements of

Y. INCY must not be zero.

*A*

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.

*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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

**Further Details:**

Level 2 Blas routine.

-- Written on 22-October-1986.

Jack Dongarra, Argonne National Lab.

Jeremy Du Croz, Nag Central Office.

Sven Hammarling, Nag Central Office.

Richard Hanson, Sandia National Labs.

## 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)¶

**ZHBMV**

**Purpose:**

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.

**Parameters**

*UPLO*

UPLO is CHARACTER*1

On entry, UPLO specifies whether the upper or lower

triangular part of the band matrix A is being supplied as

follows:

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

being supplied.

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

being supplied.

*N*

N is INTEGER

On entry, N specifies the order of the matrix A.

N must be at least zero.

*K*

K is INTEGER

On entry, K specifies the number of super-diagonals of the

matrix A. K must satisfy 0 .le. K.

*ALPHA*

ALPHA is COMPLEX*16

On entry, ALPHA specifies the scalar alpha.

*A*

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.

*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*16 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*16

On entry, BETA specifies the scalar beta.

*Y*

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.

*INCY*

INCY is INTEGER

On entry, INCY specifies the increment for the elements of

Y. INCY must not be zero.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

**Further Details:**

Level 2 Blas routine.

The vector and matrix arguments are not referenced when N = 0, or M = 0

-- Written on 22-October-1986.

Jack Dongarra, Argonne National Lab.

Jeremy Du Croz, Nag Central Office.

Sven Hammarling, Nag Central Office.

Richard Hanson, Sandia National Labs.

## 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)¶

**ZHEMV**

**Purpose:**

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.

**Parameters**

*UPLO*

UPLO is CHARACTER*1

On entry, UPLO specifies whether the upper or lower

triangular part of the array A is to be referenced as

follows:

UPLO = 'U' or 'u' Only the upper triangular part of A

is to be referenced.

UPLO = 'L' or 'l' Only the lower triangular part of A

is to be referenced.

*N*

N is INTEGER

On entry, N specifies the order of the matrix A.

N must be at least zero.

*ALPHA*

ALPHA is COMPLEX*16

On entry, ALPHA specifies the scalar alpha.

*A*

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.

*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*16 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*16

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*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.

*INCY*

INCY is INTEGER

On entry, INCY specifies the increment for the elements of

Y. INCY must not be zero.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

**Further Details:**

Level 2 Blas routine.

The vector and matrix arguments are not referenced when N = 0, or M = 0

-- Written on 22-October-1986.

Jack Dongarra, Argonne National Lab.

Jeremy Du Croz, Nag Central Office.

Sven Hammarling, Nag Central Office.

Richard Hanson, Sandia National Labs.

## subroutine zher (character UPLO, integer N, double precision ALPHA, complex*16, dimension(*) X, integer INCX, complex*16, dimension(lda,*) A, integer LDA)¶

**ZHER**

**Purpose:**

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.

**Parameters**

*UPLO*

UPLO is CHARACTER*1

On entry, UPLO specifies whether the upper or lower

triangular part of the array A is to be referenced as

follows:

UPLO = 'U' or 'u' Only the upper triangular part of A

is to be referenced.

UPLO = 'L' or 'l' Only the lower triangular part of A

is to be referenced.

*N*

N is INTEGER

On entry, N specifies the order of the matrix A.

N must be at least zero.

*ALPHA*

ALPHA is DOUBLE PRECISION.

On entry, ALPHA specifies the scalar alpha.

*X*

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.

*INCX*

INCX is INTEGER

On entry, INCX specifies the increment for the elements of

X. INCX must not be zero.

*A*

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.

*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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

**Further Details:**

Level 2 Blas routine.

-- Written on 22-October-1986.

Jack Dongarra, Argonne National Lab.

Jeremy Du Croz, Nag Central Office.

Sven Hammarling, Nag Central Office.

Richard Hanson, Sandia National Labs.

## 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)¶

**ZHER2**

**Purpose:**

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.

**Parameters**

*UPLO*

UPLO is CHARACTER*1

On entry, UPLO specifies whether the upper or lower

triangular part of the array A is to be referenced as

follows:

UPLO = 'U' or 'u' Only the upper triangular part of A

is to be referenced.

UPLO = 'L' or 'l' Only the lower triangular part of A

is to be referenced.

*N*

N is INTEGER

On entry, N specifies the order of the matrix A.

N must be at least zero.

*ALPHA*

ALPHA is COMPLEX*16

On entry, ALPHA specifies the scalar alpha.

*X*

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.

*INCX*

INCX is INTEGER

On entry, INCX specifies the increment for the elements of

X. INCX must not be zero.

*Y*

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.

*INCY*

INCY is INTEGER

On entry, INCY specifies the increment for the elements of

Y. INCY must not be zero.

*A*

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.

*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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

**Further Details:**

Level 2 Blas routine.

-- Written on 22-October-1986.

Jack Dongarra, Argonne National Lab.

Jeremy Du Croz, Nag Central Office.

Sven Hammarling, Nag Central Office.

Richard Hanson, Sandia National Labs.

## 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)¶

**ZHPMV**

**Purpose:**

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.

**Parameters**

*UPLO*

UPLO is CHARACTER*1

On entry, UPLO specifies whether the upper or lower

triangular part of the matrix A is supplied in the packed

array AP as follows:

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

supplied in AP.

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

supplied in AP.

*N*

N is INTEGER

On entry, N specifies the order of the matrix A.

N must be at least zero.

*ALPHA*

ALPHA is COMPLEX*16

On entry, ALPHA specifies the scalar alpha.

*AP*

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.

*X*

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.

*INCX*

INCX is INTEGER

On entry, INCX specifies the increment for the elements of

X. INCX must not be zero.

*BETA*

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.

*Y*

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.

*INCY*

INCY is INTEGER

On entry, INCY specifies the increment for the elements of

Y. INCY must not be zero.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

**Further Details:**

Level 2 Blas routine.

The vector and matrix arguments are not referenced when N = 0, or M = 0

-- Written on 22-October-1986.

Jack Dongarra, Argonne National Lab.

Jeremy Du Croz, Nag Central Office.

Sven Hammarling, Nag Central Office.

Richard Hanson, Sandia National Labs.

## subroutine zhpr (character UPLO, integer N, double precision ALPHA, complex*16, dimension(*) X, integer INCX, complex*16, dimension(*) AP)¶

**ZHPR**

**Purpose:**

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.

**Parameters**

*UPLO*

UPLO is CHARACTER*1

On entry, UPLO specifies whether the upper or lower

triangular part of the matrix A is supplied in the packed

array AP as follows:

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

supplied in AP.

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

supplied in AP.

*N*

N is INTEGER

On entry, N specifies the order of the matrix A.

N must be at least zero.

*ALPHA*

ALPHA is DOUBLE PRECISION.

On entry, ALPHA specifies the scalar alpha.

*X*

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.

*INCX*

INCX is INTEGER

On entry, INCX specifies the increment for the elements of

X. INCX must not be zero.

*AP*

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.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

**Further Details:**

Level 2 Blas routine.

-- Written on 22-October-1986.

Jack Dongarra, Argonne National Lab.

Jeremy Du Croz, Nag Central Office.

Sven Hammarling, Nag Central Office.

Richard Hanson, Sandia National Labs.

## 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)¶

**ZHPR2**

**Purpose:**

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.

**Parameters**

*UPLO*

UPLO is CHARACTER*1

On entry, UPLO specifies whether the upper or lower

triangular part of the matrix A is supplied in the packed

array AP as follows:

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

supplied in AP.

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

supplied in AP.

*N*

N is INTEGER

On entry, N specifies the order of the matrix A.

N must be at least zero.

*ALPHA*

ALPHA is COMPLEX*16

On entry, ALPHA specifies the scalar alpha.

*X*

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.

*INCX*

INCX is INTEGER

On entry, INCX specifies the increment for the elements of

X. INCX must not be zero.

*Y*

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.

*INCY*

INCY is INTEGER

On entry, INCY specifies the increment for the elements of

Y. INCY must not be zero.

*AP*

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.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

**Further Details:**

Level 2 Blas routine.

-- Written on 22-October-1986.

Jack Dongarra, Argonne National Lab.

Jeremy Du Croz, Nag Central Office.

Sven Hammarling, Nag Central Office.

Richard Hanson, Sandia National Labs.

## 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)¶

**ZTBMV**

**Purpose:**

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.

**Parameters**

*UPLO*

UPLO is CHARACTER*1

On entry, UPLO specifies whether the matrix is an upper or

lower triangular matrix as follows:

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

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

*TRANS*

TRANS is CHARACTER*1

On entry, TRANS specifies the operation to be performed as

follows:

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

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

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

*DIAG*

DIAG is CHARACTER*1

On entry, DIAG specifies whether or not A is unit

triangular as follows:

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

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

triangular.

*N*

N is INTEGER

On entry, N specifies the order of the matrix A.

N must be at least zero.

*K*

K is INTEGER

On entry with UPLO = 'U' or 'u', K specifies the number of

super-diagonals of the matrix A.

On entry with UPLO = 'L' or 'l', K specifies the number of

sub-diagonals of the matrix A.

K must satisfy 0 .le. K.

*A*

A is COMPLEX*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.

*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*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.

*INCX*

INCX is INTEGER

On entry, INCX specifies the increment for the elements of

X. INCX must not be zero.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

**Further Details:**

Level 2 Blas routine.

The vector and matrix arguments are not referenced when N = 0, or M = 0

-- Written on 22-October-1986.

Jack Dongarra, Argonne National Lab.

Jeremy Du Croz, Nag Central Office.

Sven Hammarling, Nag Central Office.

Richard Hanson, Sandia National Labs.

## 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)¶

**ZTBSV**

**Purpose:**

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.

**Parameters**

*UPLO*

UPLO is CHARACTER*1

On entry, UPLO specifies whether the matrix is an upper or

lower triangular matrix as follows:

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

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

*TRANS*

TRANS is CHARACTER*1

On entry, TRANS specifies the equations to be solved as

follows:

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

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

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

*DIAG*

DIAG is CHARACTER*1

On entry, DIAG specifies whether or not A is unit

triangular as follows:

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

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

triangular.

*N*

N is INTEGER

On entry, N specifies the order of the matrix A.

N must be at least zero.

*K*

K is INTEGER

On entry with UPLO = 'U' or 'u', K specifies the number of

super-diagonals of the matrix A.

On entry with UPLO = 'L' or 'l', K specifies the number of

sub-diagonals of the matrix A.

K must satisfy 0 .le. K.

*A*

A is COMPLEX*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.

*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*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.

*INCX*

INCX is INTEGER

On entry, INCX specifies the increment for the elements of

X. INCX must not be zero.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

**Further Details:**

Level 2 Blas routine.

-- Written on 22-October-1986.

Jack Dongarra, Argonne National Lab.

Jeremy Du Croz, Nag Central Office.

Sven Hammarling, Nag Central Office.

Richard Hanson, Sandia National Labs.

## subroutine ztpmv (character UPLO, character TRANS, character DIAG, integer N, complex*16, dimension(*) AP, complex*16, dimension(*) X, integer INCX)¶

**ZTPMV**

**Purpose:**

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.

**Parameters**

*UPLO*

UPLO is CHARACTER*1

On entry, UPLO specifies whether the matrix is an upper or

lower triangular matrix as follows:

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

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

*TRANS*

TRANS is CHARACTER*1

On entry, TRANS specifies the operation to be performed as

follows:

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

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

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

*DIAG*

DIAG is CHARACTER*1

On entry, DIAG specifies whether or not A is unit

triangular as follows:

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

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

triangular.

*N*

N is INTEGER

On entry, N specifies the order of the matrix A.

N must be at least zero.

*AP*

AP is COMPLEX*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.

*X*

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.

*INCX*

INCX is INTEGER

On entry, INCX specifies the increment for the elements of

X. INCX must not be zero.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

**Further Details:**

Level 2 Blas routine.

The vector and matrix arguments are not referenced when N = 0, or M = 0

-- Written on 22-October-1986.

Jack Dongarra, Argonne National Lab.

Jeremy Du Croz, Nag Central Office.

Sven Hammarling, Nag Central Office.

Richard Hanson, Sandia National Labs.

## subroutine ztpsv (character UPLO, character TRANS, character DIAG, integer N, complex*16, dimension(*) AP, complex*16, dimension(*) X, integer INCX)¶

**ZTPSV**

**Purpose:**

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.

**Parameters**

*UPLO*

UPLO is CHARACTER*1

On entry, UPLO specifies whether the matrix is an upper or

lower triangular matrix as follows:

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

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

*TRANS*

TRANS is CHARACTER*1

On entry, TRANS specifies the equations to be solved as

follows:

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

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

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

*DIAG*

DIAG is CHARACTER*1

On entry, DIAG specifies whether or not A is unit

triangular as follows:

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

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

triangular.

*N*

N is INTEGER

On entry, N specifies the order of the matrix A.

N must be at least zero.

*AP*

AP is COMPLEX*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.

*X*

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.

*INCX*

INCX is INTEGER

On entry, INCX specifies the increment for the elements of

X. INCX must not be zero.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

**Further Details:**

Level 2 Blas routine.

-- Written on 22-October-1986.

Jack Dongarra, Argonne National Lab.

Jeremy Du Croz, Nag Central Office.

Sven Hammarling, Nag Central Office.

Richard Hanson, Sandia National Labs.

## subroutine ztrmv (character UPLO, character TRANS, character DIAG, integer N, complex*16, dimension(lda,*) A, integer LDA, complex*16, dimension(*) X, integer INCX)¶

**ZTRMV**

**Purpose:**

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.

**Parameters**

*UPLO*

UPLO is CHARACTER*1

On entry, UPLO specifies whether the matrix is an upper or

lower triangular matrix as follows:

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

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

*TRANS*

TRANS is CHARACTER*1

On entry, TRANS specifies the operation to be performed as

follows:

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

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

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

*DIAG*

DIAG is CHARACTER*1

On entry, DIAG specifies whether or not A is unit

triangular as follows:

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

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

triangular.

*N*

N is INTEGER

On entry, N specifies the order of the matrix A.

N must be at least zero.

*A*

A is COMPLEX*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.

*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*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.

*INCX*

INCX is INTEGER

On entry, INCX specifies the increment for the elements of

X. INCX must not be zero.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

**Further Details:**

Level 2 Blas routine.

The vector and matrix arguments are not referenced when N = 0, or M = 0

-- Written on 22-October-1986.

Jack Dongarra, Argonne National Lab.

Jeremy Du Croz, Nag Central Office.

Sven Hammarling, Nag Central Office.

Richard Hanson, Sandia National Labs.

## subroutine ztrsv (character UPLO, character TRANS, character DIAG, integer N, complex*16, dimension(lda,*) A, integer LDA, complex*16, dimension(*) X, integer INCX)¶

**ZTRSV**

**Purpose:**

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.

**Parameters**

*UPLO*

UPLO is CHARACTER*1

On entry, UPLO specifies whether the matrix is an upper or

lower triangular matrix as follows:

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

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

*TRANS*

TRANS is CHARACTER*1

On entry, TRANS specifies the equations to be solved as

follows:

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

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

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

*DIAG*

DIAG is CHARACTER*1

On entry, DIAG specifies whether or not A is unit

triangular as follows:

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

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

triangular.

*N*

N is INTEGER

On entry, N specifies the order of the matrix A.

N must be at least zero.

*A*

A is COMPLEX*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.

*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*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.

*INCX*

INCX is INTEGER

On entry, INCX specifies the increment for the elements of

X. INCX must not be zero.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

**Further Details:**

Level 2 Blas routine.

-- Written on 22-October-1986.

Jack Dongarra, Argonne National Lab.

Jeremy Du Croz, Nag Central Office.

Sven Hammarling, Nag Central Office.

Richard Hanson, Sandia National Labs.

# Author¶

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