## table of contents

double_blas_level2(3) | LAPACK | double_blas_level2(3) |

# NAME¶

double_blas_level2

# SYNOPSIS¶

## Functions¶

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

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

**DGEMV** subroutine **dger** (M, N, ALPHA, X, INCX, Y, INCY, A, LDA)

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

**DSBMV** subroutine **dspmv** (UPLO, N, ALPHA, AP, X, INCX, BETA, Y,
INCY)

**DSPMV** subroutine **dspr** (UPLO, N, ALPHA, X, INCX, AP)

**DSPR** subroutine **dspr2** (UPLO, N, ALPHA, X, INCX, Y, INCY, AP)

**DSPR2** subroutine **dsymv** (UPLO, N, ALPHA, A, LDA, X, INCX, BETA,
Y, INCY)

**DSYMV** subroutine **dsyr** (UPLO, N, ALPHA, X, INCX, A, LDA)

**DSYR** subroutine **dsyr2** (UPLO, N, ALPHA, X, INCX, Y, INCY, A, LDA)

**DSYR2** subroutine **dtbmv** (UPLO, TRANS, DIAG, N, K, A, LDA, X,
INCX)

**DTBMV** subroutine **dtbsv** (UPLO, TRANS, DIAG, N, K, A, LDA, X,
INCX)

**DTBSV** subroutine **dtpmv** (UPLO, TRANS, DIAG, N, AP, X, INCX)

**DTPMV** subroutine **dtpsv** (UPLO, TRANS, DIAG, N, AP, X, INCX)

**DTPSV** subroutine **dtrmv** (UPLO, TRANS, DIAG, N, A, LDA, X, INCX)

**DTRMV**

# Detailed Description¶

This is the group of double LEVEL 2 BLAS routines.

# Function Documentation¶

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

**DGBMV**

**Purpose:**

DGBMV performs one of the matrix-vector operations

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

where alpha and beta are scalars, x and y are vectors and A is an

m by n band matrix, with kl sub-diagonals and ku super-diagonals.

**Parameters**

*TRANS*

TRANS is CHARACTER*1

On entry, TRANS specifies the operation to be performed as

follows:

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

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

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

*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 DOUBLE PRECISION.

On entry, ALPHA specifies the scalar alpha.

*A*

A is DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION.

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

supplied as zero then Y need not be set on input.

*Y*

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

**DGEMV**

**Purpose:**

DGEMV performs one of the matrix-vector operations

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

where alpha and beta are scalars, x and y are vectors and A is an

m by n matrix.

**Parameters**

*TRANS*

TRANS is CHARACTER*1

On entry, TRANS specifies the operation to be performed as

follows:

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

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

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

*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 DOUBLE PRECISION.

On entry, ALPHA specifies the scalar alpha.

*A*

A is DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION.

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

supplied as zero then Y need not be set on input.

*Y*

Y is DOUBLE PRECISION 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 dger (integer M, integer N, double precision ALPHA, double precision, dimension(*) X, integer INCX, double precision, dimension(*) Y, integer INCY, double precision, dimension(lda,*) A, integer LDA)¶

**DGER**

**Purpose:**

DGER performs the rank 1 operation

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

where alpha is a scalar, x is an m element vector, y is an n element

vector and A is an m by n matrix.

**Parameters**

*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 DOUBLE PRECISION.

On entry, ALPHA specifies the scalar alpha.

*X*

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

**DSBMV**

**Purpose:**

DSBMV performs the matrix-vector operation

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

where alpha and beta are scalars, x and y are n element vectors and

A is an n by n symmetric band matrix, with k super-diagonals.

**Parameters**

*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 DOUBLE PRECISION.

On entry, ALPHA specifies the scalar alpha.

*A*

A is DOUBLE PRECISION 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 symmetric matrix, supplied column by

column, with the leading diagonal of the matrix in row

( k + 1 ) of the array, the first super-diagonal starting at

position 2 in row k, and so on. The top left k by k triangle

of the array A is not referenced.

The following program segment will transfer the upper

triangular part of a symmetric band matrix from conventional

full matrix storage to band storage:

DO 20, J = 1, N

M = K + 1 - J

DO 10, I = MAX( 1, J - K ), J

A( M + I, J ) = matrix( I, J )

10 CONTINUE

20 CONTINUE

Before entry with UPLO = 'L' or 'l', the leading ( k + 1 )

by n part of the array A must contain the lower triangular

band part of the symmetric matrix, supplied column by

column, with the leading diagonal of the matrix in row 1 of

the array, the first sub-diagonal starting at position 1 in

row 2, and so on. The bottom right k by k triangle of the

array A is not referenced.

The following program segment will transfer the lower

triangular part of a symmetric band matrix from conventional

full matrix storage to band storage:

DO 20, J = 1, N

M = 1 - J

DO 10, I = J, MIN( N, J + K )

A( M + I, J ) = matrix( I, J )

10 CONTINUE

20 CONTINUE

*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 DOUBLE PRECISION 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 DOUBLE PRECISION.

On entry, BETA specifies the scalar beta.

*Y*

Y is DOUBLE PRECISION 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 dspmv (character UPLO, integer N, double precision ALPHA, double precision, dimension(*) AP, double precision, dimension(*) X, integer INCX, double precision BETA, double precision, dimension(*) Y, integer INCY)¶

**DSPMV**

**Purpose:**

DSPMV performs the matrix-vector operation

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

where alpha and beta are scalars, x and y are n element vectors and

A is an n by n symmetric matrix, supplied in packed form.

**Parameters**

*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.

*AP*

AP is DOUBLE PRECISION 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 symmetric matrix

packed sequentially, column by column, so that AP( 1 )

contains a( 1, 1 ), AP( 2 ) and AP( 3 ) contain a( 1, 2 )

and a( 2, 2 ) respectively, and so on.

Before entry with UPLO = 'L' or 'l', the array AP must

contain the lower triangular part of the symmetric matrix

packed sequentially, column by column, so that AP( 1 )

contains a( 1, 1 ), AP( 2 ) and AP( 3 ) contain a( 2, 1 )

and a( 3, 1 ) respectively, and so on.

*X*

X is DOUBLE PRECISION 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 DOUBLE PRECISION.

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

supplied as zero then Y need not be set on input.

*Y*

Y is DOUBLE PRECISION 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 dspr (character UPLO, integer N, double precision ALPHA, double precision, dimension(*) X, integer INCX, double precision, dimension(*) AP)¶

**DSPR**

**Purpose:**

DSPR performs the symmetric rank 1 operation

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

where alpha is a real scalar, x is an n element vector and A is an

n by n symmetric matrix, supplied in packed form.

**Parameters**

*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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 symmetric matrix

packed sequentially, column by column, so that AP( 1 )

contains a( 1, 1 ), AP( 2 ) and AP( 3 ) contain a( 1, 2 )

and a( 2, 2 ) respectively, and so on. On exit, the array

AP is overwritten by the upper triangular part of the

updated matrix.

Before entry with UPLO = 'L' or 'l', the array AP must

contain the lower triangular part of the symmetric matrix

packed sequentially, column by column, so that AP( 1 )

contains a( 1, 1 ), AP( 2 ) and AP( 3 ) contain a( 2, 1 )

and a( 3, 1 ) respectively, and so on. On exit, the array

AP is overwritten by the lower triangular part of the

updated matrix.

**Author**

Univ. of 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 dspr2 (character UPLO, integer N, double precision ALPHA, double precision, dimension(*) X, integer INCX, double precision, dimension(*) Y, integer INCY, double precision, dimension(*) AP)¶

**DSPR2**

**Purpose:**

DSPR2 performs the symmetric rank 2 operation

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

where alpha is a scalar, x and y are n element vectors and A is an

n by n symmetric matrix, supplied in packed form.

**Parameters**

*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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 symmetric matrix

packed sequentially, column by column, so that AP( 1 )

contains a( 1, 1 ), AP( 2 ) and AP( 3 ) contain a( 1, 2 )

and a( 2, 2 ) respectively, and so on. On exit, the array

AP is overwritten by the upper triangular part of the

updated matrix.

Before entry with UPLO = 'L' or 'l', the array AP must

contain the lower triangular part of the symmetric matrix

packed sequentially, column by column, so that AP( 1 )

contains a( 1, 1 ), AP( 2 ) and AP( 3 ) contain a( 2, 1 )

and a( 3, 1 ) respectively, and so on. On exit, the array

AP is overwritten by the lower triangular part of the

updated matrix.

**Author**

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

**DSYMV**

**Purpose:**

DSYMV performs the matrix-vector operation

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

where alpha and beta are scalars, x and y are n element vectors and

A is an n by n symmetric matrix.

**Parameters**

*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.

*A*

A is DOUBLE PRECISION 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 symmetric matrix and the strictly

lower triangular part of A is not referenced.

Before entry with UPLO = 'L' or 'l', the leading n by n

lower triangular part of the array A must contain the lower

triangular part of the symmetric matrix and the strictly

upper triangular part of A is not referenced.

*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 DOUBLE PRECISION 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 DOUBLE PRECISION.

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

supplied as zero then Y need not be set on input.

*Y*

Y is DOUBLE PRECISION 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 dsyr (character UPLO, integer N, double precision ALPHA, double precision, dimension(*) X, integer INCX, double precision, dimension(lda,*) A, integer LDA)¶

**DSYR**

**Purpose:**

DSYR performs the symmetric rank 1 operation

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

where alpha is a real scalar, x is an n element vector and A is an

n by n symmetric matrix.

**Parameters**

*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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 symmetric matrix and the strictly

lower triangular part of A is not referenced. On exit, the

upper triangular part of the array A is overwritten by the

upper triangular part of the updated matrix.

Before entry with UPLO = 'L' or 'l', the leading n by n

lower triangular part of the array A must contain the lower

triangular part of the symmetric matrix and the strictly

upper triangular part of A is not referenced. On exit, the

lower triangular part of the array A is overwritten by the

lower triangular part of the updated matrix.

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

**DSYR2**

**Purpose:**

DSYR2 performs the symmetric rank 2 operation

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

where alpha is a scalar, x and y are n element vectors and A is an n

by n symmetric matrix.

**Parameters**

*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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 symmetric matrix and the strictly

lower triangular part of A is not referenced. On exit, the

upper triangular part of the array A is overwritten by the

upper triangular part of the updated matrix.

Before entry with UPLO = 'L' or 'l', the leading n by n

lower triangular part of the array A must contain the lower

triangular part of the symmetric matrix and the strictly

upper triangular part of A is not referenced. On exit, the

lower triangular part of the array A is overwritten by the

lower triangular part of the updated matrix.

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

**DTBMV**

**Purpose:**

DTBMV performs one of the matrix-vector operations

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

where x is an n element vector and A is an n by n unit, or non-unit,

upper or lower triangular band matrix, with ( k + 1 ) diagonals.

**Parameters**

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

**DTBSV**

**Purpose:**

DTBSV solves one of the systems of equations

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

where b and x are n element vectors and A is an n by n unit, or

non-unit, upper or lower triangular band matrix, with ( k + 1 )

diagonals.

No test for singularity or near-singularity is included in this

routine. Such tests must be performed before calling this routine.

**Parameters**

*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**T*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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 dtpmv (character UPLO, character TRANS, character DIAG, integer N, double precision, dimension(*) AP, double precision, dimension(*) X, integer INCX)¶

**DTPMV**

**Purpose:**

DTPMV performs one of the matrix-vector operations

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

where x is an n element vector and A is an n by n unit, or non-unit,

upper or lower triangular matrix, supplied in packed form.

**Parameters**

*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**T*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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 dtpsv (character UPLO, character TRANS, character DIAG, integer N, double precision, dimension(*) AP, double precision, dimension(*) X, integer INCX)¶

**DTPSV**

**Purpose:**

DTPSV solves one of the systems of equations

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

where b and x are n element vectors and A is an n by n unit, or

non-unit, upper or lower triangular matrix, supplied in packed form.

No test for singularity or near-singularity is included in this

routine. Such tests must be performed before calling this routine.

**Parameters**

*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**T*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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 dtrmv (character UPLO, character TRANS, character DIAG, integer N, double precision, dimension(lda,*) A, integer LDA, double precision, dimension(*) X, integer INCX)¶

**DTRMV**

**Purpose:**

DTRMV performs one of the matrix-vector operations

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

where x is an n element vector and A is an n by n unit, or non-unit,

upper or lower triangular matrix.

**Parameters**

*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**T*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 DOUBLE PRECISION 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 DOUBLE PRECISION 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.

# Author¶

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