Provided by: liblapack-doc_3.8.0-2_all

**NAME**

double_blas_level2

**SYNOPSIS**

Functionssubroutinedgbmv(TRANS, M, N, KL, KU, ALPHA, A, LDA, X, INCX, BETA, Y, INCY)DGBMVsubroutinedgemv(TRANS, M, N, ALPHA, A, LDA, X, INCX, BETA, Y, INCY)DGEMVsubroutinedger(M, N, ALPHA, X, INCX, Y, INCY, A, LDA)DGERsubroutinedsbmv(UPLO, N, K, ALPHA, A, LDA, X, INCX, BETA, Y, INCY)DSBMVsubroutinedspmv(UPLO, N, ALPHA, AP, X, INCX, BETA, Y, INCY)DSPMVsubroutinedspr(UPLO, N, ALPHA, X, INCX, AP)DSPRsubroutinedspr2(UPLO, N, ALPHA, X, INCX, Y, INCY, AP)DSPR2subroutinedsymv(UPLO, N, ALPHA, A, LDA, X, INCX, BETA, Y, INCY)DSYMVsubroutinedsyr(UPLO, N, ALPHA, X, INCX, A, LDA)DSYRsubroutinedsyr2(UPLO, N, ALPHA, X, INCX, Y, INCY, A, LDA)DSYR2subroutinedtbmv(UPLO, TRANS, DIAG, N, K, A, LDA, X, INCX)DTBMVsubroutinedtbsv(UPLO, TRANS, DIAG, N, K, A, LDA, X, INCX)DTBSVsubroutinedtpmv(UPLO, TRANS, DIAG, N, AP, X, INCX)DTPMVsubroutinedtpsv(UPLO, TRANS, DIAG, N, AP, X, INCX)DTPSVsubroutinedtrmv(UPLO, TRANS, DIAG, N, A, LDA, X, INCX)DTRMV

**Detailed** **Description**

This is the group of double LEVEL 2 BLAS routines.

**Function** **Documentation**

subroutinedgbmv(characterTRANS,integerM,integerN,integerKL,integerKU,doubleprecisionALPHA,doubleprecision,dimension(lda,*)A,integerLDA,doubleprecision,dimension(*)X,integerINCX,doubleprecisionBETA,doubleprecision,dimension(*)Y,integerINCY)DGBMVPurpose: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:TRANSTRANS 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.MM is INTEGER On entry, M specifies the number of rows of the matrix A. M must be at least zero.NN is INTEGER On entry, N specifies the number of columns of the matrix A. N must be at least zero.KLKL is INTEGER On entry, KL specifies the number of sub-diagonals of the matrix A. KL must satisfy 0 .le. KL.KUKU is INTEGER On entry, KU specifies the number of super-diagonals of the matrix A. KU must satisfy 0 .le. KU.ALPHAALPHA is DOUBLE PRECISION. On entry, ALPHA specifies the scalar alpha.AA 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 CONTINUELDALDA 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 ).XX 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.INCXINCX is INTEGER On entry, INCX specifies the increment for the elements of X. INCX must not be zero.BETABETA is DOUBLE PRECISION. On entry, BETA specifies the scalar beta. When BETA is supplied as zero then Y need not be set on input.YY 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.INCYINCY is INTEGER On entry, INCY specifies the increment for the elements of Y. INCY must not be zero.Author:Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.Date:December 2016FurtherDetails: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.subroutinedgemv(characterTRANS,integerM,integerN,doubleprecisionALPHA,doubleprecision,dimension(lda,*)A,integerLDA,doubleprecision,dimension(*)X,integerINCX,doubleprecisionBETA,doubleprecision,dimension(*)Y,integerINCY)DGEMVPurpose: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:TRANSTRANS 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.MM is INTEGER On entry, M specifies the number of rows of the matrix A. M must be at least zero.NN is INTEGER On entry, N specifies the number of columns of the matrix A. N must be at least zero.ALPHAALPHA is DOUBLE PRECISION. On entry, ALPHA specifies the scalar alpha.AA 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.LDALDA 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 ).XX 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.INCXINCX is INTEGER On entry, INCX specifies the increment for the elements of X. INCX must not be zero.BETABETA is DOUBLE PRECISION. On entry, BETA specifies the scalar beta. When BETA is supplied as zero then Y need not be set on input.YY 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.INCYINCY is INTEGER On entry, INCY specifies the increment for the elements of Y. INCY must not be zero.Author:Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.Date:December 2016FurtherDetails: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.subroutinedger(integerM,integerN,doubleprecisionALPHA,doubleprecision,dimension(*)X,integerINCX,doubleprecision,dimension(*)Y,integerINCY,doubleprecision,dimension(lda,*)A,integerLDA)DGERPurpose: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:MM is INTEGER On entry, M specifies the number of rows of the matrix A. M must be at least zero.NN is INTEGER On entry, N specifies the number of columns of the matrix A. N must be at least zero.ALPHAALPHA is DOUBLE PRECISION. On entry, ALPHA specifies the scalar alpha.XX 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.INCXINCX is INTEGER On entry, INCX specifies the increment for the elements of X. INCX must not be zero.YY 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.INCYINCY is INTEGER On entry, INCY specifies the increment for the elements of Y. INCY must not be zero.AA 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.LDALDA is INTEGER On entry, LDA specifies the first dimension of A as declared in the calling (sub) program. LDA must be at least max( 1, m ).Author:Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.Date:December 2016FurtherDetails: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.subroutinedsbmv(characterUPLO,integerN,integerK,doubleprecisionALPHA,doubleprecision,dimension(lda,*)A,integerLDA,doubleprecision,dimension(*)X,integerINCX,doubleprecisionBETA,doubleprecision,dimension(*)Y,integerINCY)DSBMVPurpose: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:UPLOUPLO 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.NN is INTEGER On entry, N specifies the order of the matrix A. N must be at least zero.KK is INTEGER On entry, K specifies the number of super-diagonals of the matrix A. K must satisfy 0 .le. K.ALPHAALPHA is DOUBLE PRECISION. On entry, ALPHA specifies the scalar alpha.AA 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 CONTINUELDALDA 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 ).XX is DOUBLE PRECISION array, dimension at least ( 1 + ( n - 1 )*abs( INCX ) ). Before entry, the incremented array X must contain the vector x.INCXINCX is INTEGER On entry, INCX specifies the increment for the elements of X. INCX must not be zero.BETABETA is DOUBLE PRECISION. On entry, BETA specifies the scalar beta.YY 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.INCYINCY is INTEGER On entry, INCY specifies the increment for the elements of Y. INCY must not be zero.Author:Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.Date:December 2016FurtherDetails: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.subroutinedspmv(characterUPLO,integerN,doubleprecisionALPHA,doubleprecision,dimension(*)AP,doubleprecision,dimension(*)X,integerINCX,doubleprecisionBETA,doubleprecision,dimension(*)Y,integerINCY)DSPMVPurpose: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:UPLOUPLO 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.NN is INTEGER On entry, N specifies the order of the matrix A. N must be at least zero.ALPHAALPHA is DOUBLE PRECISION. On entry, ALPHA specifies the scalar alpha.APAP 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.XX 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.INCXINCX is INTEGER On entry, INCX specifies the increment for the elements of X. INCX must not be zero.BETABETA is DOUBLE PRECISION. On entry, BETA specifies the scalar beta. When BETA is supplied as zero then Y need not be set on input.YY 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.INCYINCY is INTEGER On entry, INCY specifies the increment for the elements of Y. INCY must not be zero.Author:Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.Date:December 2016FurtherDetails: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.subroutinedspr(characterUPLO,integerN,doubleprecisionALPHA,doubleprecision,dimension(*)X,integerINCX,doubleprecision,dimension(*)AP)DSPRPurpose: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:UPLOUPLO 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.NN is INTEGER On entry, N specifies the order of the matrix A. N must be at least zero.ALPHAALPHA is DOUBLE PRECISION. On entry, ALPHA specifies the scalar alpha.XX 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.INCXINCX is INTEGER On entry, INCX specifies the increment for the elements of X. INCX must not be zero.APAP 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 Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.Date:December 2016FurtherDetails: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.subroutinedspr2(characterUPLO,integerN,doubleprecisionALPHA,doubleprecision,dimension(*)X,integerINCX,doubleprecision,dimension(*)Y,integerINCY,doubleprecision,dimension(*)AP)DSPR2Purpose: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:UPLOUPLO 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.NN is INTEGER On entry, N specifies the order of the matrix A. N must be at least zero.ALPHAALPHA is DOUBLE PRECISION. On entry, ALPHA specifies the scalar alpha.XX 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.INCXINCX is INTEGER On entry, INCX specifies the increment for the elements of X. INCX must not be zero.YY 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.INCYINCY is INTEGER On entry, INCY specifies the increment for the elements of Y. INCY must not be zero.APAP 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 Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.Date:December 2016FurtherDetails: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.subroutinedsymv(characterUPLO,integerN,doubleprecisionALPHA,doubleprecision,dimension(lda,*)A,integerLDA,doubleprecision,dimension(*)X,integerINCX,doubleprecisionBETA,doubleprecision,dimension(*)Y,integerINCY)DSYMVPurpose: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:UPLOUPLO 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.NN is INTEGER On entry, N specifies the order of the matrix A. N must be at least zero.ALPHAALPHA is DOUBLE PRECISION. On entry, ALPHA specifies the scalar alpha.AA 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.LDALDA 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 ).XX 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.INCXINCX is INTEGER On entry, INCX specifies the increment for the elements of X. INCX must not be zero.BETABETA is DOUBLE PRECISION. On entry, BETA specifies the scalar beta. When BETA is supplied as zero then Y need not be set on input.YY 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.INCYINCY is INTEGER On entry, INCY specifies the increment for the elements of Y. INCY must not be zero.Author:Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.Date:December 2016FurtherDetails: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.subroutinedsyr(characterUPLO,integerN,doubleprecisionALPHA,doubleprecision,dimension(*)X,integerINCX,doubleprecision,dimension(lda,*)A,integerLDA)DSYRPurpose: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:UPLOUPLO 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.NN is INTEGER On entry, N specifies the order of the matrix A. N must be at least zero.ALPHAALPHA is DOUBLE PRECISION. On entry, ALPHA specifies the scalar alpha.XX 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.INCXINCX is INTEGER On entry, INCX specifies the increment for the elements of X. INCX must not be zero.AA 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.LDALDA is INTEGER On entry, LDA specifies the first dimension of A as declared in the calling (sub) program. LDA must be at least max( 1, n ).Author:Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.Date:December 2016FurtherDetails: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.subroutinedsyr2(characterUPLO,integerN,doubleprecisionALPHA,doubleprecision,dimension(*)X,integerINCX,doubleprecision,dimension(*)Y,integerINCY,doubleprecision,dimension(lda,*)A,integerLDA)DSYR2Purpose: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:UPLOUPLO 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.NN is INTEGER On entry, N specifies the order of the matrix A. N must be at least zero.ALPHAALPHA is DOUBLE PRECISION. On entry, ALPHA specifies the scalar alpha.XX 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.INCXINCX is INTEGER On entry, INCX specifies the increment for the elements of X. INCX must not be zero.YY 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.INCYINCY is INTEGER On entry, INCY specifies the increment for the elements of Y. INCY must not be zero.AA 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.LDALDA is INTEGER On entry, LDA specifies the first dimension of A as declared in the calling (sub) program. LDA must be at least max( 1, n ).Author:Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.Date:December 2016FurtherDetails: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.subroutinedtbmv(characterUPLO,characterTRANS,characterDIAG,integerN,integerK,doubleprecision,dimension(lda,*)A,integerLDA,doubleprecision,dimension(*)X,integerINCX)DTBMVPurpose: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:UPLOUPLO 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.TRANSTRANS 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.DIAGDIAG 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.NN is INTEGER On entry, N specifies the order of the matrix A. N must be at least zero.KK 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.AA 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.LDALDA 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 ).XX 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.INCXINCX is INTEGER On entry, INCX specifies the increment for the elements of X. INCX must not be zero.Author:Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.Date:December 2016FurtherDetails: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.subroutinedtbsv(characterUPLO,characterTRANS,characterDIAG,integerN,integerK,doubleprecision,dimension(lda,*)A,integerLDA,doubleprecision,dimension(*)X,integerINCX)DTBSVPurpose: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:UPLOUPLO 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.TRANSTRANS 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.DIAGDIAG 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.NN is INTEGER On entry, N specifies the order of the matrix A. N must be at least zero.KK 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.AA 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.LDALDA 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 ).XX 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.INCXINCX is INTEGER On entry, INCX specifies the increment for the elements of X. INCX must not be zero.Author:Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.Date:December 2016FurtherDetails: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.subroutinedtpmv(characterUPLO,characterTRANS,characterDIAG,integerN,doubleprecision,dimension(*)AP,doubleprecision,dimension(*)X,integerINCX)DTPMVPurpose: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:UPLOUPLO 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.TRANSTRANS 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.DIAGDIAG 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.NN is INTEGER On entry, N specifies the order of the matrix A. N must be at least zero.APAP 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.XX 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.INCXINCX is INTEGER On entry, INCX specifies the increment for the elements of X. INCX must not be zero.Author:Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.Date:December 2016FurtherDetails: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.subroutinedtpsv(characterUPLO,characterTRANS,characterDIAG,integerN,doubleprecision,dimension(*)AP,doubleprecision,dimension(*)X,integerINCX)DTPSVPurpose: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:UPLOUPLO 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.TRANSTRANS 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.DIAGDIAG 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.NN is INTEGER On entry, N specifies the order of the matrix A. N must be at least zero.APAP 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.XX 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.INCXINCX is INTEGER On entry, INCX specifies the increment for the elements of X. INCX must not be zero.Author:Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.Date:December 2016FurtherDetails: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.subroutinedtrmv(characterUPLO,characterTRANS,characterDIAG,integerN,doubleprecision,dimension(lda,*)A,integerLDA,doubleprecision,dimension(*)X,integerINCX)DTRMVPurpose: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:UPLOUPLO 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.TRANSTRANS 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.DIAGDIAG 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.NN is INTEGER On entry, N specifies the order of the matrix A. N must be at least zero.AA 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.LDALDA 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 ).XX 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.INCXINCX is INTEGER On entry, INCX specifies the increment for the elements of X. INCX must not be zero.Author:Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.Date:December 2016FurtherDetails: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.

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