.TH "single_blas_level2" 3 "Sat Aug 1 2020" "Version 3.9.0" "LAPACK" \" -*- nroff -*- .ad l .nh .SH NAME single_blas_level2 .SH SYNOPSIS .br .PP .SS "Functions" .in +1c .ti -1c .RI "subroutine \fBsgbmv\fP (TRANS, M, N, KL, KU, ALPHA, A, LDA, X, INCX, BETA, Y, INCY)" .br .RI "\fBSGBMV\fP " .ti -1c .RI "subroutine \fBsgemv\fP (TRANS, M, N, ALPHA, A, LDA, X, INCX, BETA, Y, INCY)" .br .RI "\fBSGEMV\fP " .ti -1c .RI "subroutine \fBsger\fP (M, N, ALPHA, X, INCX, Y, INCY, A, LDA)" .br .RI "\fBSGER\fP " .ti -1c .RI "subroutine \fBssbmv\fP (UPLO, N, K, ALPHA, A, LDA, X, INCX, BETA, Y, INCY)" .br .RI "\fBSSBMV\fP " .ti -1c .RI "subroutine \fBsspmv\fP (UPLO, N, ALPHA, AP, X, INCX, BETA, Y, INCY)" .br .RI "\fBSSPMV\fP " .ti -1c .RI "subroutine \fBsspr\fP (UPLO, N, ALPHA, X, INCX, AP)" .br .RI "\fBSSPR\fP " .ti -1c .RI "subroutine \fBsspr2\fP (UPLO, N, ALPHA, X, INCX, Y, INCY, AP)" .br .RI "\fBSSPR2\fP " .ti -1c .RI "subroutine \fBssymv\fP (UPLO, N, ALPHA, A, LDA, X, INCX, BETA, Y, INCY)" .br .RI "\fBSSYMV\fP " .ti -1c .RI "subroutine \fBssyr\fP (UPLO, N, ALPHA, X, INCX, A, LDA)" .br .RI "\fBSSYR\fP " .ti -1c .RI "subroutine \fBssyr2\fP (UPLO, N, ALPHA, X, INCX, Y, INCY, A, LDA)" .br .RI "\fBSSYR2\fP " .ti -1c .RI "subroutine \fBstbmv\fP (UPLO, TRANS, DIAG, N, K, A, LDA, X, INCX)" .br .RI "\fBSTBMV\fP " .ti -1c .RI "subroutine \fBstbsv\fP (UPLO, TRANS, DIAG, N, K, A, LDA, X, INCX)" .br .RI "\fBSTBSV\fP " .ti -1c .RI "subroutine \fBstpmv\fP (UPLO, TRANS, DIAG, N, AP, X, INCX)" .br .RI "\fBSTPMV\fP " .ti -1c .RI "subroutine \fBstpsv\fP (UPLO, TRANS, DIAG, N, AP, X, INCX)" .br .RI "\fBSTPSV\fP " .ti -1c .RI "subroutine \fBstrmv\fP (UPLO, TRANS, DIAG, N, A, LDA, X, INCX)" .br .RI "\fBSTRMV\fP " .ti -1c .RI "subroutine \fBstrsv\fP (UPLO, TRANS, DIAG, N, A, LDA, X, INCX)" .br .RI "\fBSTRSV\fP " .in -1c .SH "Detailed Description" .PP This is the group of real LEVEL 2 BLAS routines\&. .SH "Function Documentation" .PP .SS "subroutine sgbmv (character TRANS, integer M, integer N, integer KL, integer KU, real ALPHA, real, dimension(lda,*) A, integer LDA, real, dimension(*) X, integer INCX, real BETA, real, dimension(*) Y, integer INCY)" .PP \fBSGBMV\fP .PP \fBPurpose:\fP .RS 4 .PP .nf SGBMV 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. .fi .PP .RE .PP \fBParameters\fP .RS 4 \fITRANS\fP .PP .nf TRANS is CHARACTER*1 On entry, TRANS specifies the operation to be performed as follows: TRANS = 'N' or 'n' y := alpha*A*x + beta*y. TRANS = 'T' or 't' y := alpha*A**T*x + beta*y. TRANS = 'C' or 'c' y := alpha*A**T*x + beta*y. .fi .PP .br \fIM\fP .PP .nf M is INTEGER On entry, M specifies the number of rows of the matrix A. M must be at least zero. .fi .PP .br \fIN\fP .PP .nf N is INTEGER On entry, N specifies the number of columns of the matrix A. N must be at least zero. .fi .PP .br \fIKL\fP .PP .nf KL is INTEGER On entry, KL specifies the number of sub-diagonals of the matrix A. KL must satisfy 0 .le. KL. .fi .PP .br \fIKU\fP .PP .nf KU is INTEGER On entry, KU specifies the number of super-diagonals of the matrix A. KU must satisfy 0 .le. KU. .fi .PP .br \fIALPHA\fP .PP .nf ALPHA is REAL On entry, ALPHA specifies the scalar alpha. .fi .PP .br \fIA\fP .PP .nf A is REAL array, dimension ( LDA, N ) Before entry, the leading ( kl + ku + 1 ) by n part of the array A must contain the matrix of coefficients, supplied column by column, with the leading diagonal of the matrix in row ( ku + 1 ) of the array, the first super-diagonal starting at position 2 in row ku, the first sub-diagonal starting at position 1 in row ( ku + 2 ), and so on. Elements in the array A that do not correspond to elements in the band matrix (such as the top left ku by ku triangle) are not referenced. The following program segment will transfer a band matrix from conventional full matrix storage to band storage: DO 20, J = 1, N K = KU + 1 - J DO 10, I = MAX( 1, J - KU ), MIN( M, J + KL ) A( K + I, J ) = matrix( I, J ) 10 CONTINUE 20 CONTINUE .fi .PP .br \fILDA\fP .PP .nf LDA is INTEGER On entry, LDA specifies the first dimension of A as declared in the calling (sub) program. LDA must be at least ( kl + ku + 1 ). .fi .PP .br \fIX\fP .PP .nf X is REAL array, dimension at least ( 1 + ( n - 1 )*abs( INCX ) ) when TRANS = 'N' or 'n' and at least ( 1 + ( m - 1 )*abs( INCX ) ) otherwise. Before entry, the incremented array X must contain the vector x. .fi .PP .br \fIINCX\fP .PP .nf INCX is INTEGER On entry, INCX specifies the increment for the elements of X. INCX must not be zero. .fi .PP .br \fIBETA\fP .PP .nf BETA is REAL On entry, BETA specifies the scalar beta. When BETA is supplied as zero then Y need not be set on input. .fi .PP .br \fIY\fP .PP .nf Y is REAL array, dimension at least ( 1 + ( m - 1 )*abs( INCY ) ) when TRANS = 'N' or 'n' and at least ( 1 + ( n - 1 )*abs( INCY ) ) otherwise. Before entry, the incremented array Y must contain the vector y. On exit, Y is overwritten by the updated vector y. .fi .PP .br \fIINCY\fP .PP .nf INCY is INTEGER On entry, INCY specifies the increment for the elements of Y. INCY must not be zero. .fi .PP .RE .PP \fBAuthor\fP .RS 4 Univ\&. of Tennessee .PP Univ\&. of California Berkeley .PP Univ\&. of Colorado Denver .PP NAG Ltd\&. .RE .PP \fBDate\fP .RS 4 December 2016 .RE .PP \fBFurther Details:\fP .RS 4 .PP .nf Level 2 Blas routine. The vector and matrix arguments are not referenced when N = 0, or M = 0 -- Written on 22-October-1986. Jack Dongarra, Argonne National Lab. Jeremy Du Croz, Nag Central Office. Sven Hammarling, Nag Central Office. Richard Hanson, Sandia National Labs. .fi .PP .RE .PP .SS "subroutine sgemv (character TRANS, integer M, integer N, real ALPHA, real, dimension(lda,*) A, integer LDA, real, dimension(*) X, integer INCX, real BETA, real, dimension(*) Y, integer INCY)" .PP \fBSGEMV\fP .PP \fBPurpose:\fP .RS 4 .PP .nf SGEMV 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. .fi .PP .RE .PP \fBParameters\fP .RS 4 \fITRANS\fP .PP .nf TRANS is CHARACTER*1 On entry, TRANS specifies the operation to be performed as follows: TRANS = 'N' or 'n' y := alpha*A*x + beta*y. TRANS = 'T' or 't' y := alpha*A**T*x + beta*y. TRANS = 'C' or 'c' y := alpha*A**T*x + beta*y. .fi .PP .br \fIM\fP .PP .nf M is INTEGER On entry, M specifies the number of rows of the matrix A. M must be at least zero. .fi .PP .br \fIN\fP .PP .nf N is INTEGER On entry, N specifies the number of columns of the matrix A. N must be at least zero. .fi .PP .br \fIALPHA\fP .PP .nf ALPHA is REAL On entry, ALPHA specifies the scalar alpha. .fi .PP .br \fIA\fP .PP .nf A is REAL array, dimension ( LDA, N ) Before entry, the leading m by n part of the array A must contain the matrix of coefficients. .fi .PP .br \fILDA\fP .PP .nf LDA is INTEGER On entry, LDA specifies the first dimension of A as declared in the calling (sub) program. LDA must be at least max( 1, m ). .fi .PP .br \fIX\fP .PP .nf X is REAL array, dimension at least ( 1 + ( n - 1 )*abs( INCX ) ) when TRANS = 'N' or 'n' and at least ( 1 + ( m - 1 )*abs( INCX ) ) otherwise. Before entry, the incremented array X must contain the vector x. .fi .PP .br \fIINCX\fP .PP .nf INCX is INTEGER On entry, INCX specifies the increment for the elements of X. INCX must not be zero. .fi .PP .br \fIBETA\fP .PP .nf BETA is REAL On entry, BETA specifies the scalar beta. When BETA is supplied as zero then Y need not be set on input. .fi .PP .br \fIY\fP .PP .nf Y is REAL array, dimension at least ( 1 + ( m - 1 )*abs( INCY ) ) when TRANS = 'N' or 'n' and at least ( 1 + ( n - 1 )*abs( INCY ) ) otherwise. Before entry with BETA non-zero, the incremented array Y must contain the vector y. On exit, Y is overwritten by the updated vector y. .fi .PP .br \fIINCY\fP .PP .nf INCY is INTEGER On entry, INCY specifies the increment for the elements of Y. INCY must not be zero. .fi .PP .RE .PP \fBAuthor\fP .RS 4 Univ\&. of Tennessee .PP Univ\&. of California Berkeley .PP Univ\&. of Colorado Denver .PP NAG Ltd\&. .RE .PP \fBDate\fP .RS 4 December 2016 .RE .PP \fBFurther Details:\fP .RS 4 .PP .nf Level 2 Blas routine. The vector and matrix arguments are not referenced when N = 0, or M = 0 -- Written on 22-October-1986. Jack Dongarra, Argonne National Lab. Jeremy Du Croz, Nag Central Office. Sven Hammarling, Nag Central Office. Richard Hanson, Sandia National Labs. .fi .PP .RE .PP .SS "subroutine sger (integer M, integer N, real ALPHA, real, dimension(*) X, integer INCX, real, dimension(*) Y, integer INCY, real, dimension(lda,*) A, integer LDA)" .PP \fBSGER\fP .PP \fBPurpose:\fP .RS 4 .PP .nf SGER performs the rank 1 operation A := alpha*x*y**T + A, where alpha is a scalar, x is an m element vector, y is an n element vector and A is an m by n matrix. .fi .PP .RE .PP \fBParameters\fP .RS 4 \fIM\fP .PP .nf M is INTEGER On entry, M specifies the number of rows of the matrix A. M must be at least zero. .fi .PP .br \fIN\fP .PP .nf N is INTEGER On entry, N specifies the number of columns of the matrix A. N must be at least zero. .fi .PP .br \fIALPHA\fP .PP .nf ALPHA is REAL On entry, ALPHA specifies the scalar alpha. .fi .PP .br \fIX\fP .PP .nf X is REAL array, dimension at least ( 1 + ( m - 1 )*abs( INCX ) ). Before entry, the incremented array X must contain the m element vector x. .fi .PP .br \fIINCX\fP .PP .nf INCX is INTEGER On entry, INCX specifies the increment for the elements of X. INCX must not be zero. .fi .PP .br \fIY\fP .PP .nf Y is REAL array, dimension at least ( 1 + ( n - 1 )*abs( INCY ) ). Before entry, the incremented array Y must contain the n element vector y. .fi .PP .br \fIINCY\fP .PP .nf INCY is INTEGER On entry, INCY specifies the increment for the elements of Y. INCY must not be zero. .fi .PP .br \fIA\fP .PP .nf A is REAL array, dimension ( LDA, N ) Before entry, the leading m by n part of the array A must contain the matrix of coefficients. On exit, A is overwritten by the updated matrix. .fi .PP .br \fILDA\fP .PP .nf LDA is INTEGER On entry, LDA specifies the first dimension of A as declared in the calling (sub) program. LDA must be at least max( 1, m ). .fi .PP .RE .PP \fBAuthor\fP .RS 4 Univ\&. of Tennessee .PP Univ\&. of California Berkeley .PP Univ\&. of Colorado Denver .PP NAG Ltd\&. .RE .PP \fBDate\fP .RS 4 December 2016 .RE .PP \fBFurther Details:\fP .RS 4 .PP .nf Level 2 Blas routine. -- Written on 22-October-1986. Jack Dongarra, Argonne National Lab. Jeremy Du Croz, Nag Central Office. Sven Hammarling, Nag Central Office. Richard Hanson, Sandia National Labs. .fi .PP .RE .PP .SS "subroutine ssbmv (character UPLO, integer N, integer K, real ALPHA, real, dimension(lda,*) A, integer LDA, real, dimension(*) X, integer INCX, real BETA, real, dimension(*) Y, integer INCY)" .PP \fBSSBMV\fP .PP \fBPurpose:\fP .RS 4 .PP .nf SSBMV 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. .fi .PP .RE .PP \fBParameters\fP .RS 4 \fIUPLO\fP .PP .nf UPLO is CHARACTER*1 On entry, UPLO specifies whether the upper or lower triangular part of the band matrix A is being supplied as follows: UPLO = 'U' or 'u' The upper triangular part of A is being supplied. UPLO = 'L' or 'l' The lower triangular part of A is being supplied. .fi .PP .br \fIN\fP .PP .nf N is INTEGER On entry, N specifies the order of the matrix A. N must be at least zero. .fi .PP .br \fIK\fP .PP .nf K is INTEGER On entry, K specifies the number of super-diagonals of the matrix A. K must satisfy 0 .le. K. .fi .PP .br \fIALPHA\fP .PP .nf ALPHA is REAL On entry, ALPHA specifies the scalar alpha. .fi .PP .br \fIA\fP .PP .nf A is REAL 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 .fi .PP .br \fILDA\fP .PP .nf LDA is INTEGER On entry, LDA specifies the first dimension of A as declared in the calling (sub) program. LDA must be at least ( k + 1 ). .fi .PP .br \fIX\fP .PP .nf X is REAL array, dimension at least ( 1 + ( n - 1 )*abs( INCX ) ). Before entry, the incremented array X must contain the vector x. .fi .PP .br \fIINCX\fP .PP .nf INCX is INTEGER On entry, INCX specifies the increment for the elements of X. INCX must not be zero. .fi .PP .br \fIBETA\fP .PP .nf BETA is REAL On entry, BETA specifies the scalar beta. .fi .PP .br \fIY\fP .PP .nf Y is REAL array, dimension at least ( 1 + ( n - 1 )*abs( INCY ) ). Before entry, the incremented array Y must contain the vector y. On exit, Y is overwritten by the updated vector y. .fi .PP .br \fIINCY\fP .PP .nf INCY is INTEGER On entry, INCY specifies the increment for the elements of Y. INCY must not be zero. .fi .PP .RE .PP \fBAuthor\fP .RS 4 Univ\&. of Tennessee .PP Univ\&. of California Berkeley .PP Univ\&. of Colorado Denver .PP NAG Ltd\&. .RE .PP \fBDate\fP .RS 4 December 2016 .RE .PP \fBFurther Details:\fP .RS 4 .PP .nf Level 2 Blas routine. The vector and matrix arguments are not referenced when N = 0, or M = 0 -- Written on 22-October-1986. Jack Dongarra, Argonne National Lab. Jeremy Du Croz, Nag Central Office. Sven Hammarling, Nag Central Office. Richard Hanson, Sandia National Labs. .fi .PP .RE .PP .SS "subroutine sspmv (character UPLO, integer N, real ALPHA, real, dimension(*) AP, real, dimension(*) X, integer INCX, real BETA, real, dimension(*) Y, integer INCY)" .PP \fBSSPMV\fP .PP \fBPurpose:\fP .RS 4 .PP .nf SSPMV 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. .fi .PP .RE .PP \fBParameters\fP .RS 4 \fIUPLO\fP .PP .nf UPLO is CHARACTER*1 On entry, UPLO specifies whether the upper or lower triangular part of the matrix A is supplied in the packed array AP as follows: UPLO = 'U' or 'u' The upper triangular part of A is supplied in AP. UPLO = 'L' or 'l' The lower triangular part of A is supplied in AP. .fi .PP .br \fIN\fP .PP .nf N is INTEGER On entry, N specifies the order of the matrix A. N must be at least zero. .fi .PP .br \fIALPHA\fP .PP .nf ALPHA is REAL On entry, ALPHA specifies the scalar alpha. .fi .PP .br \fIAP\fP .PP .nf AP is REAL 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. .fi .PP .br \fIX\fP .PP .nf X is REAL array, dimension at least ( 1 + ( n - 1 )*abs( INCX ) ). Before entry, the incremented array X must contain the n element vector x. .fi .PP .br \fIINCX\fP .PP .nf INCX is INTEGER On entry, INCX specifies the increment for the elements of X. INCX must not be zero. .fi .PP .br \fIBETA\fP .PP .nf BETA is REAL On entry, BETA specifies the scalar beta. When BETA is supplied as zero then Y need not be set on input. .fi .PP .br \fIY\fP .PP .nf Y is REAL array, dimension at least ( 1 + ( n - 1 )*abs( INCY ) ). Before entry, the incremented array Y must contain the n element vector y. On exit, Y is overwritten by the updated vector y. .fi .PP .br \fIINCY\fP .PP .nf INCY is INTEGER On entry, INCY specifies the increment for the elements of Y. INCY must not be zero. .fi .PP .RE .PP \fBAuthor\fP .RS 4 Univ\&. of Tennessee .PP Univ\&. of California Berkeley .PP Univ\&. of Colorado Denver .PP NAG Ltd\&. .RE .PP \fBDate\fP .RS 4 December 2016 .RE .PP \fBFurther Details:\fP .RS 4 .PP .nf Level 2 Blas routine. The vector and matrix arguments are not referenced when N = 0, or M = 0 -- Written on 22-October-1986. Jack Dongarra, Argonne National Lab. Jeremy Du Croz, Nag Central Office. Sven Hammarling, Nag Central Office. Richard Hanson, Sandia National Labs. .fi .PP .RE .PP .SS "subroutine sspr (character UPLO, integer N, real ALPHA, real, dimension(*) X, integer INCX, real, dimension(*) AP)" .PP \fBSSPR\fP .PP \fBPurpose:\fP .RS 4 .PP .nf SSPR 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. .fi .PP .RE .PP \fBParameters\fP .RS 4 \fIUPLO\fP .PP .nf UPLO is CHARACTER*1 On entry, UPLO specifies whether the upper or lower triangular part of the matrix A is supplied in the packed array AP as follows: UPLO = 'U' or 'u' The upper triangular part of A is supplied in AP. UPLO = 'L' or 'l' The lower triangular part of A is supplied in AP. .fi .PP .br \fIN\fP .PP .nf N is INTEGER On entry, N specifies the order of the matrix A. N must be at least zero. .fi .PP .br \fIALPHA\fP .PP .nf ALPHA is REAL On entry, ALPHA specifies the scalar alpha. .fi .PP .br \fIX\fP .PP .nf X is REAL array, dimension at least ( 1 + ( n - 1 )*abs( INCX ) ). Before entry, the incremented array X must contain the n element vector x. .fi .PP .br \fIINCX\fP .PP .nf INCX is INTEGER On entry, INCX specifies the increment for the elements of X. INCX must not be zero. .fi .PP .br \fIAP\fP .PP .nf AP is REAL 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. .fi .PP .RE .PP \fBAuthor\fP .RS 4 Univ\&. of Tennessee .PP Univ\&. of California Berkeley .PP Univ\&. of Colorado Denver .PP NAG Ltd\&. .RE .PP \fBDate\fP .RS 4 December 2016 .RE .PP \fBFurther Details:\fP .RS 4 .PP .nf Level 2 Blas routine. -- Written on 22-October-1986. Jack Dongarra, Argonne National Lab. Jeremy Du Croz, Nag Central Office. Sven Hammarling, Nag Central Office. Richard Hanson, Sandia National Labs. .fi .PP .RE .PP .SS "subroutine sspr2 (character UPLO, integer N, real ALPHA, real, dimension(*) X, integer INCX, real, dimension(*) Y, integer INCY, real, dimension(*) AP)" .PP \fBSSPR2\fP .PP \fBPurpose:\fP .RS 4 .PP .nf SSPR2 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. .fi .PP .RE .PP \fBParameters\fP .RS 4 \fIUPLO\fP .PP .nf UPLO is CHARACTER*1 On entry, UPLO specifies whether the upper or lower triangular part of the matrix A is supplied in the packed array AP as follows: UPLO = 'U' or 'u' The upper triangular part of A is supplied in AP. UPLO = 'L' or 'l' The lower triangular part of A is supplied in AP. .fi .PP .br \fIN\fP .PP .nf N is INTEGER On entry, N specifies the order of the matrix A. N must be at least zero. .fi .PP .br \fIALPHA\fP .PP .nf ALPHA is REAL On entry, ALPHA specifies the scalar alpha. .fi .PP .br \fIX\fP .PP .nf X is REAL array, dimension at least ( 1 + ( n - 1 )*abs( INCX ) ). Before entry, the incremented array X must contain the n element vector x. .fi .PP .br \fIINCX\fP .PP .nf INCX is INTEGER On entry, INCX specifies the increment for the elements of X. INCX must not be zero. .fi .PP .br \fIY\fP .PP .nf Y is REAL array, dimension at least ( 1 + ( n - 1 )*abs( INCY ) ). Before entry, the incremented array Y must contain the n element vector y. .fi .PP .br \fIINCY\fP .PP .nf INCY is INTEGER On entry, INCY specifies the increment for the elements of Y. INCY must not be zero. .fi .PP .br \fIAP\fP .PP .nf AP is REAL 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. .fi .PP .RE .PP \fBAuthor\fP .RS 4 Univ\&. of Tennessee .PP Univ\&. of California Berkeley .PP Univ\&. of Colorado Denver .PP NAG Ltd\&. .RE .PP \fBDate\fP .RS 4 December 2016 .RE .PP \fBFurther Details:\fP .RS 4 .PP .nf Level 2 Blas routine. -- Written on 22-October-1986. Jack Dongarra, Argonne National Lab. Jeremy Du Croz, Nag Central Office. Sven Hammarling, Nag Central Office. Richard Hanson, Sandia National Labs. .fi .PP .RE .PP .SS "subroutine ssymv (character UPLO, integer N, real ALPHA, real, dimension(lda,*) A, integer LDA, real, dimension(*) X, integer INCX, real BETA, real, dimension(*) Y, integer INCY)" .PP \fBSSYMV\fP .PP \fBPurpose:\fP .RS 4 .PP .nf SSYMV 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. .fi .PP .RE .PP \fBParameters\fP .RS 4 \fIUPLO\fP .PP .nf UPLO is CHARACTER*1 On entry, UPLO specifies whether the upper or lower triangular part of the array A is to be referenced as follows: UPLO = 'U' or 'u' Only the upper triangular part of A is to be referenced. UPLO = 'L' or 'l' Only the lower triangular part of A is to be referenced. .fi .PP .br \fIN\fP .PP .nf N is INTEGER On entry, N specifies the order of the matrix A. N must be at least zero. .fi .PP .br \fIALPHA\fP .PP .nf ALPHA is REAL On entry, ALPHA specifies the scalar alpha. .fi .PP .br \fIA\fP .PP .nf A is REAL 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. .fi .PP .br \fILDA\fP .PP .nf LDA is INTEGER On entry, LDA specifies the first dimension of A as declared in the calling (sub) program. LDA must be at least max( 1, n ). .fi .PP .br \fIX\fP .PP .nf X is REAL array, dimension at least ( 1 + ( n - 1 )*abs( INCX ) ). Before entry, the incremented array X must contain the n element vector x. .fi .PP .br \fIINCX\fP .PP .nf INCX is INTEGER On entry, INCX specifies the increment for the elements of X. INCX must not be zero. .fi .PP .br \fIBETA\fP .PP .nf BETA is REAL On entry, BETA specifies the scalar beta. When BETA is supplied as zero then Y need not be set on input. .fi .PP .br \fIY\fP .PP .nf Y is REAL array, dimension at least ( 1 + ( n - 1 )*abs( INCY ) ). Before entry, the incremented array Y must contain the n element vector y. On exit, Y is overwritten by the updated vector y. .fi .PP .br \fIINCY\fP .PP .nf INCY is INTEGER On entry, INCY specifies the increment for the elements of Y. INCY must not be zero. .fi .PP .RE .PP \fBAuthor\fP .RS 4 Univ\&. of Tennessee .PP Univ\&. of California Berkeley .PP Univ\&. of Colorado Denver .PP NAG Ltd\&. .RE .PP \fBDate\fP .RS 4 December 2016 .RE .PP \fBFurther Details:\fP .RS 4 .PP .nf Level 2 Blas routine. The vector and matrix arguments are not referenced when N = 0, or M = 0 -- Written on 22-October-1986. Jack Dongarra, Argonne National Lab. Jeremy Du Croz, Nag Central Office. Sven Hammarling, Nag Central Office. Richard Hanson, Sandia National Labs. .fi .PP .RE .PP .SS "subroutine ssyr (character UPLO, integer N, real ALPHA, real, dimension(*) X, integer INCX, real, dimension(lda,*) A, integer LDA)" .PP \fBSSYR\fP .PP \fBPurpose:\fP .RS 4 .PP .nf SSYR 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. .fi .PP .RE .PP \fBParameters\fP .RS 4 \fIUPLO\fP .PP .nf UPLO is CHARACTER*1 On entry, UPLO specifies whether the upper or lower triangular part of the array A is to be referenced as follows: UPLO = 'U' or 'u' Only the upper triangular part of A is to be referenced. UPLO = 'L' or 'l' Only the lower triangular part of A is to be referenced. .fi .PP .br \fIN\fP .PP .nf N is INTEGER On entry, N specifies the order of the matrix A. N must be at least zero. .fi .PP .br \fIALPHA\fP .PP .nf ALPHA is REAL On entry, ALPHA specifies the scalar alpha. .fi .PP .br \fIX\fP .PP .nf X is REAL array, dimension at least ( 1 + ( n - 1 )*abs( INCX ) ). Before entry, the incremented array X must contain the n element vector x. .fi .PP .br \fIINCX\fP .PP .nf INCX is INTEGER On entry, INCX specifies the increment for the elements of X. INCX must not be zero. .fi .PP .br \fIA\fP .PP .nf A is REAL 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. .fi .PP .br \fILDA\fP .PP .nf LDA is INTEGER On entry, LDA specifies the first dimension of A as declared in the calling (sub) program. LDA must be at least max( 1, n ). .fi .PP .RE .PP \fBAuthor\fP .RS 4 Univ\&. of Tennessee .PP Univ\&. of California Berkeley .PP Univ\&. of Colorado Denver .PP NAG Ltd\&. .RE .PP \fBDate\fP .RS 4 December 2016 .RE .PP \fBFurther Details:\fP .RS 4 .PP .nf Level 2 Blas routine. -- Written on 22-October-1986. Jack Dongarra, Argonne National Lab. Jeremy Du Croz, Nag Central Office. Sven Hammarling, Nag Central Office. Richard Hanson, Sandia National Labs. .fi .PP .RE .PP .SS "subroutine ssyr2 (character UPLO, integer N, real ALPHA, real, dimension(*) X, integer INCX, real, dimension(*) Y, integer INCY, real, dimension(lda,*) A, integer LDA)" .PP \fBSSYR2\fP .PP \fBPurpose:\fP .RS 4 .PP .nf SSYR2 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. .fi .PP .RE .PP \fBParameters\fP .RS 4 \fIUPLO\fP .PP .nf UPLO is CHARACTER*1 On entry, UPLO specifies whether the upper or lower triangular part of the array A is to be referenced as follows: UPLO = 'U' or 'u' Only the upper triangular part of A is to be referenced. UPLO = 'L' or 'l' Only the lower triangular part of A is to be referenced. .fi .PP .br \fIN\fP .PP .nf N is INTEGER On entry, N specifies the order of the matrix A. N must be at least zero. .fi .PP .br \fIALPHA\fP .PP .nf ALPHA is REAL On entry, ALPHA specifies the scalar alpha. .fi .PP .br \fIX\fP .PP .nf X is REAL array, dimension at least ( 1 + ( n - 1 )*abs( INCX ) ). Before entry, the incremented array X must contain the n element vector x. .fi .PP .br \fIINCX\fP .PP .nf INCX is INTEGER On entry, INCX specifies the increment for the elements of X. INCX must not be zero. .fi .PP .br \fIY\fP .PP .nf Y is REAL array, dimension at least ( 1 + ( n - 1 )*abs( INCY ) ). Before entry, the incremented array Y must contain the n element vector y. .fi .PP .br \fIINCY\fP .PP .nf INCY is INTEGER On entry, INCY specifies the increment for the elements of Y. INCY must not be zero. .fi .PP .br \fIA\fP .PP .nf A is REAL 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. .fi .PP .br \fILDA\fP .PP .nf LDA is INTEGER On entry, LDA specifies the first dimension of A as declared in the calling (sub) program. LDA must be at least max( 1, n ). .fi .PP .RE .PP \fBAuthor\fP .RS 4 Univ\&. of Tennessee .PP Univ\&. of California Berkeley .PP Univ\&. of Colorado Denver .PP NAG Ltd\&. .RE .PP \fBDate\fP .RS 4 December 2016 .RE .PP \fBFurther Details:\fP .RS 4 .PP .nf Level 2 Blas routine. -- Written on 22-October-1986. Jack Dongarra, Argonne National Lab. Jeremy Du Croz, Nag Central Office. Sven Hammarling, Nag Central Office. Richard Hanson, Sandia National Labs. .fi .PP .RE .PP .SS "subroutine stbmv (character UPLO, character TRANS, character DIAG, integer N, integer K, real, dimension(lda,*) A, integer LDA, real, dimension(*) X, integer INCX)" .PP \fBSTBMV\fP .PP \fBPurpose:\fP .RS 4 .PP .nf STBMV 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. .fi .PP .RE .PP \fBParameters\fP .RS 4 \fIUPLO\fP .PP .nf UPLO is CHARACTER*1 On entry, UPLO specifies whether the matrix is an upper or lower triangular matrix as follows: UPLO = 'U' or 'u' A is an upper triangular matrix. UPLO = 'L' or 'l' A is a lower triangular matrix. .fi .PP .br \fITRANS\fP .PP .nf TRANS is CHARACTER*1 On entry, TRANS specifies the operation to be performed as follows: TRANS = 'N' or 'n' x := A*x. TRANS = 'T' or 't' x := A**T*x. TRANS = 'C' or 'c' x := A**T*x. .fi .PP .br \fIDIAG\fP .PP .nf DIAG is CHARACTER*1 On entry, DIAG specifies whether or not A is unit triangular as follows: DIAG = 'U' or 'u' A is assumed to be unit triangular. DIAG = 'N' or 'n' A is not assumed to be unit triangular. .fi .PP .br \fIN\fP .PP .nf N is INTEGER On entry, N specifies the order of the matrix A. N must be at least zero. .fi .PP .br \fIK\fP .PP .nf K is INTEGER On entry with UPLO = 'U' or 'u', K specifies the number of super-diagonals of the matrix A. On entry with UPLO = 'L' or 'l', K specifies the number of sub-diagonals of the matrix A. K must satisfy 0 .le. K. .fi .PP .br \fIA\fP .PP .nf A is REAL array, dimension ( LDA, N ) Before entry with UPLO = 'U' or 'u', the leading ( k + 1 ) by n part of the array A must contain the upper triangular band part of the matrix of coefficients, supplied column by column, with the leading diagonal of the matrix in row ( k + 1 ) of the array, the first super-diagonal starting at position 2 in row k, and so on. The top left k by k triangle of the array A is not referenced. The following program segment will transfer an upper triangular band matrix from conventional full matrix storage to band storage: DO 20, J = 1, N M = K + 1 - J DO 10, I = MAX( 1, J - K ), J A( M + I, J ) = matrix( I, J ) 10 CONTINUE 20 CONTINUE Before entry with UPLO = 'L' or 'l', the leading ( k + 1 ) by n part of the array A must contain the lower triangular band part of the matrix of coefficients, supplied column by column, with the leading diagonal of the matrix in row 1 of the array, the first sub-diagonal starting at position 1 in row 2, and so on. The bottom right k by k triangle of the array A is not referenced. The following program segment will transfer a lower triangular band matrix from conventional full matrix storage to band storage: DO 20, J = 1, N M = 1 - J DO 10, I = J, MIN( N, J + K ) A( M + I, J ) = matrix( I, J ) 10 CONTINUE 20 CONTINUE Note that when DIAG = 'U' or 'u' the elements of the array A corresponding to the diagonal elements of the matrix are not referenced, but are assumed to be unity. .fi .PP .br \fILDA\fP .PP .nf LDA is INTEGER On entry, LDA specifies the first dimension of A as declared in the calling (sub) program. LDA must be at least ( k + 1 ). .fi .PP .br \fIX\fP .PP .nf X is REAL array, dimension at least ( 1 + ( n - 1 )*abs( INCX ) ). Before entry, the incremented array X must contain the n element vector x. On exit, X is overwritten with the transformed vector x. .fi .PP .br \fIINCX\fP .PP .nf INCX is INTEGER On entry, INCX specifies the increment for the elements of X. INCX must not be zero. .fi .PP .RE .PP \fBAuthor\fP .RS 4 Univ\&. of Tennessee .PP Univ\&. of California Berkeley .PP Univ\&. of Colorado Denver .PP NAG Ltd\&. .RE .PP \fBDate\fP .RS 4 December 2016 .RE .PP \fBFurther Details:\fP .RS 4 .PP .nf Level 2 Blas routine. The vector and matrix arguments are not referenced when N = 0, or M = 0 -- Written on 22-October-1986. Jack Dongarra, Argonne National Lab. Jeremy Du Croz, Nag Central Office. Sven Hammarling, Nag Central Office. Richard Hanson, Sandia National Labs. .fi .PP .RE .PP .SS "subroutine stbsv (character UPLO, character TRANS, character DIAG, integer N, integer K, real, dimension(lda,*) A, integer LDA, real, dimension(*) X, integer INCX)" .PP \fBSTBSV\fP .PP \fBPurpose:\fP .RS 4 .PP .nf STBSV 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. .fi .PP .RE .PP \fBParameters\fP .RS 4 \fIUPLO\fP .PP .nf UPLO is CHARACTER*1 On entry, UPLO specifies whether the matrix is an upper or lower triangular matrix as follows: UPLO = 'U' or 'u' A is an upper triangular matrix. UPLO = 'L' or 'l' A is a lower triangular matrix. .fi .PP .br \fITRANS\fP .PP .nf TRANS is CHARACTER*1 On entry, TRANS specifies the equations to be solved as follows: TRANS = 'N' or 'n' A*x = b. TRANS = 'T' or 't' A**T*x = b. TRANS = 'C' or 'c' A**T*x = b. .fi .PP .br \fIDIAG\fP .PP .nf DIAG is CHARACTER*1 On entry, DIAG specifies whether or not A is unit triangular as follows: DIAG = 'U' or 'u' A is assumed to be unit triangular. DIAG = 'N' or 'n' A is not assumed to be unit triangular. .fi .PP .br \fIN\fP .PP .nf N is INTEGER On entry, N specifies the order of the matrix A. N must be at least zero. .fi .PP .br \fIK\fP .PP .nf K is INTEGER On entry with UPLO = 'U' or 'u', K specifies the number of super-diagonals of the matrix A. On entry with UPLO = 'L' or 'l', K specifies the number of sub-diagonals of the matrix A. K must satisfy 0 .le. K. .fi .PP .br \fIA\fP .PP .nf A is REAL array, dimension ( LDA, N ) Before entry with UPLO = 'U' or 'u', the leading ( k + 1 ) by n part of the array A must contain the upper triangular band part of the matrix of coefficients, supplied column by column, with the leading diagonal of the matrix in row ( k + 1 ) of the array, the first super-diagonal starting at position 2 in row k, and so on. The top left k by k triangle of the array A is not referenced. The following program segment will transfer an upper triangular band matrix from conventional full matrix storage to band storage: DO 20, J = 1, N M = K + 1 - J DO 10, I = MAX( 1, J - K ), J A( M + I, J ) = matrix( I, J ) 10 CONTINUE 20 CONTINUE Before entry with UPLO = 'L' or 'l', the leading ( k + 1 ) by n part of the array A must contain the lower triangular band part of the matrix of coefficients, supplied column by column, with the leading diagonal of the matrix in row 1 of the array, the first sub-diagonal starting at position 1 in row 2, and so on. The bottom right k by k triangle of the array A is not referenced. The following program segment will transfer a lower triangular band matrix from conventional full matrix storage to band storage: DO 20, J = 1, N M = 1 - J DO 10, I = J, MIN( N, J + K ) A( M + I, J ) = matrix( I, J ) 10 CONTINUE 20 CONTINUE Note that when DIAG = 'U' or 'u' the elements of the array A corresponding to the diagonal elements of the matrix are not referenced, but are assumed to be unity. .fi .PP .br \fILDA\fP .PP .nf LDA is INTEGER On entry, LDA specifies the first dimension of A as declared in the calling (sub) program. LDA must be at least ( k + 1 ). .fi .PP .br \fIX\fP .PP .nf X is REAL array, dimension at least ( 1 + ( n - 1 )*abs( INCX ) ). Before entry, the incremented array X must contain the n element right-hand side vector b. On exit, X is overwritten with the solution vector x. .fi .PP .br \fIINCX\fP .PP .nf INCX is INTEGER On entry, INCX specifies the increment for the elements of X. INCX must not be zero. .fi .PP .RE .PP \fBAuthor\fP .RS 4 Univ\&. of Tennessee .PP Univ\&. of California Berkeley .PP Univ\&. of Colorado Denver .PP NAG Ltd\&. .RE .PP \fBDate\fP .RS 4 December 2016 .RE .PP \fBFurther Details:\fP .RS 4 .PP .nf Level 2 Blas routine. -- Written on 22-October-1986. Jack Dongarra, Argonne National Lab. Jeremy Du Croz, Nag Central Office. Sven Hammarling, Nag Central Office. Richard Hanson, Sandia National Labs. .fi .PP .RE .PP .SS "subroutine stpmv (character UPLO, character TRANS, character DIAG, integer N, real, dimension(*) AP, real, dimension(*) X, integer INCX)" .PP \fBSTPMV\fP .PP \fBPurpose:\fP .RS 4 .PP .nf STPMV 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. .fi .PP .RE .PP \fBParameters\fP .RS 4 \fIUPLO\fP .PP .nf UPLO is CHARACTER*1 On entry, UPLO specifies whether the matrix is an upper or lower triangular matrix as follows: UPLO = 'U' or 'u' A is an upper triangular matrix. UPLO = 'L' or 'l' A is a lower triangular matrix. .fi .PP .br \fITRANS\fP .PP .nf TRANS is CHARACTER*1 On entry, TRANS specifies the operation to be performed as follows: TRANS = 'N' or 'n' x := A*x. TRANS = 'T' or 't' x := A**T*x. TRANS = 'C' or 'c' x := A**T*x. .fi .PP .br \fIDIAG\fP .PP .nf DIAG is CHARACTER*1 On entry, DIAG specifies whether or not A is unit triangular as follows: DIAG = 'U' or 'u' A is assumed to be unit triangular. DIAG = 'N' or 'n' A is not assumed to be unit triangular. .fi .PP .br \fIN\fP .PP .nf N is INTEGER On entry, N specifies the order of the matrix A. N must be at least zero. .fi .PP .br \fIAP\fP .PP .nf AP is REAL array, dimension at least ( ( n*( n + 1 ) )/2 ). Before entry with UPLO = 'U' or 'u', the array AP must contain the upper triangular matrix packed sequentially, column by column, so that AP( 1 ) contains a( 1, 1 ), AP( 2 ) and AP( 3 ) contain a( 1, 2 ) and a( 2, 2 ) respectively, and so on. Before entry with UPLO = 'L' or 'l', the array AP must contain the lower triangular matrix packed sequentially, column by column, so that AP( 1 ) contains a( 1, 1 ), AP( 2 ) and AP( 3 ) contain a( 2, 1 ) and a( 3, 1 ) respectively, and so on. Note that when DIAG = 'U' or 'u', the diagonal elements of A are not referenced, but are assumed to be unity. .fi .PP .br \fIX\fP .PP .nf X is REAL array, dimension at least ( 1 + ( n - 1 )*abs( INCX ) ). Before entry, the incremented array X must contain the n element vector x. On exit, X is overwritten with the transformed vector x. .fi .PP .br \fIINCX\fP .PP .nf INCX is INTEGER On entry, INCX specifies the increment for the elements of X. INCX must not be zero. .fi .PP .RE .PP \fBAuthor\fP .RS 4 Univ\&. of Tennessee .PP Univ\&. of California Berkeley .PP Univ\&. of Colorado Denver .PP NAG Ltd\&. .RE .PP \fBDate\fP .RS 4 December 2016 .RE .PP \fBFurther Details:\fP .RS 4 .PP .nf Level 2 Blas routine. The vector and matrix arguments are not referenced when N = 0, or M = 0 -- Written on 22-October-1986. Jack Dongarra, Argonne National Lab. Jeremy Du Croz, Nag Central Office. Sven Hammarling, Nag Central Office. Richard Hanson, Sandia National Labs. .fi .PP .RE .PP .SS "subroutine stpsv (character UPLO, character TRANS, character DIAG, integer N, real, dimension(*) AP, real, dimension(*) X, integer INCX)" .PP \fBSTPSV\fP .PP \fBPurpose:\fP .RS 4 .PP .nf STPSV 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. .fi .PP .RE .PP \fBParameters\fP .RS 4 \fIUPLO\fP .PP .nf UPLO is CHARACTER*1 On entry, UPLO specifies whether the matrix is an upper or lower triangular matrix as follows: UPLO = 'U' or 'u' A is an upper triangular matrix. UPLO = 'L' or 'l' A is a lower triangular matrix. .fi .PP .br \fITRANS\fP .PP .nf TRANS is CHARACTER*1 On entry, TRANS specifies the equations to be solved as follows: TRANS = 'N' or 'n' A*x = b. TRANS = 'T' or 't' A**T*x = b. TRANS = 'C' or 'c' A**T*x = b. .fi .PP .br \fIDIAG\fP .PP .nf DIAG is CHARACTER*1 On entry, DIAG specifies whether or not A is unit triangular as follows: DIAG = 'U' or 'u' A is assumed to be unit triangular. DIAG = 'N' or 'n' A is not assumed to be unit triangular. .fi .PP .br \fIN\fP .PP .nf N is INTEGER On entry, N specifies the order of the matrix A. N must be at least zero. .fi .PP .br \fIAP\fP .PP .nf AP is REAL array, dimension at least ( ( n*( n + 1 ) )/2 ). Before entry with UPLO = 'U' or 'u', the array AP must contain the upper triangular matrix packed sequentially, column by column, so that AP( 1 ) contains a( 1, 1 ), AP( 2 ) and AP( 3 ) contain a( 1, 2 ) and a( 2, 2 ) respectively, and so on. Before entry with UPLO = 'L' or 'l', the array AP must contain the lower triangular matrix packed sequentially, column by column, so that AP( 1 ) contains a( 1, 1 ), AP( 2 ) and AP( 3 ) contain a( 2, 1 ) and a( 3, 1 ) respectively, and so on. Note that when DIAG = 'U' or 'u', the diagonal elements of A are not referenced, but are assumed to be unity. .fi .PP .br \fIX\fP .PP .nf X is REAL array, dimension at least ( 1 + ( n - 1 )*abs( INCX ) ). Before entry, the incremented array X must contain the n element right-hand side vector b. On exit, X is overwritten with the solution vector x. .fi .PP .br \fIINCX\fP .PP .nf INCX is INTEGER On entry, INCX specifies the increment for the elements of X. INCX must not be zero. .fi .PP .RE .PP \fBAuthor\fP .RS 4 Univ\&. of Tennessee .PP Univ\&. of California Berkeley .PP Univ\&. of Colorado Denver .PP NAG Ltd\&. .RE .PP \fBDate\fP .RS 4 December 2016 .RE .PP \fBFurther Details:\fP .RS 4 .PP .nf Level 2 Blas routine. -- Written on 22-October-1986. Jack Dongarra, Argonne National Lab. Jeremy Du Croz, Nag Central Office. Sven Hammarling, Nag Central Office. Richard Hanson, Sandia National Labs. .fi .PP .RE .PP .SS "subroutine strmv (character UPLO, character TRANS, character DIAG, integer N, real, dimension(lda,*) A, integer LDA, real, dimension(*) X, integer INCX)" .PP \fBSTRMV\fP .PP \fBPurpose:\fP .RS 4 .PP .nf STRMV 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. .fi .PP .RE .PP \fBParameters\fP .RS 4 \fIUPLO\fP .PP .nf UPLO is CHARACTER*1 On entry, UPLO specifies whether the matrix is an upper or lower triangular matrix as follows: UPLO = 'U' or 'u' A is an upper triangular matrix. UPLO = 'L' or 'l' A is a lower triangular matrix. .fi .PP .br \fITRANS\fP .PP .nf TRANS is CHARACTER*1 On entry, TRANS specifies the operation to be performed as follows: TRANS = 'N' or 'n' x := A*x. TRANS = 'T' or 't' x := A**T*x. TRANS = 'C' or 'c' x := A**T*x. .fi .PP .br \fIDIAG\fP .PP .nf DIAG is CHARACTER*1 On entry, DIAG specifies whether or not A is unit triangular as follows: DIAG = 'U' or 'u' A is assumed to be unit triangular. DIAG = 'N' or 'n' A is not assumed to be unit triangular. .fi .PP .br \fIN\fP .PP .nf N is INTEGER On entry, N specifies the order of the matrix A. N must be at least zero. .fi .PP .br \fIA\fP .PP .nf A is REAL array, dimension ( LDA, N ) Before entry with UPLO = 'U' or 'u', the leading n by n upper triangular part of the array A must contain the upper triangular matrix and the strictly lower triangular part of A is not referenced. Before entry with UPLO = 'L' or 'l', the leading n by n lower triangular part of the array A must contain the lower triangular matrix and the strictly upper triangular part of A is not referenced. Note that when DIAG = 'U' or 'u', the diagonal elements of A are not referenced either, but are assumed to be unity. .fi .PP .br \fILDA\fP .PP .nf LDA is INTEGER On entry, LDA specifies the first dimension of A as declared in the calling (sub) program. LDA must be at least max( 1, n ). .fi .PP .br \fIX\fP .PP .nf X is REAL array, dimension at least ( 1 + ( n - 1 )*abs( INCX ) ). Before entry, the incremented array X must contain the n element vector x. On exit, X is overwritten with the transformed vector x. .fi .PP .br \fIINCX\fP .PP .nf INCX is INTEGER On entry, INCX specifies the increment for the elements of X. INCX must not be zero. .fi .PP .RE .PP \fBAuthor\fP .RS 4 Univ\&. of Tennessee .PP Univ\&. of California Berkeley .PP Univ\&. of Colorado Denver .PP NAG Ltd\&. .RE .PP \fBDate\fP .RS 4 December 2016 .RE .PP \fBFurther Details:\fP .RS 4 .PP .nf Level 2 Blas routine. The vector and matrix arguments are not referenced when N = 0, or M = 0 -- Written on 22-October-1986. Jack Dongarra, Argonne National Lab. Jeremy Du Croz, Nag Central Office. Sven Hammarling, Nag Central Office. Richard Hanson, Sandia National Labs. .fi .PP .RE .PP .SS "subroutine strsv (character UPLO, character TRANS, character DIAG, integer N, real, dimension(lda,*) A, integer LDA, real, dimension(*) X, integer INCX)" .PP \fBSTRSV\fP .PP \fBPurpose:\fP .RS 4 .PP .nf STRSV 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. No test for singularity or near-singularity is included in this routine. Such tests must be performed before calling this routine. .fi .PP .RE .PP \fBParameters\fP .RS 4 \fIUPLO\fP .PP .nf UPLO is CHARACTER*1 On entry, UPLO specifies whether the matrix is an upper or lower triangular matrix as follows: UPLO = 'U' or 'u' A is an upper triangular matrix. UPLO = 'L' or 'l' A is a lower triangular matrix. .fi .PP .br \fITRANS\fP .PP .nf TRANS is CHARACTER*1 On entry, TRANS specifies the equations to be solved as follows: TRANS = 'N' or 'n' A*x = b. TRANS = 'T' or 't' A**T*x = b. TRANS = 'C' or 'c' A**T*x = b. .fi .PP .br \fIDIAG\fP .PP .nf DIAG is CHARACTER*1 On entry, DIAG specifies whether or not A is unit triangular as follows: DIAG = 'U' or 'u' A is assumed to be unit triangular. DIAG = 'N' or 'n' A is not assumed to be unit triangular. .fi .PP .br \fIN\fP .PP .nf N is INTEGER On entry, N specifies the order of the matrix A. N must be at least zero. .fi .PP .br \fIA\fP .PP .nf A is REAL array, dimension ( LDA, N ) Before entry with UPLO = 'U' or 'u', the leading n by n upper triangular part of the array A must contain the upper triangular matrix and the strictly lower triangular part of A is not referenced. Before entry with UPLO = 'L' or 'l', the leading n by n lower triangular part of the array A must contain the lower triangular matrix and the strictly upper triangular part of A is not referenced. Note that when DIAG = 'U' or 'u', the diagonal elements of A are not referenced either, but are assumed to be unity. .fi .PP .br \fILDA\fP .PP .nf LDA is INTEGER On entry, LDA specifies the first dimension of A as declared in the calling (sub) program. LDA must be at least max( 1, n ). .fi .PP .br \fIX\fP .PP .nf X is REAL array, dimension at least ( 1 + ( n - 1 )*abs( INCX ) ). Before entry, the incremented array X must contain the n element right-hand side vector b. On exit, X is overwritten with the solution vector x. .fi .PP .br \fIINCX\fP .PP .nf INCX is INTEGER On entry, INCX specifies the increment for the elements of X. INCX must not be zero. .fi .PP .RE .PP \fBAuthor\fP .RS 4 Univ\&. of Tennessee .PP Univ\&. of California Berkeley .PP Univ\&. of Colorado Denver .PP NAG Ltd\&. .RE .PP \fBDate\fP .RS 4 December 2016 .RE .PP \fBFurther Details:\fP .RS 4 .PP .nf Level 2 Blas routine. -- Written on 22-October-1986. Jack Dongarra, Argonne National Lab. Jeremy Du Croz, Nag Central Office. Sven Hammarling, Nag Central Office. Richard Hanson, Sandia National Labs. .fi .PP .RE .PP .SH "Author" .PP Generated automatically by Doxygen for LAPACK from the source code\&.