--- name: "single_blas_level2" section: "3" platform: "linux" package: "liblapack-doc" version: "3.11.0-2" release: "Debian 12 (Bookworm)" platforms: ["linux"] url: "https://new.carta.tech/single_blas_level2" --- # single\_blas\_level2(3) real ## NAME single\_blas\_level2 - real ## SYNOPSIS ### Functions subroutine **sgbmv** (TRANS, M, N, KL, KU, ALPHA, A, LDA, X, INCX, BETA, Y, INCY)\ **SGBMV** subroutine **sgemv** (TRANS, M, N, ALPHA, A, LDA, X, INCX, BETA, Y, INCY)\ **SGEMV** subroutine **sger** (M, N, ALPHA, X, INCX, Y, INCY, A, LDA)\ **SGER** subroutine **ssbmv** (UPLO, N, K, ALPHA, A, LDA, X, INCX, BETA, Y, INCY)\ **SSBMV** subroutine **sspmv** (UPLO, N, ALPHA, AP, X, INCX, BETA, Y, INCY)\ **SSPMV** subroutine **sspr** (UPLO, N, ALPHA, X, INCX, AP)\ **SSPR** subroutine **sspr2** (UPLO, N, ALPHA, X, INCX, Y, INCY, AP)\ **SSPR2** subroutine **ssymv** (UPLO, N, ALPHA, A, LDA, X, INCX, BETA, Y, INCY)\ **SSYMV** subroutine **ssyr** (UPLO, N, ALPHA, X, INCX, A, LDA)\ **SSYR** subroutine **ssyr2** (UPLO, N, ALPHA, X, INCX, Y, INCY, A, LDA)\ **SSYR2** subroutine **stbmv** (UPLO, TRANS, DIAG, N, K, A, LDA, X, INCX)\ **STBMV** subroutine **stbsv** (UPLO, TRANS, DIAG, N, K, A, LDA, X, INCX)\ **STBSV** subroutine **stpmv** (UPLO, TRANS, DIAG, N, AP, X, INCX)\ **STPMV** subroutine **stpsv** (UPLO, TRANS, DIAG, N, AP, X, INCX)\ **STPSV** subroutine **strmv** (UPLO, TRANS, DIAG, N, A, LDA, X, INCX)\ **STRMV** subroutine **strsv** (UPLO, TRANS, DIAG, N, A, LDA, X, INCX)\ **STRSV** ## Detailed Description This is the group of real LEVEL 2 BLAS routines. ## Function Documentation ### 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) **SGBMV** **Purpose:** ``` 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. ``` **Parameters** *TRANS* ``` TRANS is CHARACTER*1 On entry, TRANS specifies the operation to be performed as follows: TRANS = 'N' or 'n' y := alpha*A*x + beta*y. TRANS = 'T' or 't' y := alpha*A**T*x + beta*y. TRANS = 'C' or 'c' y := alpha*A**T*x + beta*y. ``` *M* ``` M is INTEGER On entry, M specifies the number of rows of the matrix A. M must be at least zero. ``` *N* ``` N is INTEGER On entry, N specifies the number of columns of the matrix A. N must be at least zero. ``` *KL* ``` KL is INTEGER On entry, KL specifies the number of sub-diagonals of the matrix A. KL must satisfy 0 .le. KL. ``` *KU* ``` KU is INTEGER On entry, KU specifies the number of super-diagonals of the matrix A. KU must satisfy 0 .le. KU. ``` *ALPHA* ``` ALPHA is REAL On entry, ALPHA specifies the scalar alpha. ``` *A* ``` 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 ``` *LDA* ``` LDA is INTEGER On entry, LDA specifies the first dimension of A as declared in the calling (sub) program. LDA must be at least ( kl + ku + 1 ). ``` *X* ``` X is 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. ``` *INCX* ``` INCX is INTEGER On entry, INCX specifies the increment for the elements of X. INCX must not be zero. ``` *BETA* ``` BETA is REAL On entry, BETA specifies the scalar beta. When BETA is supplied as zero then Y need not be set on input. ``` *Y* ``` 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. ``` *INCY* ``` INCY 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. **Further Details:** ``` Level 2 Blas routine. The vector and matrix arguments are not referenced when N = 0, or M = 0 -- Written on 22-October-1986. Jack Dongarra, Argonne National Lab. Jeremy Du Croz, Nag Central Office. Sven Hammarling, Nag Central Office. Richard Hanson, Sandia National Labs. ``` ### subroutine 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) **SGEMV** **Purpose:** ``` 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. ``` **Parameters** *TRANS* ``` TRANS is CHARACTER*1 On entry, TRANS specifies the operation to be performed as follows: TRANS = 'N' or 'n' y := alpha*A*x + beta*y. TRANS = 'T' or 't' y := alpha*A**T*x + beta*y. TRANS = 'C' or 'c' y := alpha*A**T*x + beta*y. ``` *M* ``` M is INTEGER On entry, M specifies the number of rows of the matrix A. M must be at least zero. ``` *N* ``` N is INTEGER On entry, N specifies the number of columns of the matrix A. N must be at least zero. ``` *ALPHA* ``` ALPHA is REAL On entry, ALPHA specifies the scalar alpha. ``` *A* ``` 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. ``` *LDA* ``` LDA is INTEGER On entry, LDA specifies the first dimension of A as declared in the calling (sub) program. LDA must be at least max( 1, m ). ``` *X* ``` X is 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. ``` *INCX* ``` INCX is INTEGER On entry, INCX specifies the increment for the elements of X. INCX must not be zero. ``` *BETA* ``` BETA is REAL On entry, BETA specifies the scalar beta. When BETA is supplied as zero then Y need not be set on input. ``` *Y* ``` 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. ``` *INCY* ``` INCY 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. **Further Details:** ``` Level 2 Blas routine. The vector and matrix arguments are not referenced when N = 0, or M = 0 -- Written on 22-October-1986. Jack Dongarra, Argonne National Lab. Jeremy Du Croz, Nag Central Office. Sven Hammarling, Nag Central Office. Richard Hanson, Sandia National Labs. ``` ### subroutine sger (integer M, integer N, real ALPHA, real, dimension(\*) X, integer INCX, real, dimension(\*) Y, integer INCY, real, dimension(lda,\*) A, integer LDA) **SGER** **Purpose:** ``` 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. ``` **Parameters** *M* ``` M is INTEGER On entry, M specifies the number of rows of the matrix A. M must be at least zero. ``` *N* ``` N is INTEGER On entry, N specifies the number of columns of the matrix A. N must be at least zero. ``` *ALPHA* ``` ALPHA is REAL On entry, ALPHA specifies the scalar alpha. ``` *X* ``` 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. ``` *INCX* ``` INCX is INTEGER On entry, INCX specifies the increment for the elements of X. INCX must not be zero. ``` *Y* ``` 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. ``` *INCY* ``` INCY is INTEGER On entry, INCY specifies the increment for the elements of Y. INCY must not be zero. ``` *A* ``` 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. ``` *LDA* ``` LDA is INTEGER On entry, LDA specifies the first dimension of A as declared in the calling (sub) program. LDA must be at least max( 1, m ). ``` **Author** Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. **Further Details:** ``` Level 2 Blas routine. -- Written on 22-October-1986. Jack Dongarra, Argonne National Lab. Jeremy Du Croz, Nag Central Office. Sven Hammarling, Nag Central Office. Richard Hanson, Sandia National Labs. ``` ### subroutine 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) **SSBMV** **Purpose:** ``` 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. ``` **Parameters** *UPLO* ``` UPLO is CHARACTER*1 On entry, UPLO specifies whether the upper or lower triangular part of the band matrix A is being supplied as follows: UPLO = 'U' or 'u' The upper triangular part of A is being supplied. UPLO = 'L' or 'l' The lower triangular part of A is being supplied. ``` *N* ``` N is INTEGER On entry, N specifies the order of the matrix A. N must be at least zero. ``` *K* ``` K is INTEGER On entry, K specifies the number of super-diagonals of the matrix A. K must satisfy 0 .le. K. ``` *ALPHA* ``` ALPHA is REAL On entry, ALPHA specifies the scalar alpha. ``` *A* ``` 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 ``` *LDA* ``` LDA is INTEGER On entry, LDA specifies the first dimension of A as declared in the calling (sub) program. LDA must be at least ( k + 1 ). ``` *X* ``` X is REAL array, dimension at least ( 1 + ( n - 1 )*abs( INCX ) ). Before entry, the incremented array X must contain the vector x. ``` *INCX* ``` INCX is INTEGER On entry, INCX specifies the increment for the elements of X. INCX must not be zero. ``` *BETA* ``` BETA is REAL On entry, BETA specifies the scalar beta. ``` *Y* ``` 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. ``` *INCY* ``` INCY 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. **Further Details:** ``` Level 2 Blas routine. The vector and matrix arguments are not referenced when N = 0, or M = 0 -- Written on 22-October-1986. Jack Dongarra, Argonne National Lab. Jeremy Du Croz, Nag Central Office. Sven Hammarling, Nag Central Office. Richard Hanson, Sandia National Labs. ``` ### subroutine sspmv (character UPLO, integer N, real ALPHA, real, dimension(\*) AP, real, dimension(\*) X, integer INCX, real BETA, real, dimension(\*) Y, integer INCY) **SSPMV** **Purpose:** ``` 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. ``` **Parameters** *UPLO* ``` UPLO is CHARACTER*1 On entry, UPLO specifies whether the upper or lower triangular part of the matrix A is supplied in the packed array AP as follows: UPLO = 'U' or 'u' The upper triangular part of A is supplied in AP. UPLO = 'L' or 'l' The lower triangular part of A is supplied in AP. ``` *N* ``` N is INTEGER On entry, N specifies the order of the matrix A. N must be at least zero. ``` *ALPHA* ``` ALPHA is REAL On entry, ALPHA specifies the scalar alpha. ``` *AP* ``` 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. ``` *X* ``` 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. ``` *INCX* ``` INCX is INTEGER On entry, INCX specifies the increment for the elements of X. INCX must not be zero. ``` *BETA* ``` BETA is REAL On entry, BETA specifies the scalar beta. When BETA is supplied as zero then Y need not be set on input. ``` *Y* ``` 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. ``` *INCY* ``` INCY 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. **Further Details:** ``` Level 2 Blas routine. The vector and matrix arguments are not referenced when N = 0, or M = 0 -- Written on 22-October-1986. Jack Dongarra, Argonne National Lab. Jeremy Du Croz, Nag Central Office. Sven Hammarling, Nag Central Office. Richard Hanson, Sandia National Labs. ``` ### subroutine sspr (character UPLO, integer N, real ALPHA, real, dimension(\*) X, integer INCX, real, dimension(\*) AP) **SSPR** **Purpose:** ``` 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. ``` **Parameters** *UPLO* ``` UPLO is CHARACTER*1 On entry, UPLO specifies whether the upper or lower triangular part of the matrix A is supplied in the packed array AP as follows: UPLO = 'U' or 'u' The upper triangular part of A is supplied in AP. UPLO = 'L' or 'l' The lower triangular part of A is supplied in AP. ``` *N* ``` N is INTEGER On entry, N specifies the order of the matrix A. N must be at least zero. ``` *ALPHA* ``` ALPHA is REAL On entry, ALPHA specifies the scalar alpha. ``` *X* ``` 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. ``` *INCX* ``` INCX is INTEGER On entry, INCX specifies the increment for the elements of X. INCX must not be zero. ``` *AP* ``` 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. ``` **Author** Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. **Further Details:** ``` Level 2 Blas routine. -- Written on 22-October-1986. Jack Dongarra, Argonne National Lab. Jeremy Du Croz, Nag Central Office. Sven Hammarling, Nag Central Office. Richard Hanson, Sandia National Labs. ``` ### subroutine sspr2 (character UPLO, integer N, real ALPHA, real, dimension(\*) X, integer INCX, real, dimension(\*) Y, integer INCY, real, dimension(\*) AP) **SSPR2** **Purpose:** ``` 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. ``` **Parameters** *UPLO* ``` UPLO is CHARACTER*1 On entry, UPLO specifies whether the upper or lower triangular part of the matrix A is supplied in the packed array AP as follows: UPLO = 'U' or 'u' The upper triangular part of A is supplied in AP. UPLO = 'L' or 'l' The lower triangular part of A is supplied in AP. ``` *N* ``` N is INTEGER On entry, N specifies the order of the matrix A. N must be at least zero. ``` *ALPHA* ``` ALPHA is REAL On entry, ALPHA specifies the scalar alpha. ``` *X* ``` 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. ``` *INCX* ``` INCX is INTEGER On entry, INCX specifies the increment for the elements of X. INCX must not be zero. ``` *Y* ``` 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. ``` *INCY* ``` INCY is INTEGER On entry, INCY specifies the increment for the elements of Y. INCY must not be zero. ``` *AP* ``` 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. ``` **Author** Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. **Further Details:** ``` Level 2 Blas routine. -- Written on 22-October-1986. Jack Dongarra, Argonne National Lab. Jeremy Du Croz, Nag Central Office. Sven Hammarling, Nag Central Office. Richard Hanson, Sandia National Labs. ``` ### subroutine 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) **SSYMV** **Purpose:** ``` 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. ``` **Parameters** *UPLO* ``` UPLO is CHARACTER*1 On entry, UPLO specifies whether the upper or lower triangular part of the array A is to be referenced as follows: UPLO = 'U' or 'u' Only the upper triangular part of A is to be referenced. UPLO = 'L' or 'l' Only the lower triangular part of A is to be referenced. ``` *N* ``` N is INTEGER On entry, N specifies the order of the matrix A. N must be at least zero. ``` *ALPHA* ``` ALPHA is REAL On entry, ALPHA specifies the scalar alpha. ``` *A* ``` 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. ``` *LDA* ``` LDA is INTEGER On entry, LDA specifies the first dimension of A as declared in the calling (sub) program. LDA must be at least max( 1, n ). ``` *X* ``` X is REAL array, dimension at least ( 1 + ( n - 1 )*abs( INCX ) ). Before entry, the incremented array X must contain the n element vector x. ``` *INCX* ``` INCX is INTEGER On entry, INCX specifies the increment for the elements of X. INCX must not be zero. ``` *BETA* ``` BETA is REAL On entry, BETA specifies the scalar beta. When BETA is supplied as zero then Y need not be set on input. ``` *Y* ``` 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. ``` *INCY* ``` INCY 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. **Further Details:** ``` Level 2 Blas routine. The vector and matrix arguments are not referenced when N = 0, or M = 0 -- Written on 22-October-1986. Jack Dongarra, Argonne National Lab. Jeremy Du Croz, Nag Central Office. Sven Hammarling, Nag Central Office. Richard Hanson, Sandia National Labs. ``` ### subroutine ssyr (character UPLO, integer N, real ALPHA, real, dimension(\*) X, integer INCX, real, dimension(lda,\*) A, integer LDA) **SSYR** **Purpose:** ``` 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. ``` **Parameters** *UPLO* ``` UPLO is CHARACTER*1 On entry, UPLO specifies whether the upper or lower triangular part of the array A is to be referenced as follows: UPLO = 'U' or 'u' Only the upper triangular part of A is to be referenced. UPLO = 'L' or 'l' Only the lower triangular part of A is to be referenced. ``` *N* ``` N is INTEGER On entry, N specifies the order of the matrix A. N must be at least zero. ``` *ALPHA* ``` ALPHA is REAL On entry, ALPHA specifies the scalar alpha. ``` *X* ``` 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. ``` *INCX* ``` INCX is INTEGER On entry, INCX specifies the increment for the elements of X. INCX must not be zero. ``` *A* ``` 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. ``` *LDA* ``` LDA is INTEGER On entry, LDA specifies the first dimension of A as declared in the calling (sub) program. LDA must be at least max( 1, n ). ``` **Author** Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. **Further Details:** ``` Level 2 Blas routine. -- Written on 22-October-1986. Jack Dongarra, Argonne National Lab. Jeremy Du Croz, Nag Central Office. Sven Hammarling, Nag Central Office. Richard Hanson, Sandia National Labs. ``` ### subroutine ssyr2 (character UPLO, integer N, real ALPHA, real, dimension(\*) X, integer INCX, real, dimension(\*) Y, integer INCY, real, dimension(lda,\*) A, integer LDA) **SSYR2** **Purpose:** ``` 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. ``` **Parameters** *UPLO* ``` UPLO is CHARACTER*1 On entry, UPLO specifies whether the upper or lower triangular part of the array A is to be referenced as follows: UPLO = 'U' or 'u' Only the upper triangular part of A is to be referenced. UPLO = 'L' or 'l' Only the lower triangular part of A is to be referenced. ``` *N* ``` N is INTEGER On entry, N specifies the order of the matrix A. N must be at least zero. ``` *ALPHA* ``` ALPHA is REAL On entry, ALPHA specifies the scalar alpha. ``` *X* ``` 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. ``` *INCX* ``` INCX is INTEGER On entry, INCX specifies the increment for the elements of X. INCX must not be zero. ``` *Y* ``` 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. ``` *INCY* ``` INCY is INTEGER On entry, INCY specifies the increment for the elements of Y. INCY must not be zero. ``` *A* ``` 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. ``` *LDA* ``` LDA is INTEGER On entry, LDA specifies the first dimension of A as declared in the calling (sub) program. LDA must be at least max( 1, n ). ``` **Author** Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. **Further Details:** ``` Level 2 Blas routine. -- Written on 22-October-1986. Jack Dongarra, Argonne National Lab. Jeremy Du Croz, Nag Central Office. Sven Hammarling, Nag Central Office. Richard Hanson, Sandia National Labs. ``` ### subroutine stbmv (character UPLO, character TRANS, character DIAG, integer N, integer K, real, dimension(lda,\*) A, integer LDA, real, dimension(\*) X, integer INCX) **STBMV** **Purpose:** ``` 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. ``` **Parameters** *UPLO* ``` UPLO is CHARACTER*1 On entry, UPLO specifies whether the matrix is an upper or lower triangular matrix as follows: UPLO = 'U' or 'u' A is an upper triangular matrix. UPLO = 'L' or 'l' A is a lower triangular matrix. ``` *TRANS* ``` TRANS is CHARACTER*1 On entry, TRANS specifies the operation to be performed as follows: TRANS = 'N' or 'n' x := A*x. TRANS = 'T' or 't' x := A**T*x. TRANS = 'C' or 'c' x := A**T*x. ``` *DIAG* ``` DIAG is CHARACTER*1 On entry, DIAG specifies whether or not A is unit triangular as follows: DIAG = 'U' or 'u' A is assumed to be unit triangular. DIAG = 'N' or 'n' A is not assumed to be unit triangular. ``` *N* ``` N is INTEGER On entry, N specifies the order of the matrix A. N must be at least zero. ``` *K* ``` K is INTEGER On entry with UPLO = 'U' or 'u', K specifies the number of super-diagonals of the matrix A. On entry with UPLO = 'L' or 'l', K specifies the number of sub-diagonals of the matrix A. K must satisfy 0 .le. K. ``` *A* ``` A is 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. ``` *LDA* ``` LDA is INTEGER On entry, LDA specifies the first dimension of A as declared in the calling (sub) program. LDA must be at least ( k + 1 ). ``` *X* ``` X is 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. ``` *INCX* ``` INCX 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. **Further Details:** ``` Level 2 Blas routine. The vector and matrix arguments are not referenced when N = 0, or M = 0 -- Written on 22-October-1986. Jack Dongarra, Argonne National Lab. Jeremy Du Croz, Nag Central Office. Sven Hammarling, Nag Central Office. Richard Hanson, Sandia National Labs. ``` ### subroutine stbsv (character UPLO, character TRANS, character DIAG, integer N, integer K, real, dimension(lda,\*) A, integer LDA, real, dimension(\*) X, integer INCX) **STBSV** **Purpose:** ``` 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. ``` **Parameters** *UPLO* ``` UPLO is CHARACTER*1 On entry, UPLO specifies whether the matrix is an upper or lower triangular matrix as follows: UPLO = 'U' or 'u' A is an upper triangular matrix. UPLO = 'L' or 'l' A is a lower triangular matrix. ``` *TRANS* ``` TRANS is CHARACTER*1 On entry, TRANS specifies the equations to be solved as follows: TRANS = 'N' or 'n' A*x = b. TRANS = 'T' or 't' A**T*x = b. TRANS = 'C' or 'c' A**T*x = b. ``` *DIAG* ``` DIAG is CHARACTER*1 On entry, DIAG specifies whether or not A is unit triangular as follows: DIAG = 'U' or 'u' A is assumed to be unit triangular. DIAG = 'N' or 'n' A is not assumed to be unit triangular. ``` *N* ``` N is INTEGER On entry, N specifies the order of the matrix A. N must be at least zero. ``` *K* ``` K is INTEGER On entry with UPLO = 'U' or 'u', K specifies the number of super-diagonals of the matrix A. On entry with UPLO = 'L' or 'l', K specifies the number of sub-diagonals of the matrix A. K must satisfy 0 .le. K. ``` *A* ``` A is 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. ``` *LDA* ``` LDA is INTEGER On entry, LDA specifies the first dimension of A as declared in the calling (sub) program. LDA must be at least ( k + 1 ). ``` *X* ``` X is 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. ``` *INCX* ``` INCX 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. **Further Details:** ``` Level 2 Blas routine. -- Written on 22-October-1986. Jack Dongarra, Argonne National Lab. Jeremy Du Croz, Nag Central Office. Sven Hammarling, Nag Central Office. Richard Hanson, Sandia National Labs. ``` ### subroutine stpmv (character UPLO, character TRANS, character DIAG, integer N, real, dimension(\*) AP, real, dimension(\*) X, integer INCX) **STPMV** **Purpose:** ``` 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. ``` **Parameters** *UPLO* ``` UPLO is CHARACTER*1 On entry, UPLO specifies whether the matrix is an upper or lower triangular matrix as follows: UPLO = 'U' or 'u' A is an upper triangular matrix. UPLO = 'L' or 'l' A is a lower triangular matrix. ``` *TRANS* ``` TRANS is CHARACTER*1 On entry, TRANS specifies the operation to be performed as follows: TRANS = 'N' or 'n' x := A*x. TRANS = 'T' or 't' x := A**T*x. TRANS = 'C' or 'c' x := A**T*x. ``` *DIAG* ``` DIAG is CHARACTER*1 On entry, DIAG specifies whether or not A is unit triangular as follows: DIAG = 'U' or 'u' A is assumed to be unit triangular. DIAG = 'N' or 'n' A is not assumed to be unit triangular. ``` *N* ``` N is INTEGER On entry, N specifies the order of the matrix A. N must be at least zero. ``` *AP* ``` AP is 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. ``` *X* ``` 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. ``` *INCX* ``` INCX 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. **Further Details:** ``` Level 2 Blas routine. The vector and matrix arguments are not referenced when N = 0, or M = 0 -- Written on 22-October-1986. Jack Dongarra, Argonne National Lab. Jeremy Du Croz, Nag Central Office. Sven Hammarling, Nag Central Office. Richard Hanson, Sandia National Labs. ``` ### subroutine stpsv (character UPLO, character TRANS, character DIAG, integer N, real, dimension(\*) AP, real, dimension(\*) X, integer INCX) **STPSV** **Purpose:** ``` 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. ``` **Parameters** *UPLO* ``` UPLO is CHARACTER*1 On entry, UPLO specifies whether the matrix is an upper or lower triangular matrix as follows: UPLO = 'U' or 'u' A is an upper triangular matrix. UPLO = 'L' or 'l' A is a lower triangular matrix. ``` *TRANS* ``` TRANS is CHARACTER*1 On entry, TRANS specifies the equations to be solved as follows: TRANS = 'N' or 'n' A*x = b. TRANS = 'T' or 't' A**T*x = b. TRANS = 'C' or 'c' A**T*x = b. ``` *DIAG* ``` DIAG is CHARACTER*1 On entry, DIAG specifies whether or not A is unit triangular as follows: DIAG = 'U' or 'u' A is assumed to be unit triangular. DIAG = 'N' or 'n' A is not assumed to be unit triangular. ``` *N* ``` N is INTEGER On entry, N specifies the order of the matrix A. N must be at least zero. ``` *AP* ``` AP is 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. ``` *X* ``` 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. ``` *INCX* ``` INCX 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. **Further Details:** ``` Level 2 Blas routine. -- Written on 22-October-1986. Jack Dongarra, Argonne National Lab. Jeremy Du Croz, Nag Central Office. Sven Hammarling, Nag Central Office. Richard Hanson, Sandia National Labs. ``` ### subroutine strmv (character UPLO, character TRANS, character DIAG, integer N, real, dimension(lda,\*) A, integer LDA, real, dimension(\*) X, integer INCX) **STRMV** **Purpose:** ``` 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. ``` **Parameters** *UPLO* ``` UPLO is CHARACTER*1 On entry, UPLO specifies whether the matrix is an upper or lower triangular matrix as follows: UPLO = 'U' or 'u' A is an upper triangular matrix. UPLO = 'L' or 'l' A is a lower triangular matrix. ``` *TRANS* ``` TRANS is CHARACTER*1 On entry, TRANS specifies the operation to be performed as follows: TRANS = 'N' or 'n' x := A*x. TRANS = 'T' or 't' x := A**T*x. TRANS = 'C' or 'c' x := A**T*x. ``` *DIAG* ``` DIAG is CHARACTER*1 On entry, DIAG specifies whether or not A is unit triangular as follows: DIAG = 'U' or 'u' A is assumed to be unit triangular. DIAG = 'N' or 'n' A is not assumed to be unit triangular. ``` *N* ``` N is INTEGER On entry, N specifies the order of the matrix A. N must be at least zero. ``` *A* ``` A is 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. ``` *LDA* ``` LDA is INTEGER On entry, LDA specifies the first dimension of A as declared in the calling (sub) program. LDA must be at least max( 1, n ). ``` *X* ``` X is 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. ``` *INCX* ``` INCX 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. **Further Details:** ``` Level 2 Blas routine. The vector and matrix arguments are not referenced when N = 0, or M = 0 -- Written on 22-October-1986. Jack Dongarra, Argonne National Lab. Jeremy Du Croz, Nag Central Office. Sven Hammarling, Nag Central Office. Richard Hanson, Sandia National Labs. ``` ### subroutine strsv (character UPLO, character TRANS, character DIAG, integer N, real, dimension(lda,\*) A, integer LDA, real, dimension(\*) X, integer INCX) **STRSV** **Purpose:** ``` 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. ``` **Parameters** *UPLO* ``` UPLO is CHARACTER*1 On entry, UPLO specifies whether the matrix is an upper or lower triangular matrix as follows: UPLO = 'U' or 'u' A is an upper triangular matrix. UPLO = 'L' or 'l' A is a lower triangular matrix. ``` *TRANS* ``` TRANS is CHARACTER*1 On entry, TRANS specifies the equations to be solved as follows: TRANS = 'N' or 'n' A*x = b. TRANS = 'T' or 't' A**T*x = b. TRANS = 'C' or 'c' A**T*x = b. ``` *DIAG* ``` DIAG is CHARACTER*1 On entry, DIAG specifies whether or not A is unit triangular as follows: DIAG = 'U' or 'u' A is assumed to be unit triangular. DIAG = 'N' or 'n' A is not assumed to be unit triangular. ``` *N* ``` N is INTEGER On entry, N specifies the order of the matrix A. N must be at least zero. ``` *A* ``` 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. ``` *LDA* ``` LDA is INTEGER On entry, LDA specifies the first dimension of A as declared in the calling (sub) program. LDA must be at least max( 1, n ). ``` *X* ``` X is 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. ``` *INCX* ``` INCX 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. **Further Details:** ``` Level 2 Blas routine. -- Written on 22-October-1986. Jack Dongarra, Argonne National Lab. Jeremy Du Croz, Nag Central Office. Sven Hammarling, Nag Central Office. Richard Hanson, Sandia National Labs. ``` ## Author Generated automatically by Doxygen for LAPACK from the source code.