Provided by: liblapack-doc_3.9.0-1build1_all

**NAME**

single_blas_level3

**SYNOPSIS**

Functionssubroutinesgemm(TRANSA, TRANSB, M, N, K, ALPHA, A, LDA, B, LDB, BETA, C, LDC)SGEMMsubroutinessymm(SIDE, UPLO, M, N, ALPHA, A, LDA, B, LDB, BETA, C, LDC)SSYMMsubroutinessyr2k(UPLO, TRANS, N, K, ALPHA, A, LDA, B, LDB, BETA, C, LDC)SSYR2Ksubroutinessyrk(UPLO, TRANS, N, K, ALPHA, A, LDA, BETA, C, LDC)SSYRKsubroutinestrmm(SIDE, UPLO, TRANSA, DIAG, M, N, ALPHA, A, LDA, B, LDB)STRMMsubroutinestrsm(SIDE, UPLO, TRANSA, DIAG, M, N, ALPHA, A, LDA, B, LDB)STRSM

**Detailed** **Description**

This is the group of real LEVEL 3 BLAS routines.

**Function** **Documentation**

subroutinesgemm(characterTRANSA,characterTRANSB,integerM,integerN,integerK,realALPHA,real,dimension(lda,*)A,integerLDA,real,dimension(ldb,*)B,integerLDB,realBETA,real,dimension(ldc,*)C,integerLDC)SGEMMPurpose:SGEMM performs one of the matrix-matrix operations C := alpha*op( A )*op( B ) + beta*C, where op( X ) is one of op( X ) = X or op( X ) = X**T, alpha and beta are scalars, and A, B and C are matrices, with op( A ) an m by k matrix, op( B ) a k by n matrix and C an m by n matrix.ParametersTRANSATRANSA is CHARACTER*1 On entry, TRANSA specifies the form of op( A ) to be used in the matrix multiplication as follows: TRANSA = 'N' or 'n', op( A ) = A. TRANSA = 'T' or 't', op( A ) = A**T. TRANSA = 'C' or 'c', op( A ) = A**T.TRANSBTRANSB is CHARACTER*1 On entry, TRANSB specifies the form of op( B ) to be used in the matrix multiplication as follows: TRANSB = 'N' or 'n', op( B ) = B. TRANSB = 'T' or 't', op( B ) = B**T. TRANSB = 'C' or 'c', op( B ) = B**T.MM is INTEGER On entry, M specifies the number of rows of the matrix op( A ) and of the matrix C. M must be at least zero.NN is INTEGER On entry, N specifies the number of columns of the matrix op( B ) and the number of columns of the matrix C. N must be at least zero.KK is INTEGER On entry, K specifies the number of columns of the matrix op( A ) and the number of rows of the matrix op( B ). K must be at least zero.ALPHAALPHA is REAL On entry, ALPHA specifies the scalar alpha.AA is REAL array, dimension ( LDA, ka ), where ka is k when TRANSA = 'N' or 'n', and is m otherwise. Before entry with TRANSA = 'N' or 'n', the leading m by k part of the array A must contain the matrix A, otherwise the leading k by m part of the array A must contain the matrix A.LDALDA is INTEGER On entry, LDA specifies the first dimension of A as declared in the calling (sub) program. When TRANSA = 'N' or 'n' then LDA must be at least max( 1, m ), otherwise LDA must be at least max( 1, k ).BB is REAL array, dimension ( LDB, kb ), where kb is n when TRANSB = 'N' or 'n', and is k otherwise. Before entry with TRANSB = 'N' or 'n', the leading k by n part of the array B must contain the matrix B, otherwise the leading n by k part of the array B must contain the matrix B.LDBLDB is INTEGER On entry, LDB specifies the first dimension of B as declared in the calling (sub) program. When TRANSB = 'N' or 'n' then LDB must be at least max( 1, k ), otherwise LDB must be at least max( 1, n ).BETABETA is REAL On entry, BETA specifies the scalar beta. When BETA is supplied as zero then C need not be set on input.CC is REAL array, dimension ( LDC, N ) Before entry, the leading m by n part of the array C must contain the matrix C, except when beta is zero, in which case C need not be set on entry. On exit, the array C is overwritten by the m by n matrix ( alpha*op( A )*op( B ) + beta*C ).LDCLDC is INTEGER On entry, LDC specifies the first dimension of C as declared in the calling (sub) program. LDC must be at least max( 1, m ).AuthorUniv. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.DateDecember 2016FurtherDetails:Level 3 Blas routine. -- Written on 8-February-1989. Jack Dongarra, Argonne National Laboratory. Iain Duff, AERE Harwell. Jeremy Du Croz, Numerical Algorithms Group Ltd. Sven Hammarling, Numerical Algorithms Group Ltd.subroutinessymm(characterSIDE,characterUPLO,integerM,integerN,realALPHA,real,dimension(lda,*)A,integerLDA,real,dimension(ldb,*)B,integerLDB,realBETA,real,dimension(ldc,*)C,integerLDC)SSYMMPurpose:SSYMM performs one of the matrix-matrix operations C := alpha*A*B + beta*C, or C := alpha*B*A + beta*C, where alpha and beta are scalars, A is a symmetric matrix and B and C are m by n matrices.ParametersSIDESIDE is CHARACTER*1 On entry, SIDE specifies whether the symmetric matrix A appears on the left or right in the operation as follows: SIDE = 'L' or 'l' C := alpha*A*B + beta*C, SIDE = 'R' or 'r' C := alpha*B*A + beta*C,UPLOUPLO is CHARACTER*1 On entry, UPLO specifies whether the upper or lower triangular part of the symmetric matrix A is to be referenced as follows: UPLO = 'U' or 'u' Only the upper triangular part of the symmetric matrix is to be referenced. UPLO = 'L' or 'l' Only the lower triangular part of the symmetric matrix is to be referenced.MM is INTEGER On entry, M specifies the number of rows of the matrix C. M must be at least zero.NN is INTEGER On entry, N specifies the number of columns of the matrix C. N must be at least zero.ALPHAALPHA is REAL On entry, ALPHA specifies the scalar alpha.AA is REAL array, dimension ( LDA, ka ), where ka is m when SIDE = 'L' or 'l' and is n otherwise. Before entry with SIDE = 'L' or 'l', the m by m part of the array A must contain the symmetric matrix, such that when UPLO = 'U' or 'u', the leading m by m 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, and when UPLO = 'L' or 'l', the leading m by m 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. Before entry with SIDE = 'R' or 'r', the n by n part of the array A must contain the symmetric matrix, such that when 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, and when UPLO = 'L' or 'l', the leading n by n lower triangular part of the array A must contain the lower triangular part of the symmetric matrix and the strictly upper triangular part of A is not referenced.LDALDA is INTEGER On entry, LDA specifies the first dimension of A as declared in the calling (sub) program. When SIDE = 'L' or 'l' then LDA must be at least max( 1, m ), otherwise LDA must be at least max( 1, n ).BB is REAL array, dimension ( LDB, N ) Before entry, the leading m by n part of the array B must contain the matrix B.LDBLDB is INTEGER On entry, LDB specifies the first dimension of B as declared in the calling (sub) program. LDB must be at least max( 1, m ).BETABETA is REAL On entry, BETA specifies the scalar beta. When BETA is supplied as zero then C need not be set on input.CC is REAL array, dimension ( LDC, N ) Before entry, the leading m by n part of the array C must contain the matrix C, except when beta is zero, in which case C need not be set on entry. On exit, the array C is overwritten by the m by n updated matrix.LDCLDC is INTEGER On entry, LDC specifies the first dimension of C as declared in the calling (sub) program. LDC must be at least max( 1, m ).AuthorUniv. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.DateDecember 2016FurtherDetails:Level 3 Blas routine. -- Written on 8-February-1989. Jack Dongarra, Argonne National Laboratory. Iain Duff, AERE Harwell. Jeremy Du Croz, Numerical Algorithms Group Ltd. Sven Hammarling, Numerical Algorithms Group Ltd.subroutinessyr2k(characterUPLO,characterTRANS,integerN,integerK,realALPHA,real,dimension(lda,*)A,integerLDA,real,dimension(ldb,*)B,integerLDB,realBETA,real,dimension(ldc,*)C,integerLDC)SSYR2KPurpose:SSYR2K performs one of the symmetric rank 2k operations C := alpha*A*B**T + alpha*B*A**T + beta*C, or C := alpha*A**T*B + alpha*B**T*A + beta*C, where alpha and beta are scalars, C is an n by n symmetric matrix and A and B are n by k matrices in the first case and k by n matrices in the second case.ParametersUPLOUPLO is CHARACTER*1 On entry, UPLO specifies whether the upper or lower triangular part of the array C is to be referenced as follows: UPLO = 'U' or 'u' Only the upper triangular part of C is to be referenced. UPLO = 'L' or 'l' Only the lower triangular part of C is to be referenced.TRANSTRANS is CHARACTER*1 On entry, TRANS specifies the operation to be performed as follows: TRANS = 'N' or 'n' C := alpha*A*B**T + alpha*B*A**T + beta*C. TRANS = 'T' or 't' C := alpha*A**T*B + alpha*B**T*A + beta*C. TRANS = 'C' or 'c' C := alpha*A**T*B + alpha*B**T*A + beta*C.NN is INTEGER On entry, N specifies the order of the matrix C. N must be at least zero.KK is INTEGER On entry with TRANS = 'N' or 'n', K specifies the number of columns of the matrices A and B, and on entry with TRANS = 'T' or 't' or 'C' or 'c', K specifies the number of rows of the matrices A and B. K must be at least zero.ALPHAALPHA is REAL On entry, ALPHA specifies the scalar alpha.AA is REAL array, dimension ( LDA, ka ), where ka is k when TRANS = 'N' or 'n', and is n otherwise. Before entry with TRANS = 'N' or 'n', the leading n by k part of the array A must contain the matrix A, otherwise the leading k by n part of the array A must contain the matrix A.LDALDA is INTEGER On entry, LDA specifies the first dimension of A as declared in the calling (sub) program. When TRANS = 'N' or 'n' then LDA must be at least max( 1, n ), otherwise LDA must be at least max( 1, k ).BB is REAL array, dimension ( LDB, kb ), where kb is k when TRANS = 'N' or 'n', and is n otherwise. Before entry with TRANS = 'N' or 'n', the leading n by k part of the array B must contain the matrix B, otherwise the leading k by n part of the array B must contain the matrix B.LDBLDB is INTEGER On entry, LDB specifies the first dimension of B as declared in the calling (sub) program. When TRANS = 'N' or 'n' then LDB must be at least max( 1, n ), otherwise LDB must be at least max( 1, k ).BETABETA is REAL On entry, BETA specifies the scalar beta.CC is REAL array, dimension ( LDC, N ) Before entry with UPLO = 'U' or 'u', the leading n by n upper triangular part of the array C must contain the upper triangular part of the symmetric matrix and the strictly lower triangular part of C is not referenced. On exit, the upper triangular part of the array C 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 C must contain the lower triangular part of the symmetric matrix and the strictly upper triangular part of C is not referenced. On exit, the lower triangular part of the array C is overwritten by the lower triangular part of the updated matrix.LDCLDC is INTEGER On entry, LDC specifies the first dimension of C as declared in the calling (sub) program. LDC must be at least max( 1, n ).AuthorUniv. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.DateDecember 2016FurtherDetails:Level 3 Blas routine. -- Written on 8-February-1989. Jack Dongarra, Argonne National Laboratory. Iain Duff, AERE Harwell. Jeremy Du Croz, Numerical Algorithms Group Ltd. Sven Hammarling, Numerical Algorithms Group Ltd.subroutinessyrk(characterUPLO,characterTRANS,integerN,integerK,realALPHA,real,dimension(lda,*)A,integerLDA,realBETA,real,dimension(ldc,*)C,integerLDC)SSYRKPurpose:SSYRK performs one of the symmetric rank k operations C := alpha*A*A**T + beta*C, or C := alpha*A**T*A + beta*C, where alpha and beta are scalars, C is an n by n symmetric matrix and A is an n by k matrix in the first case and a k by n matrix in the second case.ParametersUPLOUPLO is CHARACTER*1 On entry, UPLO specifies whether the upper or lower triangular part of the array C is to be referenced as follows: UPLO = 'U' or 'u' Only the upper triangular part of C is to be referenced. UPLO = 'L' or 'l' Only the lower triangular part of C is to be referenced.TRANSTRANS is CHARACTER*1 On entry, TRANS specifies the operation to be performed as follows: TRANS = 'N' or 'n' C := alpha*A*A**T + beta*C. TRANS = 'T' or 't' C := alpha*A**T*A + beta*C. TRANS = 'C' or 'c' C := alpha*A**T*A + beta*C.NN is INTEGER On entry, N specifies the order of the matrix C. N must be at least zero.KK is INTEGER On entry with TRANS = 'N' or 'n', K specifies the number of columns of the matrix A, and on entry with TRANS = 'T' or 't' or 'C' or 'c', K specifies the number of rows of the matrix A. K must be at least zero.ALPHAALPHA is REAL On entry, ALPHA specifies the scalar alpha.AA is REAL array, dimension ( LDA, ka ), where ka is k when TRANS = 'N' or 'n', and is n otherwise. Before entry with TRANS = 'N' or 'n', the leading n by k part of the array A must contain the matrix A, otherwise the leading k by n part of the array A must contain the matrix A.LDALDA is INTEGER On entry, LDA specifies the first dimension of A as declared in the calling (sub) program. When TRANS = 'N' or 'n' then LDA must be at least max( 1, n ), otherwise LDA must be at least max( 1, k ).BETABETA is REAL On entry, BETA specifies the scalar beta.CC is REAL array, dimension ( LDC, N ) Before entry with UPLO = 'U' or 'u', the leading n by n upper triangular part of the array C must contain the upper triangular part of the symmetric matrix and the strictly lower triangular part of C is not referenced. On exit, the upper triangular part of the array C 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 C must contain the lower triangular part of the symmetric matrix and the strictly upper triangular part of C is not referenced. On exit, the lower triangular part of the array C is overwritten by the lower triangular part of the updated matrix.LDCLDC is INTEGER On entry, LDC specifies the first dimension of C as declared in the calling (sub) program. LDC must be at least max( 1, n ).AuthorUniv. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.DateDecember 2016FurtherDetails:Level 3 Blas routine. -- Written on 8-February-1989. Jack Dongarra, Argonne National Laboratory. Iain Duff, AERE Harwell. Jeremy Du Croz, Numerical Algorithms Group Ltd. Sven Hammarling, Numerical Algorithms Group Ltd.subroutinestrmm(characterSIDE,characterUPLO,characterTRANSA,characterDIAG,integerM,integerN,realALPHA,real,dimension(lda,*)A,integerLDA,real,dimension(ldb,*)B,integerLDB)STRMMPurpose:STRMM performs one of the matrix-matrix operations B := alpha*op( A )*B, or B := alpha*B*op( A ), where alpha is a scalar, B is an m by n matrix, A is a unit, or non-unit, upper or lower triangular matrix and op( A ) is one of op( A ) = A or op( A ) = A**T.ParametersSIDESIDE is CHARACTER*1 On entry, SIDE specifies whether op( A ) multiplies B from the left or right as follows: SIDE = 'L' or 'l' B := alpha*op( A )*B. SIDE = 'R' or 'r' B := alpha*B*op( A ).UPLOUPLO is CHARACTER*1 On entry, UPLO specifies whether the matrix A 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.TRANSATRANSA is CHARACTER*1 On entry, TRANSA specifies the form of op( A ) to be used in the matrix multiplication as follows: TRANSA = 'N' or 'n' op( A ) = A. TRANSA = 'T' or 't' op( A ) = A**T. TRANSA = 'C' or 'c' op( A ) = A**T.DIAGDIAG is CHARACTER*1 On entry, DIAG specifies whether or not A is unit triangular as follows: DIAG = 'U' or 'u' A is assumed to be unit triangular. DIAG = 'N' or 'n' A is not assumed to be unit triangular.MM is INTEGER On entry, M specifies the number of rows of B. M must be at least zero.NN is INTEGER On entry, N specifies the number of columns of B. N must be at least zero.ALPHAALPHA is REAL On entry, ALPHA specifies the scalar alpha. When alpha is zero then A is not referenced and B need not be set before entry.AA is REAL array, dimension ( LDA, k ), where k is m when SIDE = 'L' or 'l' and is n when SIDE = 'R' or 'r'. Before entry with UPLO = 'U' or 'u', the leading k by k 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 k by k lower triangular part of the array A must contain the lower triangular matrix and the strictly upper triangular part of A is not referenced. Note that when DIAG = 'U' or 'u', the diagonal elements of A are not referenced either, but are assumed to be unity.LDALDA is INTEGER On entry, LDA specifies the first dimension of A as declared in the calling (sub) program. When SIDE = 'L' or 'l' then LDA must be at least max( 1, m ), when SIDE = 'R' or 'r' then LDA must be at least max( 1, n ).BB is REAL array, dimension ( LDB, N ) Before entry, the leading m by n part of the array B must contain the matrix B, and on exit is overwritten by the transformed matrix.LDBLDB is INTEGER On entry, LDB specifies the first dimension of B as declared in the calling (sub) program. LDB must be at least max( 1, m ).AuthorUniv. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.DateDecember 2016FurtherDetails:Level 3 Blas routine. -- Written on 8-February-1989. Jack Dongarra, Argonne National Laboratory. Iain Duff, AERE Harwell. Jeremy Du Croz, Numerical Algorithms Group Ltd. Sven Hammarling, Numerical Algorithms Group Ltd.subroutinestrsm(characterSIDE,characterUPLO,characterTRANSA,characterDIAG,integerM,integerN,realALPHA,real,dimension(lda,*)A,integerLDA,real,dimension(ldb,*)B,integerLDB)STRSMPurpose:STRSM solves one of the matrix equations op( A )*X = alpha*B, or X*op( A ) = alpha*B, where alpha is a scalar, X and B are m by n matrices, A is a unit, or non-unit, upper or lower triangular matrix and op( A ) is one of op( A ) = A or op( A ) = A**T. The matrix X is overwritten on B.ParametersSIDESIDE is CHARACTER*1 On entry, SIDE specifies whether op( A ) appears on the left or right of X as follows: SIDE = 'L' or 'l' op( A )*X = alpha*B. SIDE = 'R' or 'r' X*op( A ) = alpha*B.UPLOUPLO is CHARACTER*1 On entry, UPLO specifies whether the matrix A 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.TRANSATRANSA is CHARACTER*1 On entry, TRANSA specifies the form of op( A ) to be used in the matrix multiplication as follows: TRANSA = 'N' or 'n' op( A ) = A. TRANSA = 'T' or 't' op( A ) = A**T. TRANSA = 'C' or 'c' op( A ) = A**T.DIAGDIAG is CHARACTER*1 On entry, DIAG specifies whether or not A is unit triangular as follows: DIAG = 'U' or 'u' A is assumed to be unit triangular. DIAG = 'N' or 'n' A is not assumed to be unit triangular.MM is INTEGER On entry, M specifies the number of rows of B. M must be at least zero.NN is INTEGER On entry, N specifies the number of columns of B. N must be at least zero.ALPHAALPHA is REAL On entry, ALPHA specifies the scalar alpha. When alpha is zero then A is not referenced and B need not be set before entry.AA is REAL array, dimension ( LDA, k ), where k is m when SIDE = 'L' or 'l' and k is n when SIDE = 'R' or 'r'. Before entry with UPLO = 'U' or 'u', the leading k by k 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 k by k lower triangular part of the array A must contain the lower triangular matrix and the strictly upper triangular part of A is not referenced. Note that when DIAG = 'U' or 'u', the diagonal elements of A are not referenced either, but are assumed to be unity.LDALDA is INTEGER On entry, LDA specifies the first dimension of A as declared in the calling (sub) program. When SIDE = 'L' or 'l' then LDA must be at least max( 1, m ), when SIDE = 'R' or 'r' then LDA must be at least max( 1, n ).BB is REAL array, dimension ( LDB, N ) Before entry, the leading m by n part of the array B must contain the right-hand side matrix B, and on exit is overwritten by the solution matrix X.LDBLDB is INTEGER On entry, LDB specifies the first dimension of B as declared in the calling (sub) program. LDB must be at least max( 1, m ).AuthorUniv. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.DateDecember 2016FurtherDetails:Level 3 Blas routine. -- Written on 8-February-1989. Jack Dongarra, Argonne National Laboratory. Iain Duff, AERE Harwell. Jeremy Du Croz, Numerical Algorithms Group Ltd. Sven Hammarling, Numerical Algorithms Group Ltd.

**Author**

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