--- name: "ctfsm" section: "3lapack" platform: "linux" package: "liblapack-doc" version: "3.2.1-8" release: "Debian 6.0 (Squeeze)" platforms: ["linux"] url: "https://new.carta.tech/ctfsm/3lapack" --- # ctfsm(3lapack) 3 BLAS like routine for A in RFP Format ## NAME CTFSM - 3 BLAS like routine for A in RFP Format ## SYNOPSIS - SUBROUTINE CTFSM( TRANSR, SIDE, UPLO, TRANS, DIAG, M, N, ALPHA, A, - \+ B, LDB ) - CHARACTER TRANSR, DIAG, SIDE, TRANS, UPLO - INTEGER LDB, M, N - COMPLEX ALPHA - COMPLEX A( 0: \* ), B( 0: LDB-1, 0: \* ) ## PURPOSE Level 3 BLAS like routine for A in RFP Format. CTFSM solves the matrix equation\ \ 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 ) = conjg( A' ).\ A is in Rectangular Full Packed (RFP) Format.\ The matrix X is overwritten on B.\ ## ARGUMENTS TRANSR - (input) CHARACTER = 'N': The Normal Form of RFP A is stored;\ \= 'C': The Conjugate-transpose Form of RFP A is stored. - SIDE - (input) CHARACTER 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. Unchanged on exit. - UPLO - (input) CHARACTER On entry, UPLO specifies whether the RFP matrix A came from an upper or lower triangular matrix as follows: UPLO = 'U' or 'u' RFP A came from an upper triangular matrix UPLO = 'L' or 'l' RFP A came from a lower triangular matrix Unchanged on exit. - TRANS - (input) CHARACTER On entry, TRANS specifies the form of op( A ) to be used in the matrix multiplication as follows: - TRANS = 'N' or 'n' op( A ) = A. - TRANS = 'C' or 'c' op( A ) = conjg( A' ). Unchanged on exit. - DIAG - (input) CHARACTER On entry, DIAG specifies whether or not RFP 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. Unchanged on exit. - M - (input) INTEGER. On entry, M specifies the number of rows of B. M must be at least zero. Unchanged on exit. - N - (input) INTEGER. On entry, N specifies the number of columns of B. N must be at least zero. Unchanged on exit. - ALPHA - (input) COMPLEX. On entry, ALPHA specifies the scalar alpha. When alpha is zero then A is not referenced and B need not be set before entry. Unchanged on exit. - A - (input) COMPLEX array, dimension ( N\*(N+1)/2 ); NT = N\*(N+1)/2. On entry, the matrix A in RFP Format. RFP Format is described by TRANSR, UPLO and N as follows:\ If TRANSR='N' then RFP A is (0:N,0:K-1) when N is even;\ K=N/2. RFP A is (0:N-1,0:K) when N is odd; K=N/2. If TRANSR = 'C' then RFP is the Conjugate-transpose of RFP A as defined when TRANSR = 'N'. The contents of RFP A are defined by UPLO as follows: If UPLO = 'U' the RFP A contains the NT elements of upper packed A either in normal or conjugate-transpose Format. If UPLO = 'L' the RFP A contains the NT elements of lower packed A either in normal or conjugate-transpose Format. The LDA of RFP A is (N+1)/2 when TRANSR = 'C'. When TRANSR is 'N' the LDA is N+1 when N is even and is N when is odd. See the Note below for more details. Unchanged on exit. - B - (input/ouptut) COMPLEX 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. - LDB - (input) 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 ). Unchanged on exit. ## FURTHER DETAILS We first consider Standard Packed Format when N is even.\ We give an example where N = 6.\ \ AP is Upper AP is Lower\ \ 00 01 02 03 04 05 00\ \ 11 12 13 14 15 10 11\ \ 22 23 24 25 20 21 22\ \ 33 34 35 30 31 32 33\ \ 44 45 40 41 42 43 44\ \ 55 50 51 52 53 54 55\ Let TRANSR = 'N'. RFP holds AP as follows:\ For UPLO = 'U' the upper trapezoid A(0:5,0:2) consists of the last three columns of AP upper. The lower triangle A(4:6,0:2) consists of conjugate-transpose of the first three columns of AP upper. For UPLO = 'L' the lower trapezoid A(1:6,0:2) consists of the first three columns of AP lower. The upper triangle A(0:2,0:2) consists of conjugate-transpose of the last three columns of AP lower. To denote conjugate we place -- above the element. This covers the case N even and TRANSR = 'N'.\ \ RFP A RFP A\ \ \-- -- --\ \ 03 04 05 33 43 53\ \ \-- --\ \ 13 14 15 00 44 54\ \ \--\ \ 23 24 25 10 11 55\ \ 33 34 35 20 21 22\ \ \--\ \ 00 44 45 30 31 32\ \ \-- --\ \ 01 11 55 40 41 42\ \ \-- -- --\ \ 02 12 22 50 51 52\ Now let TRANSR = 'C'. RFP A in both UPLO cases is just the conjugate- transpose of RFP A above. One therefore gets:\ \ RFP A RFP A\ \ \-- -- -- -- -- -- -- -- -- --\ \ 03 13 23 33 00 01 02 33 00 10 20 30 40 50\ \ \-- -- -- -- -- -- -- -- -- --\ \ 04 14 24 34 44 11 12 43 44 11 21 31 41 51\ \ \-- -- -- -- -- -- -- -- -- --\ \ 05 15 25 35 45 55 22 53 54 55 22 32 42 52\ We next consider Standard Packed Format when N is odd.\ We give an example where N = 5.\ \ AP is Upper AP is Lower\ \ 00 01 02 03 04 00\ \ 11 12 13 14 10 11\ \ 22 23 24 20 21 22\ \ 33 34 30 31 32 33\ \ 44 40 41 42 43 44\ Let TRANSR = 'N'. RFP holds AP as follows:\ For UPLO = 'U' the upper trapezoid A(0:4,0:2) consists of the last three columns of AP upper. The lower triangle A(3:4,0:1) consists of conjugate-transpose of the first two columns of AP upper. For UPLO = 'L' the lower trapezoid A(0:4,0:2) consists of the first three columns of AP lower. The upper triangle A(0:1,1:2) consists of conjugate-transpose of the last two columns of AP lower. To denote conjugate we place -- above the element. This covers the case N odd and TRANSR = 'N'.\ \ RFP A RFP A\ \ \-- --\ \ 02 03 04 00 33 43\ \ \--\ \ 12 13 14 10 11 44\ \ 22 23 24 20 21 22\ \ \--\ \ 00 33 34 30 31 32\ \ \-- --\ \ 01 11 44 40 41 42\ Now let TRANSR = 'C'. RFP A in both UPLO cases is just the conjugate- transpose of RFP A above. One therefore gets:\ \ RFP A RFP A\ \ \-- -- -- -- -- -- -- -- --\ \ 02 12 22 00 01 00 10 20 30 40 50\ \ \-- -- -- -- -- -- -- -- --\ \ 03 13 23 33 11 33 11 21 31 41 51\ \ \-- -- -- -- -- -- -- -- --\ \ 04 14 24 34 44 43 44 22 32 42 52\ \ ..\