ctfsm (l) - Linux Man Pages

ctfsm: 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( )*X alpha*B  or  X*op( 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(   or   op( conjg( Aaq ).
A is in Rectangular Full Packed (RFP) Format.
The matrix X is overwritten on B.

ARGUMENTS

TRANSR - (input) CHARACTER = aqNaq: The Normal Form of RFP A is stored;
= aqCaq: 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 = aqLaq or aqlaq op( A )*X = alpha*B. SIDE = aqRaq or aqraq 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 = aqUaq or aquaq RFP A came from an upper triangular matrix UPLO = aqLaq or aqlaq 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 = aqNaq or aqnaq op( A ) = A.
TRANS = aqCaq or aqcaq op( A ) = conjg( Aaq ).
Unchanged on exit.
DIAG - (input) CHARACTER
On entry, DIAG specifies whether or not RFP A is unit triangular as follows: DIAG = aqUaq or aquaq A is assumed to be unit triangular. DIAG = aqNaq or aqnaq 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=aqNaq 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 = aqCaq then RFP is the Conjugate-transpose of RFP A as defined when TRANSR = aqNaq. The contents of RFP A are defined by UPLO as follows: If UPLO = aqUaq the RFP A contains the NT elements of upper packed A either in normal or conjugate-transpose Format. If UPLO = aqLaq 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 = aqCaq. When TRANSR is aqNaq 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 = aqNaq. RFP holds AP as follows:
For UPLO = aqUaq 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 = aqLaq 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 = aqNaq.

 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 = aqCaq. 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 = aqNaq. RFP holds AP as follows:
For UPLO = aqUaq 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 = aqLaq 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 = aqNaq.

 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 = aqCaq. 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

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