CTFSM(1)LAPACK routine (version 3.2) CTFSM(1)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 triangu‐
lar. DIAG = 'N' or 'n' A is not assumed to be unit triangu‐
lar. 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 For‐
mat 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 fol‐
lows: 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-trans‐
pose 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-trans‐
pose 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
..
LAPACK routine (version 3.2) November 2008 CTFSM(1)
