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# stfsm.f

stfsm.f −

## SYNOPSIS

Functions/Subroutines

subroutine stfsm (TRANSR, SIDE, UPLO, TRANS, DIAG, M, N, ALPHA, A, B, LDB)
STFSM
solves a matrix equation (one operand is a triangular matrix in RFP format).

## Function/Subroutine Documentation

subroutine stfsm (characterTRANSR, characterSIDE, characterUPLO, characterTRANS, characterDIAG, integerM, integerN, realALPHA, real, dimension( 0: * )A, real, dimension( 0: ldb-1, 0: * )B, integerLDB)
STFSM
solves a matrix equation (one operand is a triangular matrix in RFP format).

Purpose:

Level 3 BLAS like routine for A in RFP Format.

STFSM 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 ) = A**T.

A is in Rectangular Full Packed (RFP) Format.

The matrix X is overwritten on B.

Parameters:

TRANSR

TRANSR is CHARACTER*1
= ’N’: The Normal Form of RFP A is stored;
= ’T’: The Transpose Form of RFP A is stored.

SIDE

SIDE 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.

Unchanged on exit.

UPLO

UPLO is CHARACTER*1
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

TRANS is CHARACTER*1
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 = ’T’ or ’t’ op( A ) = A’.

Unchanged on exit.

DIAG

DIAG is CHARACTER*1
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

M is INTEGER
On entry, M specifies the number of rows of B. M must be at
least zero.
Unchanged on exit.

N

N is INTEGER
On entry, N specifies the number of columns of B. N must be
at least zero.
Unchanged on exit.

ALPHA

ALPHA 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.
Unchanged on exit.

A

A is REAL array, dimension (NT)
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 = ’T’ then RFP is the 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
transpose Format. If UPLO = ’L’ the RFP A contains
the NT elements of lower packed A either in normal or
transpose Format. The LDA of RFP A is (N+1)/2 when
TRANSR = ’T’. 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

B 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.

LDB

LDB 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 ).
Unchanged on exit.

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

September 2012

Further Details:

We first consider Rectangular Full Packed (RFP) 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
the 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
the transpose of the last three columns of AP lower.
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 = ’T’. RFP A in both UPLO cases is just the
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 then consider Rectangular Full Packed (RFP) 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
the 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
the transpose of the last two columns of AP lower.
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 = ’T’. RFP A in both UPLO cases is just the
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

Definition at line 277 of file stfsm.f.

## Author

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