SRC/stfsm.f(3)

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💖 SRC/stfsm.f(3) 💖

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🌸 NAME 🌸

SRC/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 🎀

subroutine stfsm (character transr, character side, character uplo,
    character trans, character diag, integer m, integer n, real alpha, real,
    dimension( 0: * ) a, real, dimension( 0: ldb-1, 0: * ) b, integer ldb)

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 — CHARACTER*1. = 'N': The Normal Form of RFP A is stored; = 'T': The Transpose Form of RFP A is stored.
  • SIDE — CHARACTER*1. On entry, SIDE specifies whether op( A ) appears on the left or right of X as follows: SIDE = 'L' or 'l' means op( A )*X = alpha*B. SIDE = 'R' or 'r' means X*op( A ) = alpha*B. Unchanged on exit.
  • UPLO — 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' means RFP A came from an upper triangular matrix; UPLO = 'L' or 'l' means RFP A came from a lower triangular matrix. Unchanged on exit.
  • TRANS — CHARACTER*1. On entry, TRANS specifies the form of op( A ) to be used in the matrix multiplication as follows: TRANS = 'N' or 'n' means op( A ) = A. TRANS = 'T' or 't' means op( A ) = A'. Unchanged on exit.
  • DIAG — CHARACTER*1. On entry, DIAG specifies whether or not RFP A is unit triangular as follows: DIAG = 'U' or 'u' means A is assumed to be unit triangular. DIAG = 'N' or 'n' means A is not assumed to be unit triangular. Unchanged on exit.
  • M — INTEGER. On entry, M specifies the number of rows of B. M must be at least zero. Unchanged on exit.
  • N — INTEGER. On entry, N specifies the number of columns of B. N must be at least zero. Unchanged on exit.
  • ALPHA — 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 — REAL array, dimension (NT). NT = N*(N+1)/2 if SIDE='R' and NT = M*(M+1)/2 otherwise. 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 — 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 — 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.

🔥 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 276 of file stfsm.f.

👑 Author 👑

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