SRC/ztfsm.f(3)

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version 3.12.1 · 7 versions
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🎀 NAME 🎀

SRC/ztfsm.f — okay so this is the page, bestie, and it's from the Library Functions Manual. It's giving LAPACK documentation energy, and we're about to read every single detail. 💅

✨ SYNOPSIS ✨

Functions/Subroutines

       subroutine ztfsm (transr, side, uplo, trans, diag, m, n, alpha, a, b,
           ldb)
           ZTFSM solves a matrix equation (one operand is a triangular matrix
           in RFP format).

🦋 Function/Subroutine Documentation 🦋

   subroutine ztfsm (character transr, character side, character uplo,
       character trans, character diag, integer m, integer n, complex*16 alpha,
       complex*16, dimension( 0: * ) a, complex*16, dimension( 0: ldb-1, 0: * )
       b, integer ldb)

ZTFSM solves a matrix equation (one operand is a triangular matrix in RFP format).

💖 Purpose 💖

It's a Level 3 BLAS like routine for A in RFP Format, periodt.

ZTFSM 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**H.

A is in Rectangular Full Packed (RFP) Format.

The matrix X is overwritten on B. No cap. 💫

🌸 Parameters 🌸

TRANSR

TRANSR is CHARACTER*1.

  • = 'N': The Normal Form of RFP A is stored.
  • = 'C': The Conjugate-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 = 'C' or 'c' — op( A ) = conjg( 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 COMPLEX*16.

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 COMPLEX*16 array, dimension (N*(N+1)/2).

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 = '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

B is COMPLEX*16 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.

🌈 Further Details 🌈

Okay bestie, here's the RFP tea. First we consider Standard Packed Format when N is even. Example time: 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. Example: 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

Definition at line 297 of file ztfsm.f.

👑 Author 👑

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