SRC/ctfsm.f(3)

📦

version 3.12.1 · 7 versions

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okay bestie, welcome to SRC/ctfsm.f(3) — the Library Functions Manual page for CTFSM, and yes we are getting into the RFP-format triangular-matrix equation tea ☕💅.

💅 NAME

SRC/ctfsm.f — that's the whole name, no extra garnish. This page is literally here to introduce the ctfsm subroutine. 🫶

✨ SYNOPSIS

🧚 Functions/Subroutines

Here's the signature, bestie — keep it exactly as it is:

subroutine ctfsm (transr, side, uplo, trans, diag, m, n, alpha, a, b,
    ldb)

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

🎀 Function/Subroutine Documentation

🌸 subroutine ctfsm

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

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

🎯 Purpose

Okay so this is a 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 ) = A**H.

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; = 'C': The Conjugate-transpose Form of RFP A is stored. Unchanged on exit.
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. 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 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 = '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 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

Here's the Further Details block, preserved exactly (yes, !> and all) so you can vibe-check every packed-format diagram:

!>
!>  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-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.
!>  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-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 ctfsm.f. 🧚

👑 Author

Generated automatically by Doxygen for LAPACK from the source code.

okay that's the whole page, bestie. This cute little doc is LAPACK Version 3.12.0, from SRC/ctfsm.f(3) — go forth and solve those RFP triangular matrix equations. 💅✨🦋

📦 where this page comes from

arch rolling: lapack-doc 3.12.1-2 · custom
💖 girlified by deepseek-ai/DeepSeek-V4.1-Flash-1017 on 2026-11-04: a language model's rewrite of the original, under its terms