SRC/dtftri.f(3)

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version 3.12.1 · 7 versions
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✨ SRC/dtftri.f(3) ✨

From the Library Functions Manual, bestie — this is the page for SRC/dtftri.f, and it’s lowkey giving “tiny Fortran routine with a big RFP-format personality.” 💅

🎀 NAME 🎀

SRC/dtftri.f — the dtftri subroutine, serving triangular-inverse realness. 💖

✨ SYNOPSIS ✨

   Functions/Subroutines
       subroutine dtftri (transr, uplo, diag, n, a, info)
           DTFTRI

🦋 Function/Subroutine Documentation 🦋

subroutine dtftri (character transr, character uplo, character diag, integer
       n, double precision, dimension( 0: * ) a, integer info)

DTFTRI 💫

Purpose 💖

Okay so DTFTRI computes the inverse of a triangular matrix A stored in RFP format. No cap. This is a Level 3 BLAS version of the algorithm. Periodt. ✨

Parameters 🌸

TRANSR

TRANSR is CHARACTER*1. It’s the transr vibe check:

  • = 'N': The Normal TRANSR of RFP A is stored;
  • = 'T': The Transpose TRANSR of RFP A is stored.

UPLO

UPLO is CHARACTER*1:

  • = 'U': A is upper triangular;
  • = 'L': A is lower triangular.

DIAG

DIAG is CHARACTER*1:

  • = 'N': A is non-unit triangular;
  • = 'U': A is unit triangular.

N

N is INTEGER. The order of the matrix A. N >= 0. 💅

A

A is DOUBLE PRECISION array, dimension (0:nt-1); nt=N*(N+1)/2. On entry, the triangular factor of a Hermitian Positive Definite 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; If UPLO = 'L' the RFP A contains the nt elements of lower packed A. 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 N is odd. See the Note below for more details. 👀

On exit, the (triangular) inverse of the original matrix, in the same storage format. 🫶

INFO

INFO is INTEGER:

  • = 0: successful exit
  • < 0: if INFO = -i, the i-th argument had an illegal value
  • > 0: if INFO = i, A(i,i) is exactly zero. The triangular matrix is singular and its inverse can not be computed.
Bestie, that > 0 case is not a cute little glitch. It means A(i,i) is exactly zero, the triangular matrix is singular, and its inverse can not be computed. 💀

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 200 of file dtftri.f. 📍

Author (page generation) 🫶

Generated automatically by Doxygen for LAPACK from the source code.

Okay bestie, that’s the whole page: SRC/dtftri.f(3) from the Library Functions Manual, version LAPACK Version 3.12.0. The footer has no date stamped on it, so no invented date — periodt. Bye and slay your triangular inverses responsibly. 💜✨

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