DTFTRI(3)

🌸 DTFTRI — the girlboss guide to inverting your triangular matrices 💅

ok so bestie, gather round, because this one is from the LAPACK Library Functions Manual and it's literally about flipping a triangular matrix inside out. 🦋

The file in question lives at:

/home/abuild/rpmbuild/BUILD/lapack-static-3.12.0-build/lapack-3.12.0/SRC/dtftri.f

and that's the whole NAME section, no cap. 💖

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✨ SYNOPSIS ✨

Functions/Subroutines 🎀

subroutine DTFTRI (transr, uplo, diag, n, a, info)
    DTFTRI

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🔥 Function/Subroutine Documentation 🔥

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

🌈 Purpose:

ok so here's the tea ☕ — DTFTRI computes the inverse of a triangular matrix A that's stored in RFP format. That's it, that's the whole vibe. ✨

And it's a Level 3 BLAS version of the algorithm. iconic behaviour honestly. 👑

💜 Parameters

<purple>TRANSR</purple>

TRANSR is CHARACTER*1, and it's giving "which flavour of RFP am I storing":

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

<purple>UPLO</purple>

UPLO is CHARACTER*1 — where's the triangle, bestie?

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

<purple>DIAG</purple>

DIAG is CHARACTER*1 — is the diagonal doing anything or is it just sitting there being 1?

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

<purple>N</purple>

N is INTEGER — the order of the matrix A. Constraint: N >= 0. ✨

<purple>A</purple>

A is DOUBLE PRECISION array, dimension (0:nt-1); where nt=N*(N+1)/2. 🫶

On entry, A holds the triangular factor of a Hermitian Positive Definite matrix A in RFP format. RFP format is described by TRANSR, UPLO, and N as follows — and ngl this part is a whole paragraph so buckle up:

• 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, A holds the (triangular) inverse of the original matrix, in the same storage format. same energy, different matrix. 💫

<purple>INFO</purple>

INFO is INTEGER and it's the vibe check at the end:

• = 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 if you get INFO > 0 it's over, the matrix is singular, there is no inverse, go home. 😭

👑 Author

• Univ. of Tennessee
• Univ. of California Berkeley
• Univ. of Colorado Denver
• NAG Ltd.

🧚 Further Details:

ok so this is the part where they explain RFP format with actual pictures and honestly? iconic. let me set the scene. 🌸

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

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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

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📌 Definition at line 200 of file dtftri.f.

🎀 Author

Generated automatically by Doxygen for LAPACK from the source code.

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and that's the whole page bestie!! ✨ this absolute queen came to you from LAPACK, Version 3.12.0, and yeah the source path is literally /home/abui...RC/dtftri.f(3) — abbreviate it, we love a short king. now go invert something. 🫶💖