ZTFTRI(3)

📦 version 3.12.0, in opensuse-leap 16.0

🌸 ZTFTRI(3) — the ztftri.f page, bestie edition 🌸

Okay so real quick: this is a page from the Library Functions Manual, and it lives at the path /home/abuild/rpmbuild/BUILD/lapack-static-3.12.0-build/lapack-3.12.0/SRC/ztftri.f. That's the whole NAME section, no cap — the "name" of this one is the file path. Slay. 💅

✨ SYNOPSIS ✨

Functions/Subroutines 🫶

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

🦋 Function/Subroutine Documentation 🦋

subroutine ZTFTRI (character transr, character uplo, character diag, integer
    n, complex*16, dimension( 0: * ) a, integer info)
    ZTFTRI

💖 Purpose 💖

So what does she do? ZTFTRI computes the inverse of a triangular matrix A stored in RFP format. Iconic behaviour, honestly.

And lowkey important: this is a Level 3 BLAS version of the algorithm. Noted, periodt.

🎀 Parameters 🎀

TRANSR

TRANSR is CHARACTER*1 = 'N': The Normal TRANSR of RFP A is stored; = 'C': The Conjugate-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 COMPLEX*16 array, dimension ( N*(N+1)/2 ); On entry, the triangular 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; 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 = 'C'. 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. She ate and left no crumbs. 💅

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.
🫶 Heads up bestie: that last one is the vibe check you don't want to fail — singular matrix means the inverse literally can't be computed. No workaround, no cap.

👑 Author 👑

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

🌈 Further Details: 🌈

Alright, gather round, this is the part where she gets detailed and honestly it's giving main character energy. 💖

Even N, bestie

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

Odd N, the sequel ✨

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 220 of file ztftri.f.

🧚 Author (the meta one) 🧚

Generated automatically by Doxygen for LAPACK from the source code. Not me writing this by hand, bestie — the robots did it. 🤖✨

🍓 And that's the whole page! Footer business: this is LAPACK, Version 3.12.0, and the source string on the bottom line is /home/abui...RC/ztftri.f(3). Bye bestie, go invert something triangular! 💜🎀🦋
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