SRC/ztftri.f(3)

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🦋 SRC/ztftri.f(3) — Library Functions Manual 🦋

🌸 NAME 🌸

SRC/ztftri.f — bestie, that’s literally the page name, and we’re about to get into all of it 💅

✨ 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 💫

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

🎀 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.

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.

🫶 Author 🫶

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

🔥 Further Details 🔥

okay bestie, this next part is the whole RFP-format lore drop. Stay with me — the diagrams are iconic and they matter. 💖
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 220 of file ztftri.f.

🌟 Author 🌟

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