SRC/ztfttr.f(3)

🌸 NAME 🌸

okay bestie, we're in the Library Functions Manual and the page is SRC/ztfttr.f(3) — the star of the show is SRC/ztfttr.f, and the subroutine is ztfttr 💅✨

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✨ SYNOPSIS ✨

🧚 Functions/Subroutines 🧚

subroutine ztfttr (transr, uplo, n, arf, a, lda, info)

• subroutine ztfttr (transr, uplo, n, arf, a, lda, info) — bestie, this is the whole moment: ZTFTTR copies a triangular matrix from the rectangular full packed format (TF) to the standard full format (TR). 👑

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💖 Function/Subroutine Documentation 💖

subroutine ztfttr (character transr, character uplo, integer n, complex*16,
     dimension( 0: * ) arf, complex*16, dimension( 0: lda-1, 0: * ) a,
     integer lda, integer info)

ZTFTTR copies a triangular matrix from the rectangular full packed format (TF) to the standard full format (TR). That's the official glow-up line, and honestly? iconic. 🦋

🎯 Purpose 🎯

ZTFTTR copies a triangular matrix A from rectangular full packed format (TF) to standard full format (TR). No cap, that's the whole job. ✨

🧩 Parameters 🧩

🌟 TRANSR 🌟

• TRANSR — TRANSR is CHARACTER*1. It tells you what flavor of RFP you're holding:
  • = 'N': ARF is in Normal format;
  • = 'C': ARF is in Conjugate-transpose format;

🦋 UPLO 🦋

• UPLO — UPLO is CHARACTER*1. This is the triangle check:
  • = 'U': A is upper triangular;
  • = 'L': A is lower triangular.

🎀 N 🎀

• N — N is INTEGER. This is the order of the matrix A. And the rule is N >= 0. 💖

🍓 ARF 🍓

• ARF — ARF is COMPLEX*16 array, dimension ( N*(N+1)/2 ), On entry, this is the upper or lower triangular matrix A stored in RFP format. For a further discussion see Notes below. (Yes bestie, the juicy diagrams are down in Further Details.) 👀

👑 A 👑

• A — A is COMPLEX*16 array, dimension ( LDA, N ). On exit, this is the triangular matrix A. If UPLO = 'U', the leading N-by-N upper triangular part of the array A contains the upper triangular matrix, and the strictly lower triangular part of A is not referenced. If UPLO = 'L', the leading N-by-N lower triangular part of the array A contains the lower triangular matrix, and the strictly upper triangular part of A is not referenced. 💅

🧚 LDA 🧚

• LDA — LDA is INTEGER. The leading dimension of the array A. The rule is LDA >= max(1,N). 🫶

💀 INFO 💀

• INFO — INFO is INTEGER. The status tea:
  • = 0: successful exit
  • < 0: if INFO = -i, the i-th argument had an illegal value

👑 Author 👑

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

🌈 Further Details 🌈

First up, bestie: 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 the 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 the 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

Next we 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 the 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 the 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 215 of file ztfttr.f.

🫶 Author 🫶

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okay bye bestie — this cutie is from LAPACK Version 3.12.0, page SRC/ztfttr.f(3) 💖✨