ZTFTTP(3)

ZTFTTP(3) — the Library Functions Manual, but make it cute 💅

Okay bestie, this is ZTFTTP(3) from the Library Functions Manual. It's a LAPACK thing, and honestly the vibe is "triangular matrix glow-up from one packed format to another." Let's get into it 💖

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🎀 NAME

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

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✨ SYNOPSIS

🦋 Functions/Subroutines

subroutine ZTFTTP (transr, uplo, n, arf, ap, info)

ZTFTTP copies a triangular matrix from the rectangular full packed format (TF) to the standard packed format (TP). Slay.

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🌸 Function/Subroutine Documentation

subroutine ZTFTTP (character transr, character uplo, integer n, complex*16, dimension( 0: * ) arf, complex*16, dimension( 0: * ) ap, integer info)

ZTFTTP copies a triangular matrix from the rectangular full packed format (TF) to the standard packed format (TP).

💜 Purpose:

Okay so the whole entire point: ZTFTTP copies a triangular matrix A from rectangular full packed format (TF) to standard packed format (TP). That's it. That's the gig. No cap 💅

🎀 Parameters

• TRANSR — CHARACTER*1
  • = 'N': ARF is in Normal format;
  • = 'C': ARF is in Conjugate-transpose format;
• UPLO — CHARACTER*1
  • = 'U': A is upper triangular;
  • = 'L': A is lower triangular.
• N — INTEGER; the order of the matrix A. N >= 0.
• ARF — COMPLEX*16 array, dimension ( N*(N+1)/2 ); on entry, the upper or lower triangular matrix A stored in RFP format. For a further discussion see Notes below.
• AP — COMPLEX*16 array, dimension ( N*(N+1)/2 ); on exit, the upper or lower triangular matrix A, packed columnwise in a linear array. The j-th column of A is stored in the array AP as follows: if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j; if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.
• INFO — INTEGER
  • = 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:

Okay so first we consider Standard Packed Format when N is even. Example time: 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

Next up: Standard Packed Format when N is odd. 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 207 of file ztfttp.f.

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🧚 Author

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Signed, sealed, sparkled: LAPACK Version 3.12.0, source file /home/abui...RC/ztfttp.f(3). Periodt 💖✨