SRC/zpftrs.f(3)

🫶 SRC/zpftrs.f(3) — Library Functions Manual, but make it cute

okay so bestie, this is the SRC/zpftrs.f(3) page and it’s literally all about the zpftrs subroutine. no cap, it’s giving very specific linear-algebra main character energy ✨

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💅 NAME 💅

SRC/zpftrs.f — the ZPFTRS page, bestie 🎀

🌈 SYNOPSIS 🌈

🧚 Functions/Subroutines 🧚

subroutine zpftrs (transr, uplo, n, nrhs, a, b, ldb, info)

ZPFTRS

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🔥 Function/Subroutine Documentation 🔥

subroutine zpftrs (character transr, character uplo, integer n, integer
    nrhs, complex*16, dimension( 0: * ) a, complex*16, dimension( ldb, * )
    b, integer ldb, integer info)

ZPFTRS

🌸 Purpose 🌸

okay so the whole purpose is: ZPFTRS solves a system of linear equations A*X = B with a Hermitian positive definite matrix A using the Cholesky factorization A = U**H*U or A = L*L**H computed by ZPFTRF. slay 💅

💖 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': Upper triangle of RFP A is stored;
  • = 'L': Lower triangle of RFP A is stored.
• N — N is INTEGER
  • The order of the matrix A. N >= 0.
• NRHS — NRHS is INTEGER
  • The number of right hand sides, i.e., the number of columns of the matrix B. NRHS >= 0.
• A — A is COMPLEX*16 array, dimension ( N*(N+1)/2 );
  • The triangular factor U or L from the Cholesky factorization of RFP A = U**H*U or RFP A = L*L**H, as computed by ZPFTRF.
  • See note below for more details about RFP A.
• B — B is COMPLEX*16 array, dimension (LDB,NRHS)
  • On entry, the right hand side matrix B.
  • On exit, the solution matrix X.
• LDB — LDB is INTEGER
  • The leading dimension of the array B. LDB >= max(1,N).
• INFO — INFO is 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 bestie, here’s where it gets into the packed-format lore. 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 219 of file zpftrs.f.

📜 Author 📜

Generated automatically by Doxygen for LAPACK from the source code.

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