ZPFTRS(3)

📦 version 3.12.0, in opensuse-leap 16.0

okay bestie, you found ZPFTRS(3) in the Library Functions Manual, and this page is about the LAPACK routine that solves linear systems with a Hermitian positive definite matrix. it's giving serious Cholesky energy, no cap. 💅✨

🌸 NAME 🌸

the page source is /home/abuild/rpmbuild/BUILD/lapack-static-3.12.0-build/lapack-3.12.0/SRC/zpftrs.f, and the manual name is ZPFTRS(3). that's the whole NAME section, bestie — iconic minimalism. 💖

✨ SYNOPSIS ✨

🧚 Functions/Subroutines 🧚

  • subroutine ZPFTRS (transr, uplo, n, nrhs, a, b, ldb, info) — the one and only, a.k.a. ZPFTRS.
subroutine ZPFTRS (transr, uplo, n, nrhs, a, b, ldb, info)

🦋 Function/Subroutine Documentation 🦋

here's the full signature, exact characters and all, because we do not gatekeep:

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 🎯

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 — CHARACTER*1. = 'N': the Normal TRANSR of RFP A is stored; = 'C': the Conjugate-transpose TRANSR of RFP A is stored.
  • UPLO — CHARACTER*1. = 'U': Upper triangle of RFP A is stored; = 'L': Lower triangle of RFP A is stored.
  • N — INTEGER. The order of the matrix A. N >= 0.
  • NRHS — INTEGER. The number of right hand sides, i.e., the number of columns of the matrix B. NRHS >= 0.
  • A — 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 — COMPLEX*16 array, dimension (LDB,NRHS). On entry, the right hand side matrix B. On exit, the solution matrix X.
  • LDB — INTEGER. The leading dimension of the array B. LDB >= max(1,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 🔥

first up: 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 like this:

  • 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.

conjugates are shown with -- 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. example: 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 like this:

  • 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.

conjugates are shown with -- 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 🫶

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that's the whole page, bestie. signing off from LAPACK — Version 3.12.0 — source /home/abui...RC/zpftrs.f(3). 💖✨🎀

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