CPFTRS(3)

🎀 CPFTRS(3) — the LAPACK page for `cpftrs.f` 💅

okay so real talk: this page lives in the Library Functions Manual, and it's literally just the auto-generated docs for one file, /home/abuild/rpmbuild/BUILD/lapack-static-3.12.0-build/lapack-3.12.0/SRC/cpftrs.f. That's the whole NAME section, bestie — the name is the path. iconic behavior honestly. 🦋

✨ SYNOPSIS ✨

Functions/Subroutines

subroutine CPFTRS (transr, uplo, n, nrhs, a, b, ldb, info) — CPFTRS 🫶

💖 Function/Subroutine Documentation 💖

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

CPFTRS 🌸

Purpose 👑

CPFTRS 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 CPFTRF.

so basically: you already did the Cholesky thing with CPFTRF, congrats, and now you hand the factor to CPFTRS and it serves you the actual solutions. no cap, that's the whole vibe. ✨

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 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 CPFTRF. See note below for more details about RFP A. 👀 (and yes, the note below is long, buckle up 🎢)

B

B is COMPLEX array, dimension (LDB,NRHS)

On entry, the right hand side matrix B. On exit, the solution matrix X.

it literally transforms in front of your eyes, main character energy 💫

LDB

LDB is INTEGER

The leading dimension of the array B. LDB >= max(1,N).

bestie please respect the max(1,N) — do not come crying to me when your leading dimension is too small 😭

INFO

INFO is INTEGER

= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value

so if INFO is negative, count from the left: argument number -INFO is the one that's acting up. 💀

Author 👑

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

group project but make it ✨academia✨ — and they actually finished it, we stan. 🫶

Further Details: 🧚 the RFP lore 🧚

okay this is the part where it gets deliciously extra. RFP = Rectangular Full Packed, and the whole point is that the triangle you need is crammed into a rectangle so BLAS can eat it. let's go. 🌈

Case 1: Standard Packed Format, N is even 🍓

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

Case 2: Standard Packed Format, N is odd 🦋

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

like, look at those little -- crowns floating above the numbers. conjugate-transpose but make it fashion. 👑💅

Definition at line 219 of file `cpftrs.f`. 📍

(author of the page itself:)

Author

Generated automatically by Doxygen for LAPACK from the source code.

not a human, not a vibe, just Doxygen doing Doxygen things 🤖✨

okay that's the whole page bestie, ily, go solve your Hermitian positive definite system 🫶💖

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