SRC/spftrf.f(3)

📦
version 3.12.1 · 7 versions
, in arch rolling

💖 SRC/spftrf.f(3) — Library Functions Manual 💖

okay so this is the LAPACK page for SPFTRF, and we are gonna get you comfy with it, bestie. It's giving "block Cholesky factorization in RFP format" and honestly it ate. 🫶 iykyk.

🎀 NAME 🎀

SRC/spftrf.f — that's the file, and SPFTRF is the main character here. 💅

✨ SYNOPSIS ✨

Functions/Subroutines

SPFTRF

subroutine spftrf (character transr, character uplo, integer n, real, dimension( 0: * ) a, integer info)

💅 Function/Subroutine Documentation 💅

SPFTRF

subroutine spftrf (character transr, character uplo, integer n, real, dimension( 0: * ) a, integer info)

Purpose:

okay so SPFTRF computes the Cholesky factorization of a real symmetric positive definite matrix A. No cap. ✨

The factorization has the form

A = U**T * U,  if UPLO = 'U', or
A = L  * L**T,  if UPLO = 'L',

where U is an upper triangular matrix and L is lower triangular.

This is the block version of the algorithm, calling Level 3 BLAS. periodt. 🔥

Parameters

  • TRANSR — TRANSR is CHARACTER*1.
    • = 'N': The Normal TRANSR of RFP A is stored;
    • = 'T': The 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.
  • A — A is REAL array, dimension ( N*(N+1)/2 ).

On entry, the symmetric matrix A in RFP format. RFP format is described by TRANSR, UPLO, and N as follows: If TRANSR = 'N' then RFP A is (0:N,0:k-1) when N is even; k=N/2. RFP A is (0:N-1,0:k) when N is odd; k=N/2. IF TRANSR = 'T' then RFP is the transpose of RFP A as defined when TRANSR = 'N'. The contents of RFP A are defined by UPLO as follows: If UPLO = 'U' the RFP A contains the NT elements of upper packed A. If UPLO = 'L' the RFP A contains the elements of lower packed A. The LDA of RFP A is (N+1)/2 when TRANSR = 'T'. When TRANSR is 'N' the LDA is N+1 when N is even and N is odd. See the Note below for more details.

On exit, if INFO = 0, the factor U or L from the Cholesky factorization RFP A = U**T*U or RFP A = L*L**T.

  • INFO — INFO is INTEGER.
    • = 0: successful exit
    • < 0: if INFO = -i, the i-th argument had an illegal value 😭
    • > 0: if INFO = i, the leading principal minor of order i is not positive, and the factorization could not be completed. 🙈

Author

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

Further Details:

okay so RFP format is a whole vibe, and here's the breakdown, bestie. 💖

First up, Rectangular Full Packed (RFP) Format when N is even, with the example 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) is literally the last three columns of AP upper, and the lower triangle A(4:6,0:2) is the transpose of the first three columns of AP upper. For UPLO = 'L', the lower trapezoid A(1:6,0:2) is the first three columns of AP lower, and the upper triangle A(0:2,0:2) is the transpose of the last three columns of AP lower. 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 = 'T'. RFP A in both UPLO cases is just the transpose of RFP A above, so you get:

           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

Then we consider Rectangular Full Packed (RFP) Format when N is odd, with the 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) is the last three columns of AP upper, and the lower triangle A(3:4,0:1) is the transpose of the first two columns of AP upper. For UPLO = 'L', the lower trapezoid A(0:4,0:2) is the first three columns of AP lower, and the upper triangle A(0:1,1:2) is the transpose of the last two columns of AP lower. 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 = 'T'. RFP A in both UPLO cases is just the transpose of RFP A above, so you get:

           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 197 of file spftrf.f.

👑 Author 👑

Generated automatically by Doxygen for LAPACK from the source code.

okay that's the page, bestie 💖 — it's LAPACK Version 3.12.0, source SRC/spftrf.f(3), and the footer didn't give us a date, so no cap, that's all she wrote, fr fr. 🎀✨

📦 where this page comes from