SRC/spftri.f(3)

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version 3.12.1 · 4 versions
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🎀 SRC/spftri.f(3) — Library Functions Manual 🎀

okay so bestie, welcome to the glow-up era of spftri 💅 this page lives in the LAPACK manual and it's literally all about inverting your fave symmetric positive definite matrix. let's get into it ✨

🌸 NAME 🌸

SRC/spftri.f — no cap, that's the whole name 🫶

✨ SYNOPSIS ✨

Functions/Subroutines

  • subroutine spftri (transr, uplo, n, a, info) — the one and only SPFTRI 👑

💖 Function/Subroutine Documentation 💖

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

SPFTRI — she's iconic, she's the main character 🌟

🦋 Purpose 🦋

SPFTRI computes the inverse of a real (symmetric) positive definite matrix A using the Cholesky factorization A = U**T*U or A = L*L**T computed by SPFTRF.

so yeah, SPFTRF did the factorization homework, and SPFTRI comes in and flips it into the inverse 💫

🌈 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 A is stored ⬆️
  • 'L' — Lower triangle of A is stored ⬇️

N

N is INTEGER

The order of the matrix A. Constraint: 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, the symmetric inverse of the original matrix, in the same storage format ✨ (yes, same fit, different vibe)

INFO

INFO is INTEGER

  • = 0 — successful exit 🎉
  • < 0 — if INFO = -i, the i-th argument had an illegal value 😬
  • > 0 — if INFO = i, the (i,i) element of the factor U or L is zero, and the inverse could not be computed 💀

👑 Author 👑

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

🧚 Further Details 🧚

We first consider Rectangular Full Packed (RFP) 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 the 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 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. 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 then consider Rectangular Full Packed (RFP) 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 the 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 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. 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 190 of file spftri.f 📍

💜 Author 💜

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and that's the whole page bestie, hope you ate 🍓 this was LAPACK Version 3.12.0, page SRC/spftri.f(3) — go invert something, ily, bye 👋💖

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