SRC/spftrs.f(3)

💅 SRC/spftrs.f(3) — Library Functions Manual 💅

🌸 NAME 🌸

SRC/spftrs.f

Okay bestie, this page is for SRC/spftrs.f — specifically the SPFTRS routine, an LAPACK solver for A*X = B with a symmetric positive definite matrix A. 💖

✨ SYNOPSIS ✨

Functions/Subroutines

• subroutine spftrs (transr, uplo, n, nrhs, a, b, ldb, info) — SPFTRS

🦋 Function/Subroutine Documentation 🦋

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

SPFTRS

🎯 Purpose

SPFTRS solves a system of linear equations A*X = B with a symmetric positive definite matrix A using the Cholesky factorization A = U**T*U or A = L*L**T computed by SPFTRF. Slay. 💖

🌈 Parameters

• TRANSR — is CHARACTER*1
  • = 'N': The Normal TRANSR of RFP A is stored;
  • = 'T': The Transpose TRANSR of RFP A is stored.
• UPLO — is CHARACTER*1
  • = 'U': Upper triangle of RFP A is stored;
  • = 'L': Lower triangle of RFP A is stored.
• N — is INTEGER. The order of the matrix A. N >= 0.
• NRHS — is INTEGER. The number of right hand sides, i.e., the number of columns of the matrix B. NRHS >= 0.
• A — is REAL 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**T, as computed by SPFTRF. See note below for more details about RFP A.
• B — is REAL array, dimension (LDB,NRHS). On entry, the right hand side matrix B. On exit, the solution matrix X.
• LDB — is INTEGER. The leading dimension of the array B. LDB >= max(1,N).
• 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 so first we 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 198 of file spftrs.f.

🪄 Author

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And that's the whole vibe, bestie 💖 — this is LAPACK, Version 3.12.0, and the page is SRC/spftrs.f(3). Bye! 🎀