DPFTRS(3)

💅 DPFTRS(3) 💅

okay so bestie, this is the DPFTRS page from the Library Functions Manual, and it's lowkey a whole mood: symmetric positive definite matrix energy, Cholesky factorization, and a solve that ate. no cap. ✨

🌸 NAME 🌸

/home/abuild/rpmbuild/BUILD/lapack-static-3.12.0-build/lapack-3.12.0/SRC/dpftrs.f

That's the source file, periodt. It's giving "long path but worth it." 💖

✨ SYNOPSIS ✨

Functions/Subroutines

   Functions/Subroutines
       subroutine DPFTRS (transr, uplo, n, nrhs, a, b, ldb, info)
           DPFTRS

💖 Function/Subroutine Documentation 💖

   subroutine DPFTRS (character transr, character uplo, integer n, integer
       nrhs, double precision, dimension( 0: * ) a, double precision,
       dimension( ldb, * ) b, integer ldb, integer info)
       DPFTRS

bestie, that's the full signature, and yes it's the main character. 🫶

🎀 Purpose 🎀

DPFTRS 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 DPFTRF.

That's the whole plot, and honestly? iconic.

🧚 Parameters 🧚

• TRANSR — CHARACTER*1. Which TRANSR of RFP A is stored:
  • = 'N': The Normal TRANSR of RFP A is stored;
  • = 'T': The Transpose TRANSR of RFP A is stored.
• UPLO — CHARACTER*1. Which triangle of RFP A is stored:
  • = '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 — DOUBLE PRECISION array, dimension ( N*(N+1)/2 ). The triangular factor U or L from the Cholesky factorization of RFP A = U**T*U or RFP A = L*L**T, as computed by DPFTRF. See note below for more details about RFP A.
• B — DOUBLE PRECISION 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. Exit status tea:
  • = 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 this next part is the RFP format lore and it's a lot, but we're getting through it together. First up: 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 dpftrs.f.

🌈 Author 🌈

Generated automatically by Doxygen for LAPACK from the source code.

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okay bestie that's the whole page 💖 this was LAPACK Version 3.12.0, source path /home/abui...RC/dpftrs.f(3), and it's living in the Library Functions Manual. go solve that system, slay, periodt. 🫶