ZPFTRF(3)

💅 ZPFTRF(3) — Library Functions Manual 💅

okay bestie, welcome to the ZPFTRF(3) page from the Library Functions Manual. This is the LAPACK subroutine that does the Cholesky factorization thing, and yes it is giving main-character linear algebra energy. ✨

🌸 NAME 🌸

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

✨ SYNOPSIS ✨

🧚 Functions/Subroutines 🧚

subroutine ZPFTRF (transr, uplo, n, a, info)
    ZPFTRF

🦋 Function/Subroutine Documentation 🦋

subroutine ZPFTRF (character transr, character uplo, integer n, complex*16,
    dimension( 0: * ) a, integer info)
    ZPFTRF

🎀 Purpose 🎀

okay so ZPFTRF computes the Cholesky factorization of a complex Hermitian positive definite matrix A. The factorization has the form

A = U**H * U,  if UPLO = 'U', or
A = L  * L**H,  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. slay.

🌈 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.
• A — A is COMPLEX*16 array, dimension ( N*(N+1)/2 ); On entry, the Hermitian 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 = 'C' then RFP is the Conjugate-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 = 'C'. 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**H*U or RFP A = L*L**H.

• 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.

🧚 Further Notes on RFP Format: 🧚

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

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

👑 Author 👑

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

Definition at line 210 of file zpftrf.f.

🌟 Author 🌟

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