pftrf(3)

📦

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🦋 pftrf(3) — the triangular factor glow-up

okay so picture this: it's pftrf(3), it lives in the Library Functions Manual, and it's literally the page that gives you the Cholesky factorization but in that ✨special✨ RFP storage layout. bestie, this one is a journey, but we're gonna get through it together 💖

🌸 NAME

pftrf — pftrf: triangular factor

like, that's the whole name. short. iconic. pftrf does the triangular factor thing. periodt.

✨ SYNOPSIS

🎀 Functions

subroutine cpftrf (transr, uplo, n, a, info) — CPFTRF 💜
subroutine dpftrf (transr, uplo, n, a, info) — DPFTRF 💜
subroutine spftrf (transr, uplo, n, a, info) — SPFTRF 💜
subroutine zpftrf (transr, uplo, n, a, info) — ZPFTRF 💜

four flavors, same energy: complex, double, single, double-complex. pick your fighter 🫶

🧚 Detailed Description

okay so this is the part where we get into the actual function documentation. buckle up, bestie, there's a lot of RFP format lore and honestly? it slays once you get it. 🔥

💅 Function Documentation

⭐ subroutine `cpftrf` (character `transr`, character `uplo`, integer `n`, complex, dimension( 0: * ) `a`, integer `info`)

CPFTRF — the complex one, the hermitian one, the main character 💅

🦋 Purpose:

so here's the vibe: CPFTRF 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. no cap.

and this one? this is the block version of the algorithm, calling Level 3 BLAS. highkey the fast one. 🔥

🌈 Parameters

TRANSR — CHARACTER1*
- = 'N': the Normal TRANSR of RFP A is stored
- = 'C': the Conjugate-transpose TRANSR of RFP A is stored
UPLO — CHARACTER1*
- = '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.
A — COMPLEX 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 — 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:

============================

okay bestie, this is the part everybody skips and then gets confused about later. don't be that girl. read the notes. 💖

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 cpftrf.f.

⭐ subroutine `dpftrf` (character `transr`, character `uplo`, integer `n`, double precision, dimension( 0: * ) `a`, integer `info`)

DPFTRF — same energy, but make it real symmetric 🫶

🦋 Purpose:

DPFTRF computes the Cholesky factorization of a real symmetric positive definite matrix A.

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.

and again, this is the block version of the algorithm, calling Level 3 BLAS. ⭐

🌈 Parameters

TRANSR — CHARACTER1*
- = 'N': the Normal TRANSR of RFP A is stored
- = 'T': the Transpose TRANSR of RFP A is stored
UPLO — CHARACTER1*
- = '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.
A — DOUBLE PRECISION 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 — 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 💀

btw bestie — notice TRANSR = 'T' here instead of 'C', and **T instead of **H. that's the whole real-vs-complex moment. don't mix 'em up. 🎀

👑 Author

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

🌟 Further Details:

okay so before we even touch the code we HAVE to talk about Rectangular Full Packed (RFP) format, because that is the entire personality of this routine and honestly it's giving origami 🦋 no cap, this is the part everyone skips and then cries about later.

Case one: N is even, bestie. We're gonna use N = 6 as the example. Here's AP in upper and lower flavour:

   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, periodt:

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

Case two: N is odd. Now N = 5, because we keep it cute and slightly off-balance 🙈

    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 so:

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 197 of file dpftrf.f. 📌

💖 `subroutine spftrf`

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

🌸 Purpose

SPFTRF computes the Cholesky factorization of a real symmetric positive definite matrix A.

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. icon behaviour honestly 👑

🎀 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

Case one: N is even, bestie. We're gonna use N = 6 as the example. Here's AP in upper and lower flavour:

   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, periodt:

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

Case two: N is odd. Now N = 5, because we keep it cute and slightly off-balance 🙈

    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 so:

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 197 of file spftrf.f. 📌

🦋 `subroutine zpftrf`

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

💜 Purpose

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. yeah, Level 3 BLAS, she's fancy like that 🔥

🧚 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 🧚

Okay bestie, this is the part where the diagrams come out. No cap, they’re important.

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. 📌

Definition at line 210 of file zpftrf.f.

Author (auto-generated) 💅

Generated automatically by Doxygen for LAPACK from the source code.

Bye bestie — this page is serving LAPACK Version 3.12.0 energy, and the footer said pftrf(3), so that’s the vibe we’re leaving you with 💖✨

📦 where this page comes from

arch rolling: lapack-doc 3.12.1-2 · custom
💖 girlified by deepseek-ai/DeepSeek-V4.1-Flash-1017 on 2026-11-04: a language model's rewrite of the original, under its terms

pftrf(3)

🦋 pftrf(3) — the triangular factor glow-up

🌸 NAME

✨ SYNOPSIS

🎀 Functions

🧚 Detailed Description

💅 Function Documentation

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

🦋 Purpose:

🌈 Parameters

🍓 Further Notes on RFP Format:

👑 Author

⭐ subroutine dpftrf (character transr, character uplo, integer n, double precision, dimension( 0: * ) a, integer info)

🦋 Purpose:

🌈 Parameters

👑 Author

🌟 Further Details:

💖 subroutine spftrf

🌸 Purpose

🎀 Parameters

🌈 Author

⭐ Further Details

🦋 subroutine zpftrf

💜 Purpose

🧚 Parameters

N 🌸

A 💖

INFO 🔥

Further Notes on RFP Format 🧚

Author 👑

Definition at line 210 of file zpftrf.f. 📌

Author (auto-generated) 💅

⭐ subroutine `cpftrf` (character `transr`, character `uplo`, integer `n`, complex, dimension( 0: * ) `a`, integer `info`)

⭐ subroutine `dpftrf` (character `transr`, character `uplo`, integer `n`, double precision, dimension( 0: * ) `a`, integer `info`)

💖 `subroutine spftrf`

🦋 `subroutine zpftrf`