pftrf#

Functions

void spftrf(
    const char*          transr,
    const char*          uplo,
    const INT            n,
          f32*  restrict A,
          INT*           info
);
void spftrf(const char *transr, const char *uplo, const INT n, f32 *restrict A, INT *info)#

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.

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

Parameters

in
transr

  • 'N': The Normal TRANSR of RFP A is stored

  • 'T': The Transpose TRANSR of RFP A is stored

in
uplo

  • 'U': Upper triangle of RFP A is stored

  • 'L': Lower triangle of RFP A is stored

in
n

The order of the matrix A. n>=0.

inout
A

Array of 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 below for further 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.

out
info

  • info=0: successful exit

  • info<0: if info=-i, the i-th argument had an illegal value

  • info>0: if info=i, the leading principal minor of order i is not positive, and the factorization could not be completed.

Functions

void dpftrf(
    const char*          transr,
    const char*          uplo,
    const INT            n,
          f64*  restrict A,
          INT*           info
);
void dpftrf(const char *transr, const char *uplo, const INT n, f64 *restrict A, INT *info)#

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.

This is the block version of the algorithm, calling Level 3 BLAS.

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

Parameters

in
transr

  • 'N': The Normal TRANSR of RFP A is stored

  • 'T': The Transpose TRANSR of RFP A is stored

in
uplo

  • 'U': Upper triangle of RFP A is stored

  • 'L': Lower triangle of RFP A is stored

in
n

The order of the matrix A. n>=0.

inout
A

Array of 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 below for further 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.

out
info

  • info=0: successful exit

  • info<0: if info=-i, the i-th argument had an illegal value

  • info>0: if info=i, the leading principal minor of order i is not positive, and the factorization could not be completed.

Functions

void cpftrf(
    const char*          transr,
    const char*          uplo,
    const INT            n,
          c64*  restrict A,
          INT*           info
);
void cpftrf(const char *transr, const char *uplo, const INT n, c64 *restrict A, INT *info)#

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.

This is the block version of the algorithm, calling Level 3 BLAS.

Further Details:

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

Parameters

in
transr

  • 'N': The Normal TRANSR of RFP A is stored

  • 'C': The Conjugate-transpose TRANSR of RFP A is stored

in
uplo

  • 'U': Upper triangle of RFP A is stored

  • 'L': Lower triangle of RFP A is stored

in
n

The order of the matrix A. n>=0.

inout
A

Complex array of 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 below for further 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.

out
info

  • info=0: successful exit

  • info<0: if info=-i, the i-th argument had an illegal value

  • info>0: if info=i, the leading principal minor of order i is not positive, and the factorization could not be completed.

Functions

void zpftrf(
    const char*          transr,
    const char*          uplo,
    const INT            n,
          c128* restrict A,
          INT*           info
);
void zpftrf(const char *transr, const char *uplo, const INT n, c128 *restrict A, INT *info)#

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.

Further Details:

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

Parameters

in
transr

  • 'N': The Normal TRANSR of RFP A is stored

  • 'C': The Conjugate-transpose TRANSR of RFP A is stored

in
uplo

  • 'U': Upper triangle of RFP A is stored

  • 'L': Lower triangle of RFP A is stored

in
n

The order of the matrix A. n>=0.

inout
A

Complex array of 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 below for further 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.

out
info

  • info=0: successful exit

  • info<0: if info=-i, the i-th argument had an illegal value

  • info>0: if info=i, the leading principal minor of order i is not positive, and the factorization could not be completed.