pftrs#

spftrs

Functions

void spftrs(
    const char*          transr,
    const char*          uplo,
    const INT            n,
    const INT            nrhs,
    const f32*  restrict A,
          f32*  restrict B,
    const INT            ldb,
          INT*           info
);

void spftrs(const char *transr, const char *uplo, const INT n, const INT nrhs, const f32 *restrict A, f32 *restrict B, const INT ldb, INT *info)#

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.

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

04 05                33 43 53
14 15                00 44 54
24 25                10 11 55
34 35                20 21 22
44 45                30 31 32
11 55                40 41 42
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

13 23 33 00 01 02    33 00 10 20 30 40 50
14 24 34 44 11 12    43 44 11 21 31 41 51
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

03 04                00 33 43
13 14                10 11 44
23 24                20 21 22
33 34                30 31 32
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

12 22 00 01             00 10 20 30 40 50
13 23 33 11             33 11 21 31 41 51
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.

in

nrhs

The number of right hand sides. nrhs>=0.

in

A

Array of 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 spftrf. See below for more details about RFP A.

inout

B

Array of dimension (ldb,nrhs). On entry, the right hand side matrix B. On exit, the solution matrix X.

in

ldb

The leading dimension of the array B. ldb>=max(1,n).

out

info

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

dpftrs

Functions

void dpftrs(
    const char*          transr,
    const char*          uplo,
    const INT            n,
    const INT            nrhs,
    const f64*  restrict A,
          f64*  restrict B,
    const INT            ldb,
          INT*           info
);

void dpftrs(const char *transr, const char *uplo, const INT n, const INT nrhs, const f64 *restrict A, f64 *restrict B, const INT ldb, INT *info)#

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.

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

04 05                33 43 53
14 15                00 44 54
24 25                10 11 55
34 35                20 21 22
44 45                30 31 32
11 55                40 41 42
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

13 23 33 00 01 02    33 00 10 20 30 40 50
14 24 34 44 11 12    43 44 11 21 31 41 51
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

03 04                00 33 43
13 14                10 11 44
23 24                20 21 22
33 34                30 31 32
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

12 22 00 01             00 10 20 30 40 50
13 23 33 11             33 11 21 31 41 51
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.

in

nrhs

The number of right hand sides. nrhs>=0.

in

A

Array of 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 below for more details about RFP A.

inout

B

Array of dimension (ldb,nrhs). On entry, the right hand side matrix B. On exit, the solution matrix X.

in

ldb

The leading dimension of the array B. ldb>=max(1,n).

out

info

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

cpftrs

Functions

void cpftrs(
    const char*          transr,
    const char*          uplo,
    const INT            n,
    const INT            nrhs,
    const c64*  restrict A,
          c64*  restrict B,
    const INT            ldb,
          INT*           info
);

void cpftrs(const char *transr, const char *uplo, const INT n, const INT nrhs, const c64 *restrict A, c64 *restrict B, const INT ldb, INT *info)#

CPFTRS solves a system of linear equations A*X = B with a Hermitian positive definite matrix A using the Cholesky factorization A = U**H*U or A = L*L**H computed by CPFTRF.

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 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 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 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.

in

nrhs

The number of right hand sides. nrhs>=0.

in

A

Complex array of 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**H, as computed by cpftrf. See below for more details about RFP A.

inout

B

Complex array of dimension (ldb,nrhs). On entry, the right hand side matrix B. On exit, the solution matrix X.

in

ldb

The leading dimension of the array B. ldb>=max(1,n).

out

info

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

zpftrs

Functions

void zpftrs(
    const char*          transr,
    const char*          uplo,
    const INT            n,
    const INT            nrhs,
    const c128* restrict A,
          c128* restrict B,
    const INT            ldb,
          INT*           info
);

void zpftrs(const char *transr, const char *uplo, const INT n, const INT nrhs, const c128 *restrict A, c128 *restrict B, const INT ldb, INT *info)#

ZPFTRS solves a system of linear equations A*X = B with a Hermitian positive definite matrix A using the Cholesky factorization A = U**H*U or A = L*L**H computed by ZPFTRF.

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 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 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 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.

in

nrhs

The number of right hand sides. nrhs>=0.

in

A

Complex array of 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**H, as computed by zpftrf. See below for more details about RFP A.

inout

B

Complex array of dimension (ldb,nrhs). On entry, the right hand side matrix B. On exit, the solution matrix X.

in

ldb

The leading dimension of the array B. ldb>=max(1,n).

out

info

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