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Parallel Engineering and Scientific Subroutine Library for AIX Version 2 Release 3: Guide and Reference

PDPOTRS and PZPOTRS--Positive Definite Real Symmetric or Complex Hermitian Matrix Solve

These subroutines solve the following systems of equations for multiple right-hand sides:

AX = B

where, in the formula above:

A represents the global positive definite real symmetric or complex Hermitian submatrix Aia:ia+n-1, ja:ja+n-1 factored by Cholesky factorization.
B represents the global general submatrix Bib:ib+n-1, jb:jb+nrhs-1 containing the right-hand sides in its columns.
X represents the global general submatrix Bib:ib+n-1, jb:jb+nrhs-1 containing the solution vectors in its columns.

This subroutine uses the results of the factorization of matrix A, produced by a preceding call to PDPOTRF or PZPOTRF, respectively. For details on the factorization, see PDPOTRF and PZPOTRF--Positive Definite Real Symmetric or Complex Hermitian Matrix Factorization.

If n = 0 or nrhs = 0, no computation is performed and the subroutine returns after doing some parameter checking. See references [16], [18], [22], [36], and [37].

Table 67. Data Types

A, B Subroutine
Long-precision real PDPOTRS
Long-precision complex PZPOTRS

Syntax

Fortran CALL PDPOTRS | PZPOTRS (uplo, n, nrhs, a, ia, ja, desc_a, b, ib, jb, desc_b, info)
C and C++ pdpotrs | pzpotrs (uplo, n, nrhs, a, ia, ja, desc_a, b, ib, jb, desc_b, info);

On Entry

uplo
indicates whether the upper or lower triangular part of the global real symmetric or complex Hermitian submatrix A is referenced, where:

If uplo = 'U', the upper triangular part is referenced.

If uplo = 'L', the lower triangular part is referenced.

Scope: global

Specified as: a single character; uplo = 'U' or 'L'.

n
is the order of the factored submatrix A.

Scope: global

Specified as: a fullword integer; n >= 0.

nrhs
is the number of right-hand sides-- that is, the number of columns in submatrix B used in the computation.

Scope: global

Specified as: a fullword integer; nrhs >= 0.

a
is the local part of the global real symmetric or complex Hermitian matrix A, containing the factorization of matrix A produced by a preceding call to PDPOTRF or PZPOTRF, respectively. This identifies the first element of the local array A. This subroutine computes the location of the first element of the local subarray used, based on ia, ja, desc_a, p, q, myrow, and mycol; therefore, the leading LOCp(ia+n-1) by LOCq(ja+n-1) part of the local array A must contain the local pieces of the leading ia+n-1 by ja+n-1 part of the global matrix, and:

Scope: local

Specified as: an LLD_A by (at least) LOCq(N_A) array, containing numbers of the data type indicated in Table 67. Details about the square block-cyclic data distribution of global matrix A are stored in desc_a.

ia
is the row index of the global matrix A, identifying the first row of the submatrix A.

Scope: global

Specified as: a fullword integer; 1 <= ia <= M_A and ia+n-1 <= M_A.

ja
is the column index of the global matrix A, identifying the first column of the submatrix A.

Scope: global

Specified as: a fullword integer; 1 <= ja <= N_A and ja+n-1 <= N_A.

desc_a
is the array descriptor for global matrix A, described in the following table:
desc_a Name Description Limits Scope
1 DTYPE_A Descriptor type DTYPE_A=1 Global
2 CTXT_A BLACS context Valid value, as returned by BLACS_GRIDINIT or BLACS_GRIDMAP Global
3 M_A Number of rows in the global matrix If n = 0:
M_A >= 0
Otherwise:
M_A >= 1
Global
4 N_A Number of columns in the global matrix If n = 0:
N_A >= 0
Otherwise:
N_A >= 1
Global
5 MB_A Row block size MB_A >= 1 Global
6 NB_A Column block size NB_A >= 1 Global
7 RSRC_A The process row of the p × q grid over which the first row of the global matrix is distributed 0 <= RSRC_A < p Global
8 CSRC_A The process column of the p × q grid over which the first column of the global matrix is distributed 0 <= CSRC_A < q Global
9 LLD_A The leading dimension of the local array LLD_A >= max(1,LOCp(M_A)) Local

Specified as: an array of (at least) length 9, containing fullword integers.

b
is the local part of the global general matrix B, containing the right-hand sides of the system. This identifies the first element of the local array B. This subroutine computes the location of the first element of the local subarray used, based on ib, jb, desc_b, p, q, myrow, and mycol; therefore, the leading LOCp(ib+n-1) by LOCq(jb+nrhs-1) part of the local array B must contain the local pieces of the leading ib+n-1 by jb+nrhs-1 part of the global matrix.

Scope: local

Specified as: an LLD_B by (at least) LOCq(N_B) array, containing numbers of the data type indicated in Table 67. Details about the block-cyclic data distribution of global matrix B are stored in desc_b.

ib
is the row index of the global matrix B, identifying the first row of the submatrix B.

Scope: global

Specified as: a fullword integer; 1 <= ib <= M_B and ib+n-1 <= M_B.

jb
is the column index of the global matrix B, identifying the first column of the submatrix B.

Scope: global

Specified as: a fullword integer; 1 <= jb <= N_B and jb+nrhs-1 <= N_B.

desc_b
is the array descriptor for global matrix B, described in the following table:
desc_b Name Description Limits Scope
1 DTYPE_B Descriptor type DTYPE_B=1 Global
2 CTXT_B BLACS context Valid value, as returned by BLACS_GRIDINIT or BLACS_GRIDMAP Global
3 M_B Number of rows in the global matrix If n = 0 or nrhs = 0:
M_B >= 0
Otherwise:
M_B >= 1
Global
4 N_B Number of columns in the global matrix If n = 0 or nrhs = 0:
N_B >= 0
Otherwise:
N_B >= 1
Global
5 MB_B Row block size MB_B >= 1 Global
6 NB_B Column block size NB_B >= 1 Global
7 RSRC_B The process row of the p × q grid over which the first row of the global matrix is distributed 0 <= RSRC_B < p Global
8 CSRC_B The process column of the p × q grid over which the first column of the global matrix is distributed 0 <= CSRC_B < q Global
9 LLD_B The leading dimension of the local array LLD_B >= max(1,LOCp(M_B)) Local

Specified as: an array of (at least) length 9, containing fullword integers.

info
See On Return.

On Return

b
is the updated local part of the global matrix B, containing the solution vectors.

Scope: local

Returned as: an LLD_B by (at least) LOCq(N_B) array, containing numbers of the data type indicated in Table 67.

info
indicates that a successful computation occurred.

Scope: global

Returned as: a fullword integer; info = 0.

Notes and Coding Rules
  1. In your C program, argument info must be passed by reference.
  2. This subroutine accepts lowercase letters for the uplo argument.
  3. The matrices must have no common elements; otherwise, results are unpredictable.
  4. The scalar data specified for input argument n must be the same for both PDPOTRF/PZPOTRF and PDPOTRS/PZPOTRS.
  5. The global submatrix A input to PDPOTRS/PZPOTRS must be the same as for the corresponding output argument for PDPOTRF/PZPOTRS; and thus, the scalar data specified for ia, ja, and the contents of desc_a must also be the same.
  6. The NUMROC utility subroutine can be used to determine the values of LOCp(M_) and LOCq(N_) used in the argument descriptions above. For details, see Determining the Number of Rows and Columns in Your Local Arrays and NUMROC--Compute the Number of Rows or Columns of a Block-Cyclically Distributed Matrix Contained in a Process.
  7. For suggested block sizes, see Coding Tips for Optimizing Parallel Performance.
  8. On both input and output, matrices A and B conform to ScaLAPACK format.
  9. The following values must be equal: CTXT_A = CTXT_B.
  10. The global real symmetric or complex Hermitian matrix A must be distributed using a square block-cyclic distribution; that is, MB_A = NB_A.
  11. The following block sizes must be equal: MB_A = MB_B.
  12. The global real symmetric or complex Hermitian matrix A must be aligned on a block row boundary; that is, ia-1 must be a multiple of MB_A.
  13. The block row offset of A must be equal to the block column offset of A; that is, mod(ia-1, MB_A) = mod(ja-1, NB_A).
  14. The block row offset of A must be equal to the block row offset of B; that is, mod(ia-1, MB_A) = mod(ib-1, MB_B).
  15. In the process grid, the process row containing the first row of the submatrix A must also contain the first row of the submatrix B; that is, iarow = ibrow, where:
    iarow = mod((((ia-1)/MB_A)+RSRC_A), p)
    ibrow = mod((((ib-1)/MB_B)+RSRC_B), p)

Error Conditions

Computational Errors

None

Note:
If the factorization performed by PDPOTRF/PZPOTRF failed because of a nonpositive definite matrix A, the results returned by this subroutine are unpredictable. For details, see the info output argument for PDPOTRF/PZPOTRF.

Resource Errors

Unable to allocate work space

Input-Argument and Miscellaneous Errors

Stage 1:  

  1. DTYPE_A is invalid.
  2. DTYPE_B is invalid.

Stage 2:  

  1. CTXT_A is invalid.

Stage 3:  

  1. This subroutine was called from outside the process grid.

Stage 4:  

  1. uplo <> 'U' or 'L'
  2. n < 0
  3. nrhs < 0
  4. M_A < 0 and n = 0; M_A < 1 otherwise
  5. N_A < 0 and n = 0; N_A < 1 otherwise
  6. ia < 1
  7. ja < 1
  8. MB_A < 1
  9. NB_A < 1
  10. RSRC_A < 0 or RSRC_A >= p
  11. CSRC_A < 0 or CSRC_A >= q
  12. M_B < 0 and (n = 0 or nrhs = 0); M_B < 1 otherwise
  13. N_B < 0 and (n = 0 or nrhs = 0); N_B < 1 otherwise
  14. ib < 1
  15. jb < 1
  16. MB_B < 1
  17. NB_B < 1
  18. RSRC_B < 0 or RSRC_B >= p
  19. CSRC_B < 0 or CSRC_B >= q
  20. CTXT_A <> CTXT_B

Stage 5:  

    If n <> 0:

  1. ia > M_A
  2. ja > N_A
  3. ia+n-1 > M_A
  4. ja+n-1 > N_A

    If n <> 0 and nrhs <> 0:

  5. ib > M_B
  6. jb > N_B
  7. ib+n-1 > M_B
  8. jb+nrhs-1 > N_B

    In all cases:

  9. MB_A <> NB_A
  10. mod(ia-1, MB_A) <> mod(ja-1, NB_A)
  11. MB_B <> MB_A
  12. mod(ia-1, MB_A) <> mod(ib-1, MB_B).
  13. mod(ia-1, MB_A) <> 0
  14. In the process grid, the process row containing the first row of the submatrix A does not contain the first row of the submatrix B; that is, iarow <> ibrow, where:
    iarow = mod((((ia-1)/MB_A)+RSRC_A), p)
    ibrow = mod((((ib-1)/MB_B)+RSRC_B), p)

Stage 6:  

  1. LLD_A < max(1, LOCp(M_A))
  2. LLD_B < max(1, LOCp(M_B))

    Each of the following global input arguments are checked to determine whether its value differs from the value specified on process P00:

  3. uplo differs.
  4. n differs.
  5. nrhs differs.
  6. ia differs.
  7. ja differs.
  8. DTYPE_A differs.
  9. M_A differs.
  10. N_A differs.
  11. MB_A differs.
  12. NB_A differs.
  13. RSRC_A differs.
  14. CSRC_A differs.
  15. ib differs.
  16. jb differs.
  17. DTYPE_B differs.
  18. M_B differs.
  19. N_B differs.
  20. MB_B differs.
  21. NB_B differs.
  22. RSRC_B differs.
  23. CSRC_B differs.

Example 1

This example solves the positive definite real symmetric system AX = B with 5 right-hand sides using a 2 × 2 process grid. The transformed matrix A is the output from Example 1.

This example uses a global submatrix B within a global matrix B by specifying ib = 1 and jb = 2.

By specifying CSRC_B = 1, the columns of global matrix B are distributed over the process grid starting in the second column of the process grid.

Call Statements and Input
ORDER = 'R'
NPROW = 2
NPCOL = 2
CALL BLACS_GET (0, 0, ICONTXT)
CALL BLACS_GRIDINIT(ICONTXT, ORDER, NPROW, NPCOL)
CALL BLACS_GRIDINFO(ICONTXT, NPROW, NPCOL, MYROW, MYCOL)
 
              UPLO  N  NRHS A  IA  JA   DESC_A   B  IB  JB   DESC_B   INFO
               |    |   |   |   |   |     |      |   |   |     |       |
CALL PDPOTRS( 'L' , 9 , 5 , A , 1 , 1 , DESC_A , B , 1 , 2 , DESC_B , INFO )


Desc_A Desc_B
DTYPE_ 1 1
CTXT_ icontxt(IOBG40) icontxt(IOBG40)
M_ 9 9
N_ 9 6
MB_ 3 3
NB_ 3 2
RSRC_ 0 0
CSRC_ 0 1
LLD_ See below(EPSSL40) See below(EPSSL40)

Notes:

  1. icontxt is the output of the BLACS_GRIDINIT call.

  2. Each process should set the LLD_ as follows: 
    LLD_A = MAX(1,NUMROC(M_A, MB_A, MYROW, RSRC_A, NPROW))
LLD_B = MAX(1,NUMROC(M_B, MB_B, MYROW, RSRC_B, NPROW))

    In this example, LLD_A = LLD_B = 6 on P00 and P01, and LLD_A = LLD_B = 3 on P10 and P11.

After the global matrix B is distributed over the process grid, only a portion of the global data structure is used--that is, global submatrix B. Following is the global 9 × 5 submatrix B, starting at row 1 and column 2 in global general 9 × 6 matrix B with block size 3 × 2: 

B,D          0                 1                 2
     *                                                   *
     |     .    18.0  |    27.0   36.0  |    45.0    9.0 |
 0   |     .    34.0  |    51.0   68.0  |    85.0   17.0 |
     |     .    48.0  |    72.0   96.0  |   120.0   24.0 |
     | ---------------|-----------------|--------------- |
     |     .    60.0  |    90.0  120.0  |   150.0   30.0 |
 1   |     .    70.0  |   105.0  140.0  |   175.0   35.0 |
     |     .    78.0  |   117.0  156.0  |   195.0   39.0 |
     | ---------------|-----------------|--------------- |
     |     .    84.0  |   126.0  168.0  |   210.0   42.0 |
 2   |     .    88.0  |   132.0  176.0  |   220.0   44.0 |
     |     .    90.0  |   135.0  180.0  |   225.0   45.0 |
     *                                                   *

The following is the 2 × 2 process grid: 

B,D  |    1    | 0 2 
-----| ------- |-----
0    |   P00   |  P01
2    |         |
-----| ------- |-----
1    |   P10   |  P11
Note:
The first column of B begins in the second column of the process grid.

Local arrays for B: 

p,q  |       0        |               1
-----|----------------|------------------------------
     |   27.0   36.0  |      .    18.0   45.0    9.0
     |   51.0   68.0  |      .    34.0   85.0   17.0
     |   72.0   96.0  |      .    48.0  120.0   24.0
 0   |  126.0  168.0  |      .    84.0  210.0   42.0
     |  132.0  176.0  |      .    88.0  220.0   44.0
     |  135.0  180.0  |      .    90.0  225.0   45.0
-----|----------------|------------------------------
     |   90.0  120.0  |      .    60.0  150.0   30.0
 1   |  105.0  140.0  |      .    70.0  175.0   35.0
     |  117.0  156.0  |      .    78.0  195.0   39.0

Output:

After the global matrix B is distributed over the process grid, only a portion of the global data structure is used--that is, global submatrix B. Following is the global 9 × 5 submatrix B, starting at row 1 and column 2 in global general 9 × 6 matrix B with block size 3 × 2: 

B,D        0             1             2
     *                                       *
     |   .   2.0  |   3.0  4.0  |   5.0  1.0 |
 0   |   .   2.0  |   3.0  4.0  |   5.0  1.0 |
     |   .   2.0  |   3.0  4.0  |   5.0  1.0 |
     | -----------|-------------|----------- |
     |   .   2.0  |   3.0  4.0  |   5.0  1.0 |
 1   |   .   2.0  |   3.0  4.0  |   5.0  1.0 |
     |   .   2.0  |   3.0  4.0  |   5.0  1.0 |
     | -----------|-------------|----------- |
     |   .   2.0  |   3.0  4.0  |   5.0  1.0 |
 2   |   .   2.0  |   3.0  4.0  |   5.0  1.0 |
     |   .   2.0  |   3.0  4.0  |   5.0  1.0 |
     *                                       *

The following is the 2 × 2 process grid: 

B,D  |    1    | 0 2 
-----| ------- |-----
0    |   P00   |  P01
2    |         |
-----| ------- |-----
1    |   P10   |  P11
Note:
The first column of B begins in the second column of the process grid.

Local arrays for B: 

p,q  |     0      |           1
-----|------------|----------------------
     |  3.0  4.0  |    .   2.0  5.0  1.0
     |  3.0  4.0  |    .   2.0  5.0  1.0
     |  3.0  4.0  |    .   2.0  5.0  1.0
 0   |  3.0  4.0  |    .   2.0  5.0  1.0
     |  3.0  4.0  |    .   2.0  5.0  1.0
     |  3.0  4.0  |    .   2.0  5.0  1.0
-----|------------|----------------------
     |  3.0  4.0  |    .   2.0  5.0  1.0
 1   |  3.0  4.0  |    .   2.0  5.0  1.0
     |  3.0  4.0  |    .   2.0  5.0  1.0

The value of info is 0 on all processes.

Example 2

This example solves the positive definite complex Hermitian system AX = B with 5 right-hand sides using a 2 × 2 process grid. The transformed matrix A is the output from Example 2.

This example uses a global submatrix B within a global matrix B by specifying ib = 1 and jb = 2.

By specifying CSRC_B = 1, the columns of global matrix B are distributed over the process grid starting in the second column of the process grid.

Call Statements and Input
ORDER = 'R'
NPROW = 2
NPCOL = 2
CALL BLACS_GET (0, 0, ICONTXT)
CALL BLACS_GRIDINIT(ICONTXT, ORDER, NPROW, NPCOL)
CALL BLACS_GRIDINFO(ICONTXT, NPROW, NPCOL, MYROW, MYCOL)
 
              UPLO  N  NRHS  A  IA  JA   DESC_A   B  IB  JB   DESC_B   INFO
               |    |   |    |   |   |     |      |   |   |     |       |
CALL PZPOTRS( 'L' , 9 , 5  , A , 1 , 1 , DESC_A , B , 1 , 2 , DESC_B , INFO )


Desc_A Desc_B
DTYPE_ 1 1
CTXT_ icontxt(IOBG41) icontxt(IOBG41)
M_ 9 9
N_ 9 6
MB_ 3 3
NB_ 3 2
RSRC_ 0 0
CSRC_ 0 1
LLD_ See below(EPSSL41) See below(EPSSL41)

Notes:

  1. icontxt is the output of the BLACS_GRIDINIT call.

  2. Each process should set the LLD_ as follows: 
    LLD_A = MAX(1,NUMROC(M_A, MB_A, MYROW, RSRC_A, NPROW))
LLD_B = MAX(1,NUMROC(M_B, MB_B, MYROW, RSRC_B, NPROW))

    In this example, LLD_A = LLD_B = 6 on P00 and P01, and LLD_A = LLD_B = 3 on P10 and P11.

After the global matrix B is distributed over the process grid, only a portion of the global data structure is used--that is, global submatrix B. Following is the global 9 × 5 submatrix B, starting at row 1 and column 2 in global general 9 × 6 matrix B with block size 3 × 2: 


B,D                0                                1                                  2
     *                                                                                                  *
     |      .       (60.0, 10.0)  |      (86.0, 2.0)   (112.0, -6.0)  |   (138.0, -14.0)   (34.0, 18.0) |
 0   |      .       (86.0, 28.0)  |    (126.0, 22.0)   (166.0, 16.0)  |    (206.0, 10.0)   (46.0, 34.0) |
     |      .      (108.0, 44.0)  |    (160.0, 40.0)   (212.0, 36.0)  |    (264.0, 32.0)   (56.0, 48.0) |
     | ---------------------------|-----------------------------------|-------------------------------- |
     |      .      (126.0, 58.0)  |    (188.0, 56.0)   (250.0, 54.0)  |    (312.0, 52.0)   (64.0, 60.0) |
 1   |      .      (140.0, 70.0)  |    (210.0, 70.0)   (280.0, 70.0)  |    (350.0, 70.0)   (70.0, 70.0) |
     |      .      (150.0, 80.0)  |    (226.0, 82.0)   (302.0, 84.0)  |    (378.0, 86.0)   (74.0, 78.0) |
     | ---------------------------|-----------------------------------|-------------------------------- |
     |      .      (156.0, 88.0)  |    (236.0, 92.0)   (316.0, 96.0)  |   (396.0, 100.0)   (76.0, 84.0) |
 2   |      .      (158.0, 94.0)  |   (240.0, 100.0)  (322.0, 106.0)  |   (404.0, 112.0)   (76.0, 88.0) |
     |      .      (156.0, 98.0)  |   (238.0, 106.0)  (320.0, 114.0)  |   (402.0, 122.0)   (74.0, 90.0) |
     *                                                                                                  *

The following is the 2 × 2 process grid: 

B,D  |    1    | 0 2 
-----| ------- |-----
0    |   P00   |  P01
2    |         |
-----| ------- |-----
1    |   P10   |  P11
Note:
The first column of B begins in the second column of the process grid.

Local arrays for B: 


p,q  |                0                 |                             1
-----|----------------------------------|-----------------------------------------------------------
     |     (86.0, 2.0)   (112.0, -6.0)  |       .       (60.0, 10.0)  (138.0, -14.0)   (34.0, 18.0)
     |   (126.0, 22.0)   (166.0, 16.0)  |       .       (86.0, 28.0)   (206.0, 10.0)   (46.0, 34.0)
     |   (160.0, 40.0)   (212.0, 36.0)  |       .      (108.0, 44.0)   (264.0, 32.0)   (56.0, 48.0)
 0   |   (236.0, 92.0)   (316.0, 96.0)  |       .      (156.0, 88.0)  (396.0, 100.0)   (76.0, 84.0)
     |  (240.0, 100.0)  (322.0, 106.0)  |       .      (158.0, 94.0)  (404.0, 112.0)   (76.0, 88.0)
     |  (238.0, 106.0)  (320.0, 114.0)  |       .      (156.0, 98.0)  (402.0, 122.0)   (74.0, 90.0)
-----|----------------------------------|-----------------------------------------------------------
     |   (188.0, 56.0)   (250.0, 54.0)  |       .      (126.0, 58.0)   (312.0, 52.0)   (64.0, 60.0)
 1   |   (210.0, 70.0)   (280.0, 70.0)  |       .      (140.0, 70.0)   (350.0, 70.0)   (70.0, 70.0)
     |   (226.0, 82.0)   (302.0, 84.0)  |       .      (150.0, 80.0)   (378.0, 86.0)   (74.0, 78.0)

Output:

After the global matrix B is distributed over the process grid, only a portion of the global data structure is used--that is, global submatrix B. Following is the global 9 × 5 submatrix B, starting at row 1 and column 2 in global general 9 × 6 matrix B with block size 3 × 2: 


B,D           0                        1                           2
     *                                                                          *
     |   .   (2.0, 1.0)  |   (3.0, 1.0)  (4.0, 1.0)  |   (5.0, 1.0)  (1.0, 1.0) |
 0   |   .   (2.0, 1.0)  |   (3.0, 1.0)  (4.0, 1.0)  |   (5.0, 1.0)  (1.0, 1.0) |
     |   .   (2.0, 1.0)  |   (3.0, 1.0)  (4.0, 1.0)  |   (5.0, 1.0)  (1.0, 1.0) |
     | ------------------|---------------------------|------------------------- |
     |   .   (2.0, 1.0)  |   (3.0, 1.0)  (4.0, 1.0)  |   (5.0, 1.0)  (1.0, 1.0) |
 1   |   .   (2.0, 1.0)  |   (3.0, 1.0)  (4.0, 1.0)  |   (5.0, 1.0)  (1.0, 1.0) |
     |   .   (2.0, 1.0)  |   (3.0, 1.0)  (4.0, 1.0)  |   (5.0, 1.0)  (1.0, 1.0) |
     | ------------------|---------------------------|------------------------- |
     |   .   (2.0, 1.0)  |   (3.0, 1.0)  (4.0, 1.0)  |   (5.0, 1.0)  (1.0, 1.0) |
 2   |   .   (2.0, 1.0)  |   (3.0, 1.0)  (4.0, 1.0)  |   (5.0, 1.0)  (1.0, 1.0) |
     |   .   (2.0, 1.0)  |   (3.0, 1.0)  (4.0, 1.0)  |   (5.0, 1.0)  (1.0, 1.0) |
     *                                                                          *

The following is the 2 × 2 process grid: 

B,D  |    1    | 0 2 
-----| ------- |-----
0    |   P00   |  P01
2    |         |
-----| ------- |-----
1    |   P10   |  P11
Note:
The first column of B begins in the second column of the process grid.

Local arrays for B: 


p,q  |            0             |                     1
-----|--------------------------|-------------------------------------------
     |  (3.0, 4.0)  (3.0, 4.0)  |    .   (2.0, 1.0)  (5.0, 1.0)  (1.0, 1.0)
     |  (3.0, 4.0)  (3.0, 4.0)  |    .   (2.0, 1.0)  (5.0, 1.0)  (1.0, 1.0)
     |  (3.0, 4.0)  (3.0, 4.0)  |    .   (2.0, 1.0)  (5.0, 1.0)  (1.0, 1.0)
 0   |  (3.0, 4.0)  (3.0, 4.0)  |    .   (2.0, 1.0)  (5.0, 1.0)  (1.0, 1.0)
     |  (3.0, 4.0)  (3.0, 4.0)  |    .   (2.0, 1.0)  (5.0, 1.0)  (1.0, 1.0)
     |  (3.0, 4.0)  (3.0, 4.0)  |    .   (2.0, 1.0)  (5.0, 1.0)  (1.0, 1.0)
-----|--------------------------|-------------------------------------------
     |  (3.0, 4.0)  (3.0, 4.0)  |    .   (2.0, 1.0)  (5.0, 1.0)  (1.0, 1.0)
 1   |  (3.0, 4.0)  (3.0, 4.0)  |    .   (2.0, 1.0)  (5.0, 1.0)  (1.0, 1.0)
     |  (3.0, 4.0)  (3.0, 4.0)  |    .   (2.0, 1.0)  (5.0, 1.0)  (1.0, 1.0)

The value of info is 0 on all processes.

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