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

PDPTSV--Positive Definite Symmetric Tridiagonal Matrix Factorization and Solve

This subroutine solves the tridiagonal systems of linear equations, AX = B, where the positive definite symmetric tridiagonal matrix A is stored in parallel-symmetric-tridiagonal storage mode. In this description:

• A represents the global positive definite symmetric tridiagonal submatrix A_{ia:ia+n-1, ia:ia+n-1}.
• B represents the global general submatrix B_{ib:ib+n-1, 1:nrhs} containing the right-hand sides in its columns.
• X represents the global general submatrix B_{ib:ib+n-1, 1:nrhs} containing the output solution vectors in its columns.

If n = 0, no computation is performed and the subroutine returns after doing some parameter checking. See reference [51].

Table 86. Data Types

Syntax

On Entry

n

is the order of the positive definite symmetric tridiagonal matrix A and the number of rows in the general submatrix B, which contains the multiple right-hand sides.

Scope: global

Specified as: a fullword integer, where:

• If (the process grid is p × 1 and DTYPE_A = 1) or DTYPE_A = 502, 0 <= n <= (MB_A)(p)-mod(ia-1,MB_A).
• If (the process grid is 1 × p and DTYPE_A = 1) or DTYPE_A = 501, 0 <= n <= (NB_A)(p)-mod(ia-1,NB_A).

where p is the number of processes in a process grid.

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.

d

is the local part of the global vector d. This identifies the first element of the local array D. This subroutine computes the location of the first element of the local subarray used, based on ia, desc_a, and p; therefore, the leading LOCp(ia+n-1) part of the local array D contains the local pieces of the leading ia+n-1 part of the global vector.

The global vector d contains the main diagonal of the global positive definite symmetric tridiagonal submatrix A in elements ia through ia+n-1.

Scope: local

Specified as: a one-dimensional array of (at least) length LOCp(ia+n-1) containing numbers of the data type indicated in Table 86. Details about block-cyclic data distribution of global matrix A are stored in desc_a.

On output, D is overwritten; that is, the original input is not preserved.

e

is the local part of the global vector e. This identifies the first element of the local array E. This subroutine computes the location of the first element of the local subarray used, based on ia, desc_a, and p; therefore, the leading LOCp(ia+n-1) part of the local array E contains the local pieces of the leading ia+n-1 part of the global vector.

The global vector e contains the off-diagonal of the global positive definite symmetric tridiagonal submatrix A in elements ia through ia+n-2.

Scope: local

Specified as: a one-dimensional array of (at least) length LOCp(ia+n-1), containing numbers of the data type indicated in Table 86. Details about block-cyclic data distribution of global matrix A are stored in desc_a.

On output, E is overwritten; that is, the original input is not preserved.

ia

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

Scope: global

Specified as: a fullword integer, where:

• If (the process grid is p × 1 and DTYPE_A = 1) or DTYPE_A = 502, 1 <= ia <= M_A and ia+n-1 <= M_A.
• If (the process grid is 1 × p and DTYPE_A = 1) or DTYPE_A = 501, 1 <= ia <= N_A and ia+n-1 <= N_A.

desc_a

is the array descriptor for global matrix A. Because vectors are one-dimensional data structures, you may use a type-502, type-501, or type-1 array descriptor regardless of whether the process grid is p × 1 or 1 × p. For a type-502 array descriptor, the process grid is used as if it is a p × 1 process grid. For a type-501 array descriptor, the process grid is used as if it is a 1 × p process grid. For a type-1 array descriptor, the process grid is used as if it is either a p × 1 process grid or a 1 × p process grid. The following tables describe three types of array descriptors. For rules on using array descriptors, see Notes and Coding Rules.

Table 87. Type-502 Array Descriptor

desc_a	Name	Description	Limits	Scope
1	DTYPE_A	Descriptor Type	DTYPE_A=502 for p × 1 or 1 × p where p is the number of processes in a process grid.	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	MB_A	Row block size	MB_A >= 1 and 0 <= n <= (MB_A)(p)-mod(ia-1,MB_A)	Global
5	RSRC_A	The process row over which the first row of the global matrix is distributed	0 <= RSRC_A < p	Global
6	--	Not used by this subroutine.	--	--
7	--	Reserved	--	--

Specified as: an array of (at least) length 7, containing fullword integers.

Table 88. Type-1 Array Descriptor (p × 1 Process Grid)

desc_a	Name	Description	Limits	Scope
1	DTYPE_A	Descriptor Type	DTYPE_A = 1 for p × 1 where p is the number of processes in a process grid.	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	N_A = 1	Global
5	MB_A	Row block size	MB_A >= 1 and 0 <= n <= (MB_A)(p)-mod(ia-1,MB_A)	Global
6	NB_A	Column block size	NB_A >= 1	Global
7	RSRC_A	The process row over which the first row of the global matrix is distributed	0 <= RSRC_A < p	Global
8	CSRC_A	The process column over which the first column of the global matrix is distributed	CSRC_A = 0	Global
9	--	Not used by this subroutine.	--	--

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

Table 89. Type-501 Array Descriptor

desc_a	Name	Description	Limits	Scope
1	DTYPE_A	Descriptor Type	DTYPE_A=501 for 1 × p or p × 1 where p is the number of processes in a process grid.	Global
2	CTXT_A	BLACS context	Valid value, as returned by BLACS_GRIDINIT or BLACS_GRIDMAP	Global
3	N_A	Number of columns in the global matrix	If n = 0: N_A >= 0 Otherwise: N_A >= 1	Global
4	NB_A	Column block size	NB_A >= 1 and 0 <= n <= (NB_A)(p)-mod(ia-1,NB_A)	Global
5	CSRC_A	The process column over which the first column of the global matrix is distributed	0 <= CSRC_A < p	Global
6	--	Not used by this subroutine.	--	--
7	--	Reserved	--	--

Specified as: an array of (at least) length 7, containing fullword integers.

Table 90. Type-1 Array Descriptor (1 × p Process Grid)

desc_a	Name	Description	Limits	Scope
1	DTYPE_A	Descriptor type	DTYPE_A = 1 for 1 × p where p is the number of processes in a process grid.	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	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 and 0 <= n <= (NB_A)(p)-mod(ia-1,NB_A)	Global
7	RSRC_A	The process row over which the first row of the global matrix is distributed	RSRC_A=0	Global
8	CSRC_A	The process column over which the first column of the global matrix is distributed	0 <= CSRC_A < p	Global
9	--	Not used by this subroutine.	--	--

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 multiple 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, desc_b, and p; therefore, the leading LOCp(ib+n-1) by nrhs part of the local array B must contain the local pieces of the leading ib+n-1 by nrhs part of the global matrix.

Scope: local

Specified as: an LLD_B by (at least) nrhs array, containing numbers of the data type indicated in Table 86. 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.

desc_b

is the array descriptor for global matrix B, which may be type 502 or type 1, as described in the following tables. For type-502 array descriptor, the process grid is used as if it is a p × 1 process grid. For rules on using array descriptors, see Notes and Coding Rules.

desc_b	Name	Description	Limits	Scope
1	DTYPE_B	Descriptor type	DTYPE_B = 502 for p × 1 or 1 × p where p is the number of processes in a process grid.	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: M_B >= 0 Otherwise: M_B >= 1	Global
4	MB_B	Row block size	MB_B >= 1 and 0 <= n <= (MB_B)(p)-mod(ia-1,MB_B)	Global
5	RSRC_B	The process row over which the first row of the global matrix is distributed	0 <= RSRC_B < p	Global
6	LLD_B	Leading dimension	LLD_B >= max(1, LOCp(MB_B))	Local
7	--	Reserved	--	--

Specified as: an array of (at least) length 7, containing fullword integers.

desc_b	Name	Description	Limits	Scope
1	DTYPE_B	Descriptor type	DTYPE_B = 1 for p × 1 where p is the number of processes in a process grid.	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: M_B >= 0 Otherwise: M_B >= 1	Global
4	N_B	Number of columns in the global matrix	N_B >= nrhs	Global
5	MB_B	Row block size	MB_B >= 1 and 0 <= n <= (MB_B)(p)-mod(ib-1,MB_B)	Global
6	NB_B	Column block size	NB_B >= 1	Global
7	RSRC_B	The process row over which the first row of the global matrix is distributed	0 <= RSRC_B < p	Global
8	CSRC_B	The process column over which the first column of the global matrix is distributed	CSRC_B = 0	Global
9	LLD_B	Leading dimension	LLD_B >= max(1,LOCp(MB_B))	Local

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

work

has the following meaning:

If lwork = 0, work is ignored.

If lwork <> 0, work is the work area used by this subroutine, where:

• If lwork <> -1, the size of work is (at least) of length lwork.
• If lwork = -1, the size of work is (at least) of length 1.

Scope: local

Specified as: an area of storage containing numbers of data type indicated in Table 86.

lwork

is the number of elements in array WORK.

Scope:

• If lwork >= 0, lwork is local
• If lwork = -1, lwork is global

Specified as: a fullword integer; where:

• If lwork = 0, PDPTSV dynamically allocates the work area used by the subroutine. The work area is deallocated before control is returned to the calling program. This option is an extension to the ScaLAPACK standard.
• If lwork = -1, PDPTSV performs a work area query and return the optimum size of work in work₁. No computation is performed and the subroutine returns after error checking is complete.
• Otherwise, if (the process grid is p × 1 and DTYPE_A = 1) or DTYPE_A = 502:
  • If nrhs <= 1, lwork >= 10P+MB_A+10.
  • If nrhs > 1, (lwork>=(20+2min(100,nrhs))P+3(MB_A)+4(nrhs)).

  where, in the above formulas, P is the actual number of processes containing data.

  If (the process grid is 1 × p and DTYPE_A = 1) or DTYPE_A = 501, you would substitute NB_A in place of MB_A in the formulas above.

  Note:

  In ScaLAPACK 1.5, PDPTSV requires lwork = 22P+3MB_A+2min(100,nrhs)P+4(nrhs). This value is greater than or equal to the value required by Parallel ESSL.

info

See On Return.

On Return

d

is overwritten; that is, the original input is not preserved.

e

is overwritten; that is, the original input is not preserved.

b

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

Scope: local

Returned as: an LLD_B by (at least) nrhs array, containing numbers of the data type indicated in Table 86.

work

is the work area used by this subroutine if lwork <> 0, where:

If lwork <> 0 and lwork <> -1, the size of work is (at least) of length lwork.

If lwork = -1, the size of work is (at least) of length 1.

Scope: local

Returned as: an area of storage, containing numbers of the data type indicated in Table 86, where:

• If lwork >= 1, work₁ is set to the minimum lwork value needed.
• If lwork = -1, work₁ is set to the optimum lwork value needed.

Except for work₁, the contents of work are overwritten on return.

info

has the following meaning:

If info = 0, global submatrix A is positive definite, and the factorization completed successfully or the work area query completed successfully.

If 1 <= info <= p, the portion of global submatrix A stored on process info-1 and factored locally, is not positive definite. A pivot element whose value is less than or equal to a small positive number was detected.

If info > p, the portion of global submatrix A stored on process info-p-1 representing interactions with other processes, is not positive definite. A pivot element whose value is less than or equal to a small positive number was detected.

If info > 0, the results of the computation are unpredictable.

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. If n > 0 and nrhs = 0, only the factorization is completed.
3. d, e, B, and work must have no common elements; otherwise, results are unpredictable.
4. In all cases, follow these rules:
   • ia = ib
   • CTXT_A = CTXT_B
   • If (the process grid is p × 1 and DTYPE_A = 1) or DTYPE_A = 502, MB_A = MB_B.
   • If (the process grid is 1 × p and DTYPE_A = 1) or DTYPE_A = 501, NB_A = MB_B.
   • If DTYPE_A=1, then:
     • For a p × 1 process grid (where p>1), N_A=1, NB_A>=1, and CSRC_A=0.
     • For a 1 × p process grid (where p>1), M_A=1, MB_A>=1, and RSRC_A=0.
     • For a 1 × 1 process grid:
       • If N_A=1, NB_A>=1 and CSRC_A=0.
       • If M_A=1, MB_A>=1 and RSRC_A=0.
   • If DTYPE_B=1, N_B>=nrhs, NB_B>=1, and CSRC_B=0.
   • Following are the consistent combinations of array descriptor types and process grids, where p is the number of processes in the process grid:
     

     DTYPE_A	DTYPE_B	Process Grid
     501	502	p × 1 or 1 × p
     502	502	p × 1 or 1 × p
     501	1	p × 1
     502	1	p × 1
     1	502	p × 1 or 1 × p
     1	1	p × 1

5. To determine the values of LOCp(n) used in the argument descriptions, see Determining the Number of Rows and Columns in Your Local Arrays for descriptor type-1 or Determining the Number of Rows or Columns in Your Local Arrays for descriptor type-501 and type-502.
6. The global symmetric tridiagonal matrix A must be positive definite. This subroutine uses the info argument to provide information about A, like ScaLAPACK. However, this subroutine also issues an error message, which differs from ScaLAPACK.
7. The global positive definite symmetric tridiagonal matrix A must be stored in parallel-symmetric-tridiagonal storage mode and distributed over a one-dimensional process grid, using block-cyclic data distribution. See the section on block-cyclically distributing a tridiagonal matrix in Matrices.

   For more information on using block-cyclic data distribution, see Specifying Block-Cyclically-Distributed Matrices for the Banded Linear Algebraic Equations.

8. Matrix B must be distributed over a one-dimensional process grid, using block-cyclic data distribution. For more information using block-cyclic data distribution, see Specifying Block-Cyclically-Distributed Matrices for the Banded Linear Algebraic Equations. Also, see the section on distributing the right-hand side matrix in Matrices.
9. If lwork = -1 on any process, it must equal -1 on all processes. That is, if a subset of the processes specifies -1 for the work area size, they must all specify -1.
10. Although global matrices A and B may be block-cyclically distributed on a 1 × p or p × 1 process grid, the values of n, ia, ib, MB_A (if (the process grid is p × 1 and DTYPE_A = 1) or DTYPE_A = 502), NB_A (if (the process grid is 1 × p and DTYPE_A = 1) or DTYPE_A = 501), must be chosen so that each process has at most one full or partial block of each of the global submatrices A and B.
11. For global tridiagonal matrix A, use of the type-1 array descriptor is an extension to ScaLAPACK 1.5. If your application needs to run with both Parallel ESSL and ScaLAPACK 1.5, it is suggested that you use either a type-501 or a type-502 array descriptor for the matrix A.

Error Conditions

Computational Errors

Matrix A is not positive definite. For details, see the description of the info argument.

Resource Errors

Unable to allocate workspace

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:  

Note:

In the following error conditions:

• If M_A = 1 and DTYPE_A = 1, a 1 × 1 process grid is treated as a 1 × p process grid.
• If N_A = 1 and DTYPE_A = 1, a 1 × 1 process grid is treated as a p × 1 process grid.

1. The process grid is not 1 × p or p × 1.
2. CTXT_A <> CTXT_B
3. n < 0
4. ia < 1
5. DTYPE_A = 1 and M_A <>1 and N_A <> 1

   If (the process grid is 1 × p and DTYPE_A = 1) or DTYPE_A = 501:

6. N_A < 0 and (n = 0); N_A < 1 otherwise
7. NB_A < 1
8. n > (NB_A)(p)-mod(ia,NB_A)
9. ia > N_A and (n > 0)
10. ia+n-1 > N_A and (n > 0)
11. CSRC_A < 0 or CSRC_A >= p
12. NB_A <> MB_B
13. CSRC_A <> RSRC_B

    If the process grid is 1 × p and DTYPE_A = 1:

14. M_A <> 1
15. MB_A < 1
16. RSRC_A <> 0

    If (the process grid is p × 1 and DTYPE_A = 1) or DTYPE_A = 502:

17. M_A < 0 and (n = 0); M_A < 1 otherwise
18. MB_A < 1
19. n > (MB_A)(p)-mod(ia,MB_A)
20. ia > M_A and (n > 0)
21. ia+n-1 > M_A and (n > 0)
22. RSRC_A < 0 or RSRC_A >= p
23. MB_A <> MB_B
24. RSRC_A <> RSRC_B

    If the process grid is p × 1 and DTYPE_A = 1:

25. N_A <> 1
26. NB_A < 1
27. CSRC_A <> 0

    In all cases:

28. ia <> ib
29. DTYPE_B = 1 and the process grid is 1 × p and p > 1
30. nrhs < 0
31. ib < 1
32. M_B < 0 and (n = 0); M_B < 1 otherwise
33. MB_B < 1
34. ib > M_B and (n > 0)
35. ib+n-1 > M_B and (n > 0)
36. RSRC_B <= 0 or RSRC_B >= p
37. LLD_B < max(1,LOCp(M_B))

    If DTYPE_B = 1:

38. N_B < 0 and (nrhs = 0); N_B < 1 otherwise
39. N_B < nrhs
40. NB_B < 1
41. CSRC_B <> 0

    In all cases:

42. lwork <> 0, lwork <> -1, and lwork < (minimum value) (For the minimum value, see the lwork argument description.)

Stage 5:  

Each of the following global input arguments are checked to determine whether its value is the same on all processes in the process grid:

1. n differs.
2. nrhs differs.
3. ia differs.
4. ib differs.
5. DTYPE_A differs.

   If DTYPE_A = 1 on all processes:

6. M_A differs.
7. N_A differs.
8. MB_A differs.
9. NB_A differs.
10. RSRC_A differs.
11. CSRC_A differs.

    If DTYPE_A = 501 on all processes:

12. N_A differs.
13. NB_A differs.
14. CSRC_A differs.

    If DTYPE_A = 502 on all processes:

15. M_A differs.
16. MB_A differs.
17. RSRC_A differs.

    In all cases:

18. DTYPE_B differs.

    If DTYPE_B = 1 on all processes:

19. M_B differs.
20. N_B differs.
21. MB_B differs.
22. NB_B differs.
23. RSRC_B differs.
24. CSRC_B differs.

    If DTYPE_B = 502 on all processes:

25. M_B differs.
26. MB_B differs.
27. RSRC_B differs.

    Also:

28. lwork = -1 on a subset of processes.

Example

This example shows a factorization of the positive definite symmetric tridiagonal matrix A of order 12:

      *                                                            *
      | 4.0  2.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0 |
      | 2.0  5.0  2.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0 |
      | 0.0  2.0  5.0  2.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0 |
      | 0.0  0.0  2.0  5.0  2.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0 |
      | 0.0  0.0  0.0  2.0  5.0  2.0  0.0  0.0  0.0  0.0  0.0  0.0 |
      | 0.0  0.0  0.0  0.0  2.0  5.0  2.0  0.0  0.0  0.0  0.0  0.0 |
      | 0.0  0.0  0.0  0.0  0.0  2.0  5.0  2.0  0.0  0.0  0.0  0.0 |
      | 0.0  0.0  0.0  0.0  0.0  0.0  2.0  5.0  2.0  0.0  0.0  0.0 |
      | 0.0  0.0  0.0  0.0  0.0  0.0  0.0  2.0  5.0  2.0  0.0  0.0 |
      | 0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0  2.0  5.0  2.0  0.0 |
      | 0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0  2.0  5.0  2.0 |
      | 0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0  2.0  5.0 |
      *                                                            *

Matrix A is stored in parallel-symmetric-tridiagonal storage mode and is distributed over a 1 × 3 process grid using block-cyclic distribution.

Notes:

1. On output, the vectors d and e are overwritten by this subroutine.

2. Notice only one process grid was created, even though, DTYPE_A = 501 and DTYPE_B = 502.

3. Because lwork = 0, this subroutine dynamically allocates the work area used by this subroutine.

Call Statements and Input

ORDER = 'R'
NPROW = 1
NPCOL = 3
CALL BLACS_GET (0, 0, ICONTXT)
CALL BLACS_GRIDINIT(ICONTXT, ORDER, NPROW, NPCOL)
CALL BLACS_GRIDINFO(ICONTXT, NPROW, NPCOL, MYROW, MYCOL)
 
              N  NRHS D   E  IA   DESC_A   B  IB   DESC_B   WORK LWORK INFO
              |   |   |   |   |      |     |   |      |      |     |     |
CALL PDPTSV( 12 , 3 , D , E , 1 , DESC_A , B , 1 , DESC_B , WORK , 0 , INFO)

Global vector d with block size of 4:

B,D             0                       1                       2
     *                                                                     *
 0   |  4.0  5.0  5.0  5.0  |   5.0  5.0  5.0  5.0  |   5.0  5.0  5.0  5.0 |
     *                                                                     *

Global vector e with block size of 4:

B,D             0                       1                      2
     *                                                                    *
 0   |  2.0  2.0  2.0  2.0  |   2.0  2.0  2.0  2.0  |   2.0  2.0  2.0   . |
     *                                                                    *

The following is the 1 × 3 process grid:

B,D  |    0    |    1    |    2    
-----| ------- | ------- |------- 
0    |   P00   |   P01   |   P02

Local array D with block size of 4:

p,q  |          0           |           1           |           2
-----|----------------------|-----------------------|----------------------
 0   |  4.0  5.0  5.0  5.0  |   5.0  5.0  5.0  5.0  |   5.0  5.0  5.0  5.0

Local array E with block size of 4:

p,q  |          0           |           1           |          2
-----|----------------------|-----------------------|---------------------
 0   |  2.0  2.0  2.0  2.0  |   2.0  2.0  2.0  2.0  |   2.0  2.0  2.0   .

Global matrix B with a block size of 4:

p,q  |       0
-----|----------------
     | 70.0  8.0  6.0
     | 99.0 18.0  9.0
 0   | 90.0 27.0  9.0
     | 81.0 36.0  9.0
-----|----------------
     | 72.0 45.0  9.0
     | 63.0 54.0  9.0
 1   | 54.0 63.0  9.0
     | 45.0 72.0  9.0
-----|----------------
     | 36.0 81.0  9.0
     | 27.0 90.0  9.0
 2   | 18.0 99.0  9.0
     |  9.0 82.0  7.0

The following is the 1 × 3 process grid:

B,D  |    0    |    1    |    2    
-----| ------- | ------- |------- 
0    |   P00   |   P01   |   P02

Local matrix B with a block size of 4:

p,q  |       0         |        1         |        2
-----|-----------------|------------------|-----------------
     | 70.0  8.0  6.0  |  72.0 45.0  9.0  |  36.0 81.0  9.0
     | 99.0 18.0  9.0  |  63.0 54.0  9.0  |  27.0 90.0  9.0
 0   | 90.0 27.0  9.0  |  54.0 63.0  9.0  |  18.0 99.0  9.0
     | 81.0 36.0  9.0  |  45.0 72.0  9.0  |   9.0 82.0  7.0

Output:

Global matrix B with a block size of 4:

p,q  |       0
-----|----------------
     | 12.0  1.0  1.0
     | 11.0  2.0  1.0
 0   | 10.0  3.0  1.0
     |  9.0  4.0  1.0
-----|----------------
     |  8.0  5.0  1.0
     |  7.0  6.0  1.0
 1   |  6.0  7.0  1.0
     |  5.0  8.0  1.0
-----|----------------
     |  4.0  9.0  1.0
     |  3.0 10.0  1.0
 2   |  2.0 11.0  1.0
     |  1.0 12.0  1.0

The following is the 1 × 3 process grid:

B,D  |    0    |    1    |   2    
-----| ------- | ------- |----- 
0    |   P00   |   P01   |  P02

Local matrix B with a block size of 4:

p,q  |       0         |        1         |        2
-----|-----------------|------------------|-----------------
     | 12.0  1.0  1.0  |   8.0  5.0  1.0  |   4.0  9.0  1.0
     | 11.0  2.0  1.0  |   7.0  6.0  1.0  |   3.0 10.0  1.0
 0   | 10.0  3.0  1.0  |   6.0  7.0  1.0  |   2.0 11.0  1.0
     |  9.0  4.0  1.0  |   5.0  8.0  1.0  |   1.0 12.0  1.0

The value of info is 0 on all processes.