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

PDGTTRF and PDDTTRF--General Tridiagonal Matrix Factorization

PDGTTRF factors the general tridiagonal matrix A, stored in tridiagonal storage mode, using Gaussian elimination with partial pivoting.

PDDTTRF factors the diagonally dominant general tridiagonal matrix A, stored in tridiagonal storage mode, using Gaussian elimination.

In these subroutine descriptions, A represents the global square general tridiagonal submatrix A_{ia:ia+n-1, ia:ia+n-1}.

To solve a tridiagonal system of linear equations with multiple right-hand sides, follow the call to PDGTTRF or PDDTTRF with one or more calls to PDGTTRS or PDDTTRS, respectively. The output from these factorization subroutines should be used only as input to the solve subroutines PDGTTRS and PDDTTRS, respectively.

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

Table 76. Data Types

Syntax

On Entry

n

is the order of the general tridiagonal matrix A and the number of elements in vector ipiv used in the computation.

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.

dl

is the local part of the global vector dl. This identifies the first element of the local array DL. These subroutines compute 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 DL contains the local pieces of the leading ia+n-1 part of the global vector.

The global vector dl contains the subdiagonal of the global general tridiagonal submatrix A in elements ia+1 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 76. Details about block-cyclic data distribution of global matrix A are stored in desc_a.

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

d

is the local part of the global vector d. This identifies the first element of the local array D. These subroutines compute 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 general 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 76. 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.

du

is the local part of the global vector du. This identifies the first element of the local array DU. These subroutines compute 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 DU contains the local pieces of the leading ia+n-1 part of the global vector.

The global vector du contains the superdiagonal of the global general 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 76. Details about block-cyclic data distribution of global matrix A are stored in desc_a.

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

du2

See On Return.

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 77. 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 these subroutines.	--	--
7	--	Reserved	--	--

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

Table 78. 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
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 these subroutines.	--	--

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

Table 79. 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 these subroutines.	--	--
7	--	Reserved	--	--

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

Table 80. 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 these subroutines.	--	--

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

ipiv

See On Return.

af

See On Return.

laf

is the number of elements in array AF.

Scope: local

Specified as: a fullword integer, where:

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

• For PDGTTRF, laf >= 12P+3(MB_A)
• For PDDTTRF, laf >= 12P+2(MB_A).

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, PDDTTRF requires laf = 12P+3NB_A. This value is greater than or equal to the value required by Parallel ESSL.

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 76.

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, PDGTTRF and PDDTTRF dynamically allocate 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, PDGTTRF and PDDTTRF perform 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, lwork must have the following value:

  For PDGTTRF, lwork >= 10P

  For PDDTTRF, lwork >= 8P

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

info

See On Return.

On Return

dl

dl is the updated local part of the global vector dl, containing part of the factorization.

Scope: local

Returned as: a one-dimensional array of (at least) LOCp(ia+n-1), containing numbers of the data type indicated in Table 76.

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

d

d is the updated local part of the global vector d, containing part of the factorization.

Scope: local

Returned as: a one-dimensional array of (at least) length LOCp(ia+n-1), containing numbers of the data type indicated in Table 76.

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

du

du is the updated local part of the global vector du, containing part of the factorization.

Scope: local

Returned as: a one-dimensional array of (at least) length LOCp(ia+n-1), containing numbers of the data type indicated in Table 76.

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

du2

is the local part of the global vector du2, containing part of the factorization.

Scope: local

Returned as: a one-dimensional array of (at least) length LOCp(ia+n-1), containing numbers of the data type indicated in Table 76.

ipiv

is the local part of the global vector ipiv, containing the pivot information needed by PDGTTRS. This identifies the first element of the local array IPIV. These subroutines compute 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 IPIV contains the local pieces of the leading ia+n-1 part of the global vector.

Scope: local

Returned as: an array of (at least) length LOCp(ia+n-1), containing fullword integers. There is no array descriptor for ipiv. The details about the block data distribution of global vector ipiv are stored in desc_a.

af

is a work area used by these subroutines and contains part of the factorization. Its size is specified by laf.

Scope: local

Returned as: a one-dimensional array of (at least) length laf, containing numbers of the data type indicated in Table 76.

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 data type indicated in Table 76, where:

• If lwork >= 1, the work₁ is set to the minimum lwork value needed.
• If lwork = -1, the 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, the factorization or work area query completed successfully.

Note:

For PDDTTRF, if the input matrix A is not diagonally dominant, the subroutine may still complete the factorization; however, results are unpredictable.

If 1 <= info <= p, the portion of the global submatrix A stored on process info-1 and factored locally, is singular or reducible (for PDGTTRF), or not diagonally dominant (for PDDTTRF). The magnitude of a pivot element was zero or too small.

If info > p, the portion of the global submatrix A stored on process info-p-1 representing interactions with other processes, is singular or reducible (for PDGTTRF), or not diagonally dominant (for PDDTTRF). The magnitude of a pivot element was zero or too small.

If info > 0, the factorization is completed; however, if you call PDGTTRS/PDDTTRS with these factors, results 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. The output from these factorization subroutines should be used only as input to the solve subroutines PDGTTRS and PDDTTRS, respectively.

   The factored matrix A is stored in an internal format that depends on the number of processes.

   The format of the output from PDDTTRF has changed. Therefore, the factorization and solve must be performed using Parallel ESSL Version 2 Release 1.2, or later.

   The scalar data specified for input argument n must be the same for both PDGTTRF/PDDTTRF and PDGTTRS/PDDTTRS.

   The global vectors for dl, d, du, du2, and af input to PDGTTRS/PDDTTRS must be the same as the corresponding output arguments for PDGTTRF/PDDTTRF; and thus, the scalar data specified for ia, desc_a, and laf must also be the same.

3. In all cases, follow these rules:
   • ia = ib
   • 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.
   • 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	Process Grid
     501	p × 1 or 1 × p
     502	p × 1 or 1 × p
     1	p × 1 or 1 × p

4. 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.
5. dl, d, du, du2, ipiv, af, and work must have no common elements; otherwise, results are unpredictable.
6. For PDGTTRF, the global general tridiagonal matrix A must be non-singular and irreducible. For PDDTTRF, the global general tridiagonal matrix A must be diagonally dominant to ensure numerical accuracy, because no pivoting is performed. These subroutines use the info argument to provide information about A, like ScaLAPACK. However, these subroutines also issue an error message, which differs from ScaLAPACK.
7. The global general tridiagonal matrix A must be stored in 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. 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.
9. Although global matrix A may be block-cyclically distributed on a 1 × p or p × 1 process grid, the values of n, ia, 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 global submatrix A.
10. 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 a singular or reducible matrix (for PDGTTRF), or not diagonally dominant (for PDDTTRF). 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.

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. n < 0
3. ia < 1
4. 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:

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

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

11. M_A <> 1
12. MB_A < 1
13. RSRC_A <> 0

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

14. M_A < 0 and (n = 0); M_A < 1 otherwise
15. MB_A < 1
16. n > (MB_A)(p)-mod(ia-1,MB_A)
17. ia > M_A and (n > 0)
18. ia+n-1 > M_A and (n > 0)
19. RSRC_A < 0 or RSRC_A >= p

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

20. N_A <> 1
21. NB_A < 1
22. CSRC_A <> 0

    In all cases:

23. laf < (minimum value) (For the minimum value, see the laf argument description.)
24. 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. ia differs.
3. DTYPE_A differs.

   If DTYPE_A = 1 on all processes:

4. M_A differs.
5. N_A differs.
6. MB_A differs.
7. NB_A differs.
8. RSRC_A differs.
9. CSRC_A differs.

   If DTYPE_A = 501 on all processes:

10. N_A differs.
11. NB_A differs.
12. CSRC_A differs.

    If DTYPE_A = 502 on all processes:

13. M_A differs.
14. MB_A differs.
15. RSRC_A differs.

    Also:

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

Example 1

This example shows a factorization of the general tridiagonal matrix A of order 12.

      *                                                            *
      | 2.0  2.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0 |
      | 1.0  3.0  2.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0 |
      | 0.0  1.0  3.0  2.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0 |
      | 0.0  0.0  1.0  3.0  2.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0 |
      | 0.0  0.0  0.0  1.0  3.0  2.0  0.0  0.0  0.0  0.0  0.0  0.0 |
      | 0.0  0.0  0.0  0.0  1.0  3.0  2.0  0.0  0.0  0.0  0.0  0.0 |
      | 0.0  0.0  0.0  0.0  0.0  1.0  3.0  2.0  0.0  0.0  0.0  0.0 |
      | 0.0  0.0  0.0  0.0  0.0  0.0  1.0  3.0  2.0  0.0  0.0  0.0 |
      | 0.0  0.0  0.0  0.0  0.0  0.0  0.0  1.0  3.0  2.0  0.0  0.0 |
      | 0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0  1.0  3.0  2.0  0.0 |
      | 0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0  1.0  3.0  2.0 |
      | 0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0  1.0  3.0 |
      *                                                            *

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

Notes:

1. The vectors dl, d, and du, output from PDGTTRF, are stored in an internal format that depends on the number of processes. These vectors are passed, unchanged, to the solve subroutine PDGTTRS.

2. The contents of the du2 and af vectors, output from PDGTTRF, are not shown. These vectors are passed, unchanged, to the solve subroutine PDGTTRS.

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

Call Statements and Input

ORDER = 'R'
NPROW = 3
NPCOL = 1
CALL BLACS_GET (0, 0, ICONTXT)
CALL BLACS_GRIDINIT(ICONTXT, ORDER, NPROW, NPCOL)
CALL BLACS_GRIDINFO(ICONTXT, NPROW, NPCOL, MYROW, MYCOL)
 
              N    DL   D   DU   DU2  IA   DESC_A   IPIV   AF   LAF  WORK LWORK INFO
              |    |    |    |    |    |      |      |     |    |      |    |    |
CALL PDGTTRF( 12 , DL , D , DU , DU2 , 1 , DESC_A , IPIV , AF , 48 , WORK , 0 , INFO )

Global vector dl with block size of 4:

B,D     0
     *     *
     |  .  |
     | 1.0 |
 0   | 1.0 |
     | 1.0 |
     | --- |
     | 1.0 |
     | 1.0 |
 1   | 1.0 |
     | 1.0 |
     | --- |
     | 1.0 |
     | 1.0 |
 2   | 1.0 |
     | 1.0 |
     *     *

Global vector d with block size of 4:

B,D     0
     *     *
     | 2.0 |
     | 3.0 |
 0   | 3.0 |
     | 3.0 |
     | --- |
     | 3.0 |
     | 3.0 |
 1   | 3.0 |
     | 3.0 |
     | --- |
     | 3.0 |
     | 3.0 |
 2   | 3.0 |
     | 3.0 |
     *     *

Global vector du with block size of 4:

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

The following is the 3 × 1 process grid:

B,D  |    0    
-----| -------  
0    |   P00
-----| ------- 
1    |   P10
-----| ------- 
2    |   P20
-----| ------- 

Local array DL with block size of 4:

p,q  |  0
-----|-----
     |  .
     | 1.0
 0   | 1.0
     | 1.0
-----|-----
     | 1.0
     | 1.0
 1   | 1.0
     | 1.0
-----|-----
     | 1.0
     | 1.0
 2   | 1.0
     | 1.0

Local array D with block size of 4:

p,q  |  0
-----|-----
     | 2.0
     | 3.0
 0   | 3.0
     | 3.0
-----|-----
     | 3.0
     | 3.0
 1   | 3.0
     | 3.0
-----|-----
     | 3.0
     | 3.0
 2   | 3.0
     | 3.0

Local array DU with block size of 4:

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

Output:

Global vector dl with block size of 4:

B,D     0
     *      *
     |  .   |
     | 0.5  |
 0   | 0.5  |
     | 0.5  |
     | ---- |
     | 1.0  |
     | 0.33 |
 1   | 0.43 |
     | 0.47 |
     | ---- |
     | 1.0  |
     | 1.0  |
 2   | 1.0  |
     | 1.0  |
     *      *

Global vector d with block size of 4:

B,D     0
     *      *
     | 0.5  |
     | 0.5  |
 0   | 0.5  |
     | 2.0  |
     | ---- |
     | 0.33 |
     | 0.43 |
 1   | 0.47 |
     | 2.07 |
     | ---- |
     | 2.07 |
     | 0.47 |
 2   | 0.43 |
     | 0.33 |
     *      *

Global vector du with block size of 4:

B,D     0
     *      *
     | 2.0  |
     | 2.0  |
 0   | 2.0  |
     | 2.0  |
     | ---- |
     | 2.0  |
     | 2.0  |
 1   | 2.0  |
     | 2.0  |
     | ---- |
     | 0.93 |
     | 0.86 |
 2   | 0.67 |
     |  .   |
     *      *

Global vector ipiv with block size of 4:

B,D    0
     *   *
     | 0 |
     | 0 |
 0   | 0 |
     | 0 |
     | - |
     | 0 |
     | 0 |
 1   | 0 |
     | 0 |
     | - |
     | 0 |
     | 0 |
 2   | 0 |
     | 0 |
     *   *

The following is the 3 × 1 process grid:

B,D  |    0    
-----| -------  
0    |   P00
-----| -------  
1    |   P10
-----| -------  
2    |   P20

Local array DL with block size of 4:

p,q  |  0
-----|------
     |  .
     | 0.5
 0   | 0.5
     | 0.5
-----|------
     | 1.0
     | 0.33
 1   | 0.43
     | 0.47
-----|------
     | 1.0
     | 1.0
 2   | 1.0
     | 1.0

Local array D with block size of 4:

p,q  |  0
-----|------
     | 0.5
     | 0.5
 0   | 0.5
     | 2.0
-----|------
     | 0.33
     | 0.43
 1   | 0.47
     | 2.07
-----|------
     | 2.07
     | 0.47
 2   | 0.43
     | 0.33

Local array DU with block size of 4:

p,q  |  0
-----|------
     | 2.0
     | 2.0
 0   | 2.0
     | 2.0
-----|------
     | 2.0
     | 2.0
 1   | 2.0
     | 2.0
-----|------
     | 0.93
     | 0.86
 2   | 0.67
     |  .

Local array IPIV with block size of 4:

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

The value of info is 0 on all processes.

Example 2

This example shows a factorization of the diagonally dominant general tridiagonal matrix A of order 12. Matrix A is stored in tridiagonal storage mode and distributed over a 3 × 1 process grid using block-cyclic distribution.

Matrix A and the input and/or output values for dl, d, du, desc_a, and info in this example are the same as shown for Example 1.

Notes:

1. The vectors dl, d, and du, output from PDDTTRF, are stored in an internal format that depends on the number of processes. These vectors are passed, unchanged, to the solve subroutine PDDTTRS.

2. The contents of vector af, output from PDDTTRF, are not shown. This vector is passed, unchanged, to the solve subroutine PDDTTRS.

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

Call Statements and Input

ORDER = 'R'
NPROW = 3
NPCOL = 1
CALL BLACS_GET (0, 0, ICONTXT)
CALL BLACS_GRIDINIT(ICONTXT, ORDER, NPROW, NPCOL)
CALL BLACS_GRIDINFO(ICONTXT, NPROW, NPCOL, MYROW, MYCOL)
 
              N    DL   D   DU   IA  DESC_A   AF   LAF  WORK LWORK  INFO
              |    |    |    |   |      |      |    |     |    |     |
CALL PDDTTRF( 12 , DL , D , DU , 1 , DESC_A , AF , 44 , WORK , 0  , INFO )