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

PDSYRK, PZSYRK, and PZHERK--Rank-K Update of a Real or Complex Symmetric or a Complex Hermitian Matrix

PDSYRK and PZSYRK compute one of the following rank-k updates:

1. C<--alphaAA^T+betaC

2. C<--alphaA^TA+betaC

PZHERK computes one of the following rank-k updates:

3. C<--alphaAA^H+betaC

4. C<--alphaA^HA+betaC

where, in the formulas above:

A represents the global general submatrix:

For trans = 'N', it is A_{ia:ia+n-1,
ja:ja+k-1}.
For trans = 'T' or 'C', it is A_{ia:ia+k-1,
ja:ja+n-1}.

C represents the global submatrix C_{ic:ic+n-1,
jc:jc+n-1}.

and:

For PDSYRK, submatrix C is real symmetric.
For PZSYRK, submatrix C is complex symmetric.
For PZHERK, submatrix C is complex Hermitian.

Note:: No data should be moved to form A^T or A^H; that is, the matrix A should always be stored in its untransposed form.

In the following two cases, no computation is performed and the subroutine returns after doing some parameter checking:

n = 0
beta is one, and alpha is zero or k = 0.

See references [14] and [15].

Table 51. Data Types

A, C	alpha, beta	Subprogram
Long-precision real	Long-precision real	PDSYRK
Long-precision complex	Long-precision complex	PZSYRK
Long-precision complex	Long-precision real	PZHERK

Syntax

Fortran	CALL PDSYRK \| PZSYRK \| PZHERK (`uplo`, `trans`, `n`, `k`, `alpha`, `a`, `ia`, `ja`, `desc_a`, `beta`, `c`, `ic`, `jc`, `desc_c`)
C and C++	pdsyrk \| pzsyrk \| pzherk (`uplo`, `trans`, `n`, `k`, `alpha`, `a`, `ia`, `ja`, `desc_a`, `beta`, `c`, `ic`, `jc`, `desc_c`);

On Entry

uplo

indicates whether the upper or lower triangular part of the global submatrix C 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'.

trans

indicates which computation is performed, where:

If trans = 'N', A is used.

If trans = 'T', A^T is used.

If trans = 'C', A^H is used.

Scope: global

Specified as: a single character, where:

For PDSYRK, it must be 'N', 'T', or 'C'.

For PZSYRK, it must be 'N' or 'T'.

For PZHERK, it must be 'N' or 'C'.

n

is the order of the global submatrix C used in the computation, and:

If trans = 'N', it is the number of rows in submatrix A used in the computation.

If trans = 'T' or 'C', it is the number of columns in submatrix A used in the computation.

Scope: global

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

k

has the following meaning:

If trans = 'N', it is the number of columns in submatrix A used in the computation.

If trans = 'T' or 'C', it is the number of rows in submatrix A used in the computation.

Scope: global

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

alpha

is the scalar alpha.

Scope: global

Specified as: a number of the data type indicated in Table 51.

a

is the local part of the global general matrix A. 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:

If trans = 'N', the leading LOCp(ia+n-1) by LOCq(ja+k-1) part of the local array A must contain the local pieces of the leading ia+n-1 by ja+k-1 part of the global matrix.
If trans = 'T' or 'C', the leading LOCp(ia+k-1) by LOCq(ja+n-1) part of the local array A must contain the local pieces of the leading ia+k-1 by ja+n-1 part of the global matrix.

Note:: No data should be moved to form A^T or A^H; that is, the matrix A should always be stored in its untransposed form.

Scope: local

Specified as: an LLD_A by (at least) LOCq(N_A) array, containing numbers of the data type indicated in Table 51. Details about the 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:

If trans = 'N', then ia+n-1 <= M_A.

If trans = 'T' or 'C', then ia+k-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:

If trans = 'N', then ja+k-1 <= N_A.

If trans = 'T' or 'C', then 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 or `k` = 0: M_A >= 0 Otherwise: M_A >= 1	Global
4	N_A	Number of columns in the global matrix	If `n` = 0 or `k` = 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.

beta

is the scalar beta.

Scope: global

Specified as: a number of the data type indicated in Table 51.

c

is the local part of the global real or complex symmetric or complex Hermitian matrix C. This identifies the first element of the local array C. This subroutine computes the location of the first element of the local subarray used, based on ic, jc, desc_c, p, q, myrow, and mycol; therefore, the leading LOCp(ic+n-1) by LOCq(jc+n-1) part of the local array C must contain the local pieces of the leading ic+n-1 by jc+n-1 part of the global matrix, and:

If uplo = 'U', the leading n × n upper triangular part of the global submatrix C_{ic:ic+n-1,
jc:jc+n-1} must contain the upper triangular part of the submatrix, and the strictly lower triangular part is not referenced.
If uplo = 'L', the leading n × n lower triangular part of the global submatrix C_{ic:ic+n-1,
jc:jc+n-1} must contain the lower triangular part of the submatrix, and the strictly upper triangular part is not referenced.

When beta is zero, C need not be set on input.

Scope: local

Specified as: an LLD_C by (at least) LOCq(N_C) array, containing numbers of the data type indicated in Table 51. Details about the block-cyclic data distribution of global matrix C are stored in desc_c.

ic

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

Scope: global

Specified as: a fullword integer; 1 <= ic <= M_C and ic+n-1 <= M_C.

jc

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

Scope: global

Specified as: a fullword integer; 1 <= jc <= N_C and jc+n-1 <= N_C.

desc_c

is the array descriptor for global matrix C, described in the following table:

`desc_c`	Name	Description	Limits	Scope
1	DTYPE_C	Descriptor type	DTYPE_C=1	Global
2	CTXT_C	BLACS context	Valid value, as returned by BLACS_GRIDINIT or BLACS_GRIDMAP	Global
3	M_C	Number of rows in the global matrix	If `n` = 0: M_C >= 0 Otherwise: M_C >= 1	Global
4	N_C	Number of columns in the global matrix	If `n` = 0: N_C >= 0 Otherwise: N_C >= 1	Global
5	MB_C	Row block size	MB_C >= 1	Global
6	NB_C	Column block size	NB_C >= 1	Global
7	RSRC_C	The process row of the `p` × `q` grid over which the first row of the global matrix is distributed	0 <= RSRC_C < `p`	Global
8	CSRC_C	The process column of the `p` × `q` grid over which the first column of the global matrix is distributed	0 <= CSRC_C < `q`	Global
9	LLD_C	The leading dimension of the local array	LLD_C >= max(1,LOCp(M_C))	Local

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

On Return

c

is the updated local part of the global real or complex symmetric or complex Hermitian matrix C, containing the results of the computation.

Scope: local

Returned as: an LLD_C by (at least) LOCq(N_C) array, containing numbers of the data type indicated in Table 51.

Notes and Coding Rules

These subroutines accept lowercase letters for the uplo and trans arguments.
For PDSYRK, if you specify 'C' for the trans argument, it is interpreted as though you specified 'T'.
The imaginary parts of the diagonal elements of a complex Hermitian matrix C are assumed to be zero, so you do not have to set these values. On output, they are set to zero, except when beta is one and alpha or k is zero, in which case no computation is performed.
The matrices must have no common elements; otherwise, results are unpredictable.
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.
For suggested block sizes, see Coding Tips for Optimizing Parallel Performance.
The following values must be equal: CTXT_A = CTXT_C.
If C is not contained within a single block, that is:

n+mod(ic-1, MB_C) > MB_C
n+mod(jc-1, NB_C) > NB_C

then:
- The global matrix C must be distributed using a square block-cyclic distribution; that is, MB_C = NB_C.
- The global matrix C must be aligned on a block boundary, that is:
  
  ic-1 must be a multiple of MB_C.
  jc-1 must be a multiple of NB_C.
If trans = 'N':
- If C is not contained within a single block, then:
  - The following block sizes must be equal: MB_A = NB_C.
  - The global matrix A must be aligned on a block row boundary; that is, ia-1 must be a multiple of MB_A.
- In the process grid, the process row containing the first row of the submatrix C must also contain the first row of the submatrix A; that is, icrow = iarow, where:
  
  icrow = mod((((ic-1)/MB_C)+RSRC_C), p)
  iarow = mod((((ia-1)/MB_A)+RSRC_A), p)
If trans = 'T' or 'C':
- If C is not contained within a single block, then:
  - The following block sizes must be equal: NB_A = MB_C.
  - The global matrix A must be aligned on a block column boundary; that is, ja-1 must be a multiple of NB_A.
- In the process grid, the process column containing the first column of the submatrix C must also contain the first column of the submatrix A; that is, iccol = iacol, where:
  
  iccol = mod((((jc-1)/NB_C)+CSRC_C), q)
  iacol = mod((((ja-1)/NB_A)+CSRC_A), q)
If C is contained within a single block:
- If trans = 'N', A must be a block row matrix; that is, if p > 1:
  
  n+mod(ia-1, MB_A) <= MB_A
- If trans = 'T' or 'C', A must be a block column matrix; that is, if q > 1:
  
  n+mod(ja-1, NB_A) <= NB_A

DTYPE_A is invalid.
DTYPE_C is invalid.

Stage 2:

CTXT_A is invalid.

Stage 3:

This subroutine was called from outside the process grid.

Stage 4:

uplo <> 'U' or 'L'
trans <>
- 'N', 'T', or 'C' for PDSYRK
- 'N' or 'T' for PZSYRK
- 'N' or 'C' for PZHERK
n < 0 and trans = 'N'
n < 0 and trans = 'T' or 'C'
n < 0 and trans is invalid.
k < 0 and trans = 'N'
k < 0 and trans = 'T' or 'C'
k < 0 and trans is invalid.
M_A < 0 and (n = 0 or k = 0); M_A < 1 otherwise
N_A < 0 and (n = 0 or k = 0); N_A < 1 otherwise
MB_A < 1
NB_A < 1
RSRC_A < 0 or RSRC_A >= p
CSRC_A < 0 or CSRC_A >= q
ia < 1
ja < 1
M_C < 0 and n = 0; M_C < 1 otherwise
N_C < 0 and n = 0; N_C < 1 otherwise
MB_C < 1
NB_C < 1
RSRC_C < 0 or RSRC_C >= p
CSRC_C < 0 or CSRC_C >= q
ic < 1
jc < 1
CTXT_A <> CTXT_C

If n <> 0 and k <> 0:

ia > M_A
ja > N_A
trans = 'N' and ia+n-1 > M_A
trans = 'N' and ja+k-1 > N_A
trans = 'T' or 'C' and ia+k-1 > M_A
trans = 'T' or 'C' and ja+n-1 > N_A

If n <> 0:

ic > M_C
jc > N_C
ic+n-1 > M_C
jc+n-1 > N_C

Stage 5:

If C is not contained within a single block, that is:

n+mod(ic-1, MB_C) > MB_C
n+mod(jc-1, NB_C) > NB_C

and NB_C <> MB_C.
trans = 'N' and NB_C <> MB_A.
trans = 'T' or 'C' and MB_C <> NB_A.

If C is not contained within a single block:

mod(ic-1, MB_C) <> 0
mod(jc-1, NB_C) <> 0
trans = 'N' and mod(ia-1, MB_A) <> 0
trans = 'T' or 'C' and mod(ja-1, NB_A) <> 0

Stage 6:

LLD_A < max(1, LOCp(M_A))
LLD_C < max(1, LOCp(M_C))
If trans = 'N', then (in the process grid) the process row containing the first row of the submatrix C does not contain the first row of the submatrix A; that is, icrow <> iarow, where:

icrow = mod((((ic-1)/MB_C)+RSRC_C), p)
iarow = mod((((ia-1)/MB_A)+RSRC_A), p)
If trans = 'T' or 'C', then (in the process grid) the process column containing the first column of the submatrix C does not contain the first column of the submatrix A; that is, iccol <> iacol, where:

iccol = mod((((jc-1)/NB_C)+CSRC_C), q)
iacol = mod((((ja-1)/NB_A)+CSRC_A), q)

If C is contained within a single block:

If trans = 'N':

p > 1 and n+mod(ia-1, MB_A) > MB_A
If trans = 'T' or 'C':

q > 1 and n+mod(ja-1, NB_A) > NB_A

Example 1

This example computes C = alphaAA^T+betaC using a 2 × 3 process grid.

Call Statements and Input

 ORDER = 'R'
 NPROW = 2
 NPCOL = 3
 CALL BLACS_GET (0, 0, ICONTXT)
 CALL BLACS_GRIDINIT(ICONTXT, ORDER, NPROW, NPCOL)
 CALL BLACS_GRIDINFO(ICONTXT, NPROW, NPCOL, MYROW, MYCOL)
 
             UPLO   TRANS   N    K     ALPHA    A  IA  JA    DESC_A    BETA
               |      |     |    |       |      |   |   |      |         |
 CALL PDSYRK( 'L' ,  'N' ,  8  , 5  ,  1.0D0  , A , 1 , 1 ,  DESC_A ,  1.0D0 ,
 
               C  IC  JC   DESC_C
               |   |   |     |
               C , 1 , 1 , DESC_C )

	Desc_A	Desc_C
DTYPE_	1	1
CTXT_	`icontxt`^(IOBG26)	`icontxt`^(IOBG26)
M_	8	8
N_	5	8
MB_	2	2
NB_	2	2
RSRC_	0	0
CSRC_	0	0
LLD_	See below^(EPSSL26)	See below^(EPSSL26)
Notes: `icontxt` is the output of the BLACS_GRIDINIT call. Each process should set the LLD_ as follows: LLD_A = MAX(1,NUMROC(M_A, MB_A, MYROW, RSRC_A, NPROW)) LLD_C = MAX(1,NUMROC(M_C, MB_C, MYROW, RSRC_C, NPROW)) In this example, LLD_A = LLD_C = 4 on all processes.

Global general 8 × 5 matrix A with block size 2 × 2:

B,D         0                1             2
     *                                         *
 0   |   0.0    8.0  |   16.0   24.0  |   32.0 |
     |   1.0    9.0  |   17.0   25.0  |   33.0 |
     | --------------|----------------|------- |
 1   |   2.0   10.0  |   18.0   26.0  |   34.0 |
     |   3.0   11.0  |   19.0   27.0  |   35.0 |
     | --------------|----------------|------- |
 2   |   4.0   12.0  |   20.0   28.0  |   36.0 |
     |   5.0   13.0  |   21.0   29.0  |   37.0 |
     | --------------|----------------|------- |
 3   |   6.0   14.0  |   22.0   30.0  |   38.0 |
     |   7.0   15.0  |   23.0   31.0  |   39.0 |
     *                                         *

The following is the 2 × 3 process grid:

B,D  |   0   |   1   |   2   
-----|-------|-------|------- 
0    |   P₀₀   |   P₀₁   |   P₀₂
2    |       |       |
-----|-------|-------|------- 
1    |   P₁₀   |   P₁₁   |   P₁₂
3    |       |       |

Local arrays for A:

p,q  |     0       |       1        |    2
-----|-------------|----------------|--------
     |  0.0   8.0  |   16.0   24.0  |   32.0
     |  1.0   9.0  |   17.0   25.0  |   33.0
 0   |  4.0  12.0  |   20.0   28.0  |   36.0
     |  5.0  13.0  |   21.0   29.0  |   37.0
-----|-------------|----------------|--------
     |  2.0  10.0  |   18.0   26.0  |   34.0
     |  3.0  11.0  |   19.0   27.0  |   35.0
 1   |  6.0  14.0  |   22.0   30.0  |   38.0
     |  7.0  15.0  |   23.0   31.0  |   39.0

Global real symmetric matrix C of order 8 block size 2 × 2:

B,D         0               1               2               3
     *                                                             *
 0   |   0.0    .   |     .     .   |     .     .   |     .     .  |
     |   1.0   8.0  |     .     .   |     .     .   |     .     .  |
     | -------------|---------------|---------------|------------- |
 1   |   2.0   9.0  |   15.0    .   |     .     .   |     .     .  |
     |   3.0  10.0  |   16.0  21.0  |     .     .   |     .     .  |
     | -------------|---------------|---------------|------------- |
 2   |   4.0  11.0  |   17.0  22.0  |   26.0    .   |     .     .  |
     |   5.0  12.0  |   18.0  23.0  |   27.0  30.0  |     .     .  |
     | -------------|---------------|---------------|------------- |
 3   |   6.0  13.0  |   19.0  24.0  |   28.0  31.0  |   33.0    .  |
     |   7.0  14.0  |   20.0  25.0  |   29.0  32.0  |   34.0  35.0 |
     *                                                             *

The following is the 2 × 3 process grid:

B,D  |   0   |   1   |   2   
-----|-------|-------|------- 
0    |   P₀₀   |   P₀₁   |   P₀₂
2    |       |       |
-----|-------|-------|------- 
1    |   P₁₀   |   P₁₁   |   P₁₂
3    |       |       |

Local arrays for C:

p,q  |            0             |       1       |       2
-----|--------------------------|---------------|--------------
     |   0.0    .     .     .   |     .     .   |     .     .
     |   1.0   8.0    .     .   |     .     .   |     .     .
 0   |   4.0  11.0    .     .   |   17.0  22.0  |   26.0    .
     |   5.0  12.0    .     .   |   18.0  23.0  |   27.0  30.0
-----|--------------------------|---------------|--------------
     |   2.0   9.0    .     .   |   15.0    .   |     .     .
     |   3.0  10.0    .     .   |   16.0  21.0  |     .     .
 1   |   6.0  13.0  33.0    .   |   19.0  24.0  |   28.0  31.0
     |   7.0  14.0  34.0  35.0  |   20.0  25.0  |   29.0  32.0

Output:

Global real symmetric matrix C of order 8 with block size 2 × 2:

B,D           0                   1                   2                   3
     *                                                                             *
 0   |  1920.0      .   |       .       .   |       .       .   |       .       .  |
     |  2001.0  2093.0  |       .       .   |       .       .   |       .       .  |
     | -----------------|-------------------|-------------------|----------------- |
 1   |  2082.0  2179.0  |   2275.0      .   |       .       .   |       .       .  |
     |  2163.0  2265.0  |   2366.0  2466.0  |       .       .   |       .       .  |
     | -----------------|-------------------|-------------------|----------------- |
 2   |  2244.0  2351.0  |   2457.0  2562.0  |   2666.0      .   |       .       .  |
     |  2325.0  2437.0  |   2548.0  2658.0  |   2767.0  2875.0  |       .       .  |
     | -----------------|-------------------|-------------------|----------------- |
 3   |  2406.0  2523.0  |   2639.0  2754.0  |   2868.0  2981.0  |   3093.0      .  |
     |  2487.0  2609.0  |   2730.0  2850.0  |   2969.0  3087.0  |   3204.0  3320.0 |
     *                                                                             *

The following is the 2 × 3 process grid:

B,D  |   0   |   1   |   2   
-----|-------|-------|------- 
0    |   P₀₀   |   P₀₁   |   P₀₂
2    |       |       |
-----|-------|-------|------- 
1    |   P₁₀   |   P₁₁   |   P₁₂
3    |       |       |

Local arrays for C:

p,q  |                0                 |         1         |         2
-----|----------------------------------|-------------------|------------------
     |  1920.0      .       .       .   |       .       .   |       .       .
     |  2001.0  2093.0      .       .   |       .       .   |       .       .
 0   |  2244.0  2351.0      .       .   |   2457.0  2562.0  |   2666.0      .
     |  2325.0  2437.0      .       .   |   2548.0  2658.0  |   2767.0  2875.0
-----|----------------------------------|-------------------|------------------
     |  2082.0  2179.0      .       .   |   2275.0      .   |       .       .
     |  2163.0  2265.0      .       .   |   2366.0  2466.0  |       .       .
 1   |  2406.0  2523.0  3093.0      .   |   2639.0  2754.0  |   2868.0  2981.0
     |  2487.0  2609.0  3204.0  3320.0  |   2730.0  2850.0  |   2969.0  3087.0

Example 2

This example computes C = alphaAA^T+betaC using a 2 × 3 process grid.

Call Statements and Input

 ORDER = 'R'
 NPROW = 2
 NPCOL = 3
 CALL BLACS_GET (0, 0, ICONTXT)
 CALL BLACS_GRIDINIT(ICONTXT, ORDER, NPROW, NPCOL)
 CALL BLACS_GRIDINFO(ICONTXT, NPROW, NPCOL, MYROW, MYCOL)
 
             UPLO   TRANS   N    K     ALPHA    A  IA  JA    DESC_A    BETA
               |      |     |    |       |      |   |   |      |         |
 CALL PZSYRK( 'U' ,  'N' ,  3  , 5  ,  ALPHA  , A , 1 , 1 ,  DESC_A ,  BETA ,
 
               C  IC  JC   DESC_C
               |   |   |     |
               C , 1 , 1 , DESC_C )
 
               ALPHA = (1.0, 1.0)
 
               BETA  = (1.0, 1.0)

	Desc_A	Desc_C
DTYPE_	1	1
CTXT_	`icontxt`^(IOBG27)	`icontxt`^(IOBG27)
M_	3	3
N_	5	3
MB_	2	2
NB_	2	2
RSRC_	0	0
CSRC_	0	0
LLD_	See below^(EPSSL27)	See below^(EPSSL27)
Notes: `icontxt` is the output of the BLACS_GRIDINIT call. Each process should set the LLD_ as follows: LLD_A = MAX(1,NUMROC(M_A, MB_A, MYROW, RSRC_A, NPROW)) LLD_C = MAX(1,NUMROC(M_C, MB_C, MYROW, RSRC_C, NPROW)) In this example: LLD_A = LLD_C = 2 on P₀₀, P₀₁, and P₀₂ LLD_A = LLD_C = 1 on P₁₀, P₁₁, and P₁₂

Global general 3 × 5 matrix A with block size 2 × 2:

 B,D             0                     1                2
     *                                                       *
 0   | (2.0,0.0) (3.0,2.0) | (4.0,1.0) (1.0,7.0) | (0.0,0.0) |
     | (30.,3.0) (8.0,0.0) | (2.0,5.0) (2.0,4.0) | (1.0,2.0) |
     | --------------------|---------------------|---------- |
 1   | (1.0,3.0) (2.0,1.0) | (6.0,0.0) (3.0,2.0) | (2.0,2.0) |
     *                                                       *

The following is the 2 × 3 process grid:

B,D  |   0   |   1   |   2   
-----|-------|-------|------- 
0    |   P₀₀   |   P₀₁   |   P₀₂
-----|-------|-------|------- 
1    |   P₁₀   |   P₁₁   |   P₁₂

Local arrays for A:

p,q  |          0          |          1          |     2
-----|---------------------|---------------------|-----------
     | (2.0,0.0) (3.0,2.0) | (4.0,1.0) (1.0,7.0) | (0.0,0.0)
 0   | (3.0,3.0) (8.0,0.0) | (2.0,5.0) (2.0,4.0) | (1.0,2.0)
-----|---------------------|---------------------|-----------
 1   | (1.0,3.0) (2.0,1.0) | (6.0,0.0) (3.0,2.0) | (2.0,2.0)

Global complex symmetric matrix C of order 3 with block size 2 × 2:

B,D             0                1
     *                                 *
 0   | (2.0,1.0) (1.0,9.0) | (4.0,5.0) |
     |     .     (3.0,1.0) | (6.0,7.0) |
     | --------------------|---------- |
 1   |     .         .     | (8.0,1.0) |
     *                                 *

The following is the 2 × 3 process grid:

B,D  |   0   |   1   |   2   
-----|-------|-------|------- 
0    |   P₀₀   |   P₀₁   |   P₀₂
-----|-------|-------|------- 
1    |   P₁₀   |   P₁₁   |   P₁₂

Local arrays for C:

p,q  |          0          |     1     |     2
-----|---------------------|-----------|-----------
     | (2.0,1.0) (1.0,9.0) | (4.0,5.0) |     .
 0   |     .     (3.0,1.0) | (6.0,7.0) |     .
-----|---------------------|-----------|-----------
 1   |     .         .     | (8.0,1.0) |     .

Output:

Global complex symmetric matrix C of order 3 with block size 2 × 2:

B,D                 0                      1
     *                                             *
 0   | (-57.0, 13.0) (-63.0, 79.0) | (-24.0, 70.0) | 
     |       .       (-28.0, 90.0) | (-55.0,103.0) |
     | ----------------------------|---------------|
 1   |       .             .       | ( 13.0, 75.0) |
     *                                             *

The following is the 2 × 3 process grid:

B,D  |   0   |   1   |   2   
-----|-------|-------|------- 
0    |   P₀₀   |   P₀₁   |   P₀₂
-----|-------|-------|------- 
1    |   P₁₀   |   P₁₁   |   P₁₂

Local arrays for C:

p,q  |              0              |       1       |       2
-----|-----------------------------|---------------|---------------
     | (-57.0, 13.0) (-63.0, 79.0) | (-24.0, 70.0) |       .   
 0   |       .       (-28.0, 90.0) | (-55.0,103.0) |       .
-----|-----------------------------|---------------|---------------
 1   |       .             .       | ( 13.0, 75.0) |       .

Example 3

This example computes C = alphaA^HA+betaC using a 3 × 2 process grid.

Note:: On output, the imaginary parts of the diagonal elements of a complex Hermitian matrix are set to zero, except when beta is one and alpha or k is zero.

Call Statements and Input

 ORDER = 'R'
 NPROW = 3
 NPCOL = 2
 CALL BLACS_GET (0, 0, ICONTXT)
 CALL BLACS_GRIDINIT(ICONTXT, ORDER, NPROW, NPCOL)
 CALL BLACS_GRIDINFO(ICONTXT, NPROW, NPCOL, MYROW, MYCOL)
 
             UPLO   TRANS   N    K     ALPHA    A  IA  JA    DESC_A    BETA
               |      |     |    |       |      |   |   |      |         |
 CALL PZHERK( 'L' ,  'C' ,  3  , 5  ,  ALPHA  , A , 1 , 1 ,  DESC_A ,  BETA ,
 
               C  IC  JC   DESC_C
               |   |   |     |
               C , 1 , 1 , DESC_C )
 
               ALPHA = 1.0
 
               BETA  = 1.0

	Desc_A	Desc_C
DTYPE_	1	1
CTXT_	`icontxt`^(IOBG28)	`icontxt`^(IOBG28)
M_	5	3
N_	3	3
MB_	2	2
NB_	2	2
RSRC_	0	0
CSRC_	0	0
LLD_	See below^(EPSSL28)	See below^(EPSSL28)
Notes: `icontxt` is the output of the BLACS_GRIDINIT call. Each process should set the LLD_ as follows: LLD_A = MAX(1,NUMROC(M_A, MB_A, MYROW, RSRC_A, NPROW)) LLD_C = MAX(1,NUMROC(M_C, MB_C, MYROW, RSRC_C, NPROW)) In this example: LLD_A = 2 on P₀₀, P₀₁, P₁₀, and P₁₁ LLD_A = 1 on P₂₀ and P₂₁ LLD_C = 2 on P₀₀ and P₀₁ LLD_C = 1 on P₁₀ and P₁₁

Global general 5 × 3 matrix A with block size 2 × 2:

B,D             0                1
     *                                 *
 0   | (2.0,0.0) (3.0,2.0) | (4.0,1.0) |
     | (3.0,3.0) (8.0,0.0) | (2.0,5.0) |
     | --------------------|-----------|
 1   | (1.0,3.0) (2.0,1.0) | (6.0,0.0) |
     | (3.0,3.0) (8.0,0.0) | (2.0,5.0) |
     | --------------------|-----------|
 2   | (1.0,9.0) (3.0,0.0) | (6.0,7.0) |
     *                                 *

The following is the 3 × 2 process grid:

B,D  |   0   | 1 
-----|-------|-----
0    |   P₀₀   |  P₀₁
-----|-------|-----
1    |   P₁₀   |  P₁₁
-----|-------|-----
2    |   P₂₀   |  P₂₁

Local arrays for A:

p,q  |          0          |     1     |
-----|---------------------|-----------|
     | (2.0,0.0) (3.0,2.0) | (4.0,1.0) |
 0   | (3.0,3.0) (8.0,0.0) | (2.0,5.0) |
-----|---------------------|-----------|
     | (1.0,3.0) (2.0,1.0) | (6.0,0.0) |
 1   | (3.0,3.0) (8.0,0.0) | (2.0,5.0) |
-----|---------------------|-----------|
 2   | (1.0,9.0) (3.0,0.0) | (6.0,7.0) |
-----|---------------------|-----------|

Global complex Hermitian matrix C of order 3 with block size 2 × 2:

B,D             0              1
     *                                  *
 0   | (6.0,0.0)      .     |     .     |
     | (3.0,4.0) (10.0,0.0) |     .     |
     | ---------------------|-----------|
 1   | (9.0,1.0) (12.0,2.0) | (3.0,0.0) |
     *                                  *

The following is the 3 × 2 process grid:

B,D  |   0   | 1 
-----|-------|-----
0    |   P₀₀   |  P₀₁
-----|-------|-----
1    |   P₁₀   |  P₁₁
-----|-------|-----
--   |   P₂₀   |  P₂₁

Local arrays for C:

p,q  |          0           |     1     |
-----|----------------------|-----------|
     | (6.0, . )      .     |     .     |
 0   | (3.0,4.0) (10.0, . ) |     .     |
-----|----------------------|-----------|
 1   | (9.0,1.0) (12.0,2.0) | (3.0, . ) |
-----|----------------------|-----------|
 2   |     .          .     |     .     |

Output:

Global complex Hermitian matrix C of order 3 with block size 2 × 2:

B,D                 0                      1
     *                                             *
 0   | (138.0,  0.0)       .       |       .       |
     | ( 65.0, 80.0) (165.0,  0.0) |       .       |
     | ----------------------------|---------------|
 1   | (134.0, 46.0) ( 88.0,-88.0) | (199.0,  0.0) |
     *                                             *

The following is the 3 × 2 process grid:

B,D  |   0   | 1 
-----|-------|-----
0    |   P₀₀   |  P₀₁
-----|-------|-----
1    |   P₁₀   |  P₁₁
-----|-------|-----
--   |   P₂₀   |  P₂₁

Local arrays for C:

p,q  |              0              |       1       |
-----|-----------------------------|---------------|
     | (138.0,  0.0)       .       |       .       |
 0   | ( 65.0, 80.0) (165.0,  0.0) |       .       |
-----|-----------------------------|---------------|
 1   | (134.0, 46.0) ( 88.0,-88.0) | (199.0,  0.0) |
-----|-----------------------------|---------------|
 2   |       .             .       |       .       |

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

PDSYRK, PZSYRK, and PZHERK--Rank-K Update of a Real or Complex Symmetric or a Complex Hermitian Matrix

Syntax

On Entry

On Return

Notes and Coding Rules

Error Conditions

Computational Errors

Resource Errors

Input-Argument and Miscellaneous Errors

Example 1

Call Statements and Input

Example 2

Call Statements and Input

Example 3

Call Statements and Input