PDGEBAL(3) ScaLAPACK routine of NEC Numeric Library Collection PDGEBAL(3)
NAME
PDGEBAL - balances a general real matrix A
SYNOPSIS
SUBROUTINE PDGEBAL( JOB, N, A, DESCA, ILO, IHI, SCALE, INFO )
CHARACTER JOB
INTEGER IHI, ILO, INFO, N
INTEGER DESCA( * )
DOUBLE PRECISION A( * ), SCALE( * )
PURPOSE
PDGEBAL balances a general real matrix A. This involves, first, per-
muting A by a similarity transformation to isolate eigenvalues in the
first 1 to ILO-1 and last IHI+1 to N elements on the diagonal; and sec-
ond, applying a diagonal similarity transformation to rows and columns
ILO to IHI to make the rows and columns as close in norm as possible.
Both steps are optional.
Balancing may reduce the 1-norm of the matrix, and improve the accuracy
of the computed eigenvalues and/or eigenvectors.
Notes
=====
Each global data object is described by an associated description vec-
tor. This vector stores the information required to establish the map-
ping between an object element and its corresponding process and memory
location.
Let A be a generic term for any 2D block cyclicly distributed array.
Such a global array has an associated description vector DESCA. In the
following comments, the character _ should be read as "of the global
array".
NOTATION STORED IN EXPLANATION
--------------- -------------- --------------------------------------
DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case,
DTYPE_A = 1.
CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating
the BLACS process grid A is distribu-
ted over. The context itself is glo-
bal, but the handle (the integer
value) may vary.
M_A (global) DESCA( M_ ) The number of rows in the global
array A.
N_A (global) DESCA( N_ ) The number of columns in the global
array A.
MB_A (global) DESCA( MB_ ) The blocking factor used to distribute
the rows of the array.
NB_A (global) DESCA( NB_ ) The blocking factor used to distribute
the columns of the array.
RSRC_A (global) DESCA( RSRC_ ) The process row over which the first
row of the array A is distributed.
CSRC_A (global) DESCA( CSRC_ ) The process column over which the
first column of the array A is
distributed.
LLD_A (local) DESCA( LLD_ ) The leading dimension of the local
array. LLD_A >= MAX(1,LOCr(M_A)).
Let K be the number of rows or columns of a distributed matrix, and
assume that its process grid has dimension p x q.
LOCr( K ) denotes the number of elements of K that a process would
receive if K were distributed over the p processes of its process col-
umn.
Similarly, LOCc( K ) denotes the number of elements of K that a process
would receive if K were distributed over the q processes of its process
row.
The values of LOCr() and LOCc() may be determined via a call to the
ScaLAPACK tool function, NUMROC:
LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ),
LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ).
An upper bound for these quantities may be computed by:
LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A
LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A
ARGUMENTS
JOB (global input) CHARACTER*1
Specifies the operations to be performed on A:
= 'N': none: simply set ILO = 1, IHI = N, SCALE(I) = 1.0
for i = 1,...,N;
= 'P': permute only;
= 'S': scale only;
= 'B': both permute and scale.
N (global input) INTEGER
The order of the matrix A. N >= 0.
A (local input/output) DOUBLE PRECISION array, dimension
(DESCA(LLD_,LOCc(N))
On entry, the input matrix A.
On exit, A is overwritten by the balanced matrix.
If JOB = 'N', A is not referenced.
See Further Details.
DESCA (global and local input) INTEGER array of dimension DLEN_.
The array descriptor for the distributed matrix A.
ILO (global output) INTEGER
IHI (global output) INTEGER
ILO and IHI are set to integers such that on exit
A(i,j) = 0 if i > j and j = 1,...,ILO-1 or I = IHI+1,...,N.
If JOB = 'N' or 'S', ILO = 1 and IHI = N.
SCALE (global output) DOUBLE PRECISION array, dimension (N)
Details of the permutations and scaling factors applied to A.
If P(j) is the index of the row and column interchanged with
row and column j and D(j) is the scaling factor applied to row
and column j, then
SCALE(j) = P(j) for j = 1,...,ILO-1
= D(j) for j = ILO,...,IHI
= P(j) for j = IHI+1,...,N.
The order in which the interchanges are made is N to IHI+1,
then 1 to ILO-1.
INFO (global output) INTEGER
= 0: successful exit.
< 0: if INFO = -i, the i-th argument had an illegal value.
FURTHER DETAILS
The permutations consist of row and column interchanges which put the
matrix in the form
( T1 X Y )
P A P = ( 0 B Z )
( 0 0 T2 )
where T1 and T2 are upper triangular matrices whose eigenvalues lie
along the diagonal. The column indices ILO and IHI mark the starting
and ending columns of the submatrix B. Balancing consists of applying a
diagonal similarity transformation inv(D) * B * D to make the 1-norms
of each row of B and its corresponding column nearly equal. The output
matrix is
( T1 X*D Y )
( 0 inv(D)*B*D inv(D)*Z ).
( 0 0 T2 )
Information about the permutations P and the diagonal matrix D is
returned in the vector SCALE.
This subroutine is based on the EISPACK routine BALANC. In principle,
the parallelism is extracted by using PBLAS and BLACS routines for the
permutation and balancing.
ScaLAPACK routine 31 October 2017 PDGEBAL(3)