PZLACON - estimate the 1-norm of a square, complex distributed matrix A

SUBROUTINE PZLACON(

N, V, IV, JV, DESCV, X, IX, JX, DESCX, EST, KASE )

INTEGER IV, IX, JV, JX, KASE, N

DOUBLE PRECISION EST

INTEGER DESCV( * ), DESCX( * )

COMPLEX*16 V( * ), X( * )

PZLACON estimates the 1-norm of a square, complex distributed matrix A. Reverse communication is used for evaluating matrix-vector products. X and V are aligned with the distributed matrix A, this information is implicitly contained within IV, IX, DESCV, and DESCX.

Notes

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Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping 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 column.

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

N (global input) INTEGER

The length of the distributed vectors V and X. N >= 0.

V (local workspace) COMPLEX*16 pointer into the local

memory to an array of dimension LOCr(N+MOD(IV-1,MB_V)). On the final return, V = A*W, where EST = norm(V)/norm(W) (W is not returned).

IV (global input) INTEGER

The row index in the global array V indicating the first row of sub( V ).

JV (global input) INTEGER

The column index in the global array V indicating the first column of sub( V ).

DESCV (global and local input) INTEGER array of dimension DLEN_.

The array descriptor for the distributed matrix V.

X (local input/local output) COMPLEX*16 pointer into the

local memory to an array of dimension LOCr(N+MOD(IX-1,MB_X)). On an intermediate return, X should be overwritten by A * X, if KASE=1, A' * X, if KASE=2, where A' is the conjugate transpose of A, and PZLACON must be re-called with all the other parameters unchanged.

IX (global input) INTEGER

The row index in the global array X indicating the first row of sub( X ).

JX (global input) INTEGER

The column index in the global array X indicating the first column of sub( X ).

DESCX (global and local input) INTEGER array of dimension DLEN_.

The array descriptor for the distributed matrix X.

EST (global output) DOUBLE PRECISION

An estimate (a lower bound) for norm(A).

KASE (local input/local output) INTEGER

On the initial call to PZLACON, KASE should be 0. On an intermediate return, KASE will be 1 or 2, indicating whether X should be overwritten by A * X or A' * X. On the final return from PZLACON, KASE will again be 0.

The serial version ZLACON has been contributed by Nick Higham, University of Manchester. It was originally named SONEST, dated March 16, 1988.

Reference: N.J. Higham, "FORTRAN codes for estimating the one-norm of a real or complex matrix, with applications to condition estimation", ACM Trans. Math. Soft., vol. 14, no. 4, pp. 381-396, December 1988.