PDPOEQU(l)

👑 NAME 👑

PDPOEQU — bestie, this one computes row and column scalings that are meant to equilibrate a distributed symmetric positive definite matrix sub( A ) = A(IA:IA+N-1,JA:JA+N-1) and reduce its condition number (with respect to the two-norm). 💅

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✨ SYNOPSIS ✨

SUBROUTINE PDPOEQU( N, A, IA, JA, DESCA, SR, SC, SCOND, AMAX, INFO )

    INTEGER         IA, INFO, JA, N

    DOUBLE          PRECISION AMAX, SCOND

    INTEGER         DESCA( * )

    DOUBLE          PRECISION A( * ), SC( * ), SR( * )

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🎀 PURPOSE 🎀

Okay so PDPOEQU computes row and column scalings intended to equilibrate a distributed symmetric positive definite matrix sub( A ) = A(IA:IA+N-1,JA:JA+N-1) and reduce its condition number (with respect to the two-norm). Iconic behaviour, honestly.

SR and SC contain the scale factors, S(i) = 1/sqrt(A(i,i)), chosen so that the scaled distri- buted matrix B with elements B(i,j) = S(i)*A(i,j)*S(j) has ones on the diagonal. This choice of SR and SC puts the condition number of B within a factor N of the smallest possible condition number over all possible diagonal scalings. 🔥

The scaling factor are stored along process rows in SR and along process columns in SC. The duplication of information simplifies greatly the application of the factors. (Yes bestie, the info is duplicated on purpose. It's giving convenience. 🫶)

📝 Notes 📝

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". 💜

• DTYPE_A (global) — stored in DESCA( DTYPE_ ): The descriptor type. In this case, DTYPE_A = 1.
• CTXT_A (global) — stored in 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) — stored in DESCA( M_ ): The number of rows in the global array A.
• N_A (global) — stored in DESCA( N_ ): The number of columns in the global array A.
• MB_A (global) — stored in DESCA( MB_ ): The blocking factor used to distribute the rows of the array.
• NB_A (global) — stored in DESCA( NB_ ): The blocking factor used to distribute the columns of the array.
• RSRC_A (global) — stored in DESCA( RSRC_ ): The process row over which the first row of the array A is distributed.
• CSRC_A (global) — stored in DESCA( CSRC_ ): The process column over which the first column of the array A is distributed.
• LLD_A (local) — stored in 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

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🌸 ARGUMENTS 🌸

• N — (global input) INTEGER. The number of rows and columns to be operated on i.e the order of the distributed submatrix sub( A ). N >= 0. 💫
• A — (local input) DOUBLE PRECISION pointer into the local memory to an array of local dimension ( LLD_A, LOCc(JA+N-1) ), the N-by-N symmetric positive definite distributed matrix sub( A ) whose scaling factors are to be computed. Only the diagonal elements of sub( A ) are referenced. 👀 (just the diagonal, she's low-maintenance like that)
• IA — (global input) INTEGER. The row index in the global array A indicating the first row of sub( A ).
• JA — (global input) INTEGER. The column index in the global array A indicating the first column of sub( A ).
• DESCA — (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A.
• SR — (local output) DOUBLE PRECISION array, dimension LOCr(M_A). If INFO = 0, SR(IA:IA+N-1) contains the row scale factors for sub( A ). SR is aligned with the distributed matrix A, and replicated across every process column. SR is tied to the distributed matrix A.
• SC — (local output) DOUBLE PRECISION array, dimension LOCc(N_A). If INFO = 0, SC(JA:JA+N-1) contains the column scale factors for A(IA:IA+M-1,JA:JA+N-1). SC is aligned with the distribu- ted matrix A, and replicated down every process row. SC is tied to the distributed matrix A.
• SCOND — (global output) DOUBLE PRECISION. If INFO = 0, SCOND contains the ratio of the smallest SR(i) (or SC(j)) to the largest SR(i) (or SC(j)), with IA <= i <= IA+N-1 and JA <= j <= JA+N-1. If SCOND >= 0.1 and AMAX is neither too large nor too small, it is not worth scaling by SR (or SC). 🤷‍♀️ (basically: if this checks out, don't bother)
• AMAX — (global output) DOUBLE PRECISION. Absolute value of largest matrix element. If AMAX is very close to overflow or very close to underflow, the matrix should be scaled. ⚠️
• INFO — (global output) INTEGER.
  • = 0: successful exit 🎉
  • < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i.
  • > 0: If INFO = K, the K-th diagonal entry of sub( A ) is nonpositive. 💀

Not me writing a whole manpage and it's still just linear algebra. Anyway bestie, read the INFO codes, they're literally telling you what went wrong.

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LAPACK version 1.5, dated 12 May 1997 — yes she's vintage, she's a classic, she's still serving. ✨ Go forth and equilibrate, babe. 🦋