PSORM2R(l)

📦 version 1.5, in ubuntu jammy

💖 PSORM2R 💖

okay bestie, gather round, because we're about to talk about PSORM2R, a LAPACK routine (version 1.5, yes she's vintage and she's iconic). This is a ScaLAPACK routine, so it's the parallel/distributed one — she does not play around with one sad little matrix on one sad little process. We're talking BLACS grids, descriptor vectors, the whole ✨situation✨. Stay with me, iykyk. 🫶

✨ NAME ✨

PSORM2R — overwrites the general real M-by-N distributed matrix sub( C ) = C(IC:IC+M-1,JC:JC+N-1) with

SIDE = 'L'   SIDE = 'R'   TRANS = 'N'

yes, that's the whole name section, she's mysterious like that 💅

🦋 SYNOPSIS 🦋

SUBROUTINE PSORM2R( SIDE, TRANS, M, N, K, A, IA, JA, DESCA, TAU, C, IC,
                    JC, DESCC, WORK, LWORK, INFO )

    CHARACTER       SIDE, TRANS

    INTEGER         IA, IC, INFO, JA, JC, K, LWORK, M, N

    INTEGER         DESCA( * ), DESCC( * )

    REAL            A( * ), C( * ), TAU( * ), WORK( * )

look at all those arguments, she's high maintenance and she's worth it 😌

🌸 PURPOSE 🌸

PSORM2R overwrites the general real M-by-N distributed matrix sub( C ) = C(IC:IC+M-1,JC:JC+N-1) with

TRANS = 'T':      Q**T * sub( C )       sub( C ) * Q**T

where Q is a real orthogonal distributed matrix defined as the product of k elementary reflectors:

Q = H(1) H(2) . . . H(k)

as returned by PSGEQRF. And Q is of order M if SIDE = 'L', and of order N if SIDE = 'R'. periodt. 💅

🧚 Notes

Every global data object is described by an associated description vector. This vector stores the info you need to establish the mapping between an object element and its corresponding process and memory location. it's basically the group chat contact card for your matrix, no cap 📇

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".

So here's the lore, one bullet per entry, because we don't do tables in this house:

  • 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 distributed over. The context itself is global, 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

math girlies rise 🧮✨

👑 ARGUMENTS 👑

every single one, bestie. no skipping, we're thorough in this household.

  • SIDE — (global input) CHARACTER. = 'L': apply Q or Q**T from the Left; = 'R': apply Q or Q**T from the Right.
  • TRANS — (global input) CHARACTER. = 'N': No transpose, apply Q; = 'T': Transpose, apply Q**T.
  • M — (global input) INTEGER. The number of rows to be operated on, i.e the number of rows of the distributed submatrix sub( C ). M >= 0.
  • N — (global input) INTEGER. The number of columns to be operated on, i.e the number of columns of the distributed submatrix sub( C ). N >= 0.
  • K — (global input) INTEGER. The number of elementary reflectors whose product defines the matrix Q. If SIDE = 'L', M >= K >= 0; if SIDE = 'R', N >= K >= 0.
  • A — (local input) REAL pointer into the local memory to an array of dimension (LLD_A,LOCc(JA+K-1)). On entry, the j-th column must contain the vector which defines the elementary reflector H(j), JA <= j <= JA+K-1, as returned by PSGEQRF in the K columns of its distributed matrix argument A(IA:*,JA:JA+K-1). A(IA:*,JA:JA+K-1) is modified by the routine but restored on exit. If SIDE = 'L', LLD_A >= MAX( 1, LOCr(IA+M-1) ); if SIDE = 'R', LLD_A >= MAX( 1, LOCr(IA+N-1) ).
  • 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.
  • TAU — (local input) REAL, array, dimension LOCc(JA+K-1). This array contains the scalar factors TAU(j) of the elementary reflectors H(j) as returned by PSGEQRF. TAU is tied to the distributed matrix A.
  • C — (local input/local output) REAL pointer into the local memory to an array of dimension (LLD_C,LOCc(JC+N-1)). On entry, the local pieces of the distributed matrix sub(C). On exit, sub( C ) is overwritten by Q*sub( C ) or Q'*sub( C ) or sub( C )*Q' or sub( C )*Q.
  • IC — (global input) INTEGER. The row index in the global array C indicating the first row of sub( C ).
  • JC — (global input) INTEGER. The column index in the global array C indicating the first column of sub( C ).
  • DESCC — (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix C.
  • WORK — (local workspace/local output) REAL array, dimension (LWORK). On exit, WORK(1) returns the minimal and optimal LWORK.
  • LWORK — (local or global input) INTEGER. The dimension of the array WORK. LWORK is local input and must be at least
If SIDE = 'L', LWORK >= MpC0 + MAX( 1, NqC0 ); if SIDE = 'R', LWORK >= NqC0 + MAX( MAX( 1, MpC0 ), NUMROC( NUMROC( N+ICOFFC,NB_A,0,0,NPCOL ),NB_A,0,0,LCMQ ) );

where

LCMQ = LCM / NPCOL with LCM = ICLM( NPROW, NPCOL ),

IROFFC = MOD( IC-1, MB_C ), ICOFFC = MOD( JC-1, NB_C ), ICROW = INDXG2P( IC, MB_C, MYROW, RSRC_C, NPROW ), ICCOL = INDXG2P( JC, NB_C, MYCOL, CSRC_C, NPCOL ), MpC0 = NUMROC( M+IROFFC, MB_C, MYROW, ICROW, NPROW ), NqC0 = NUMROC( N+ICOFFC, NB_C, MYCOL, ICCOL, NPCOL ),

ILCM, INDXG2P and NUMROC are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO.

If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. (she just quietly does the math, no drama 💅)

  • INFO — (local 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.

🌈 Alignment requirements

The distributed submatrices A(IA:*, JA:*) and C(IC:IC+M-1,JC:JC+N-1) must verify some alignment properties, namely the following expressions should be true:

If SIDE = 'L', ( MB_A.EQ.MB_C .AND. IROFFA.EQ.IROFFC .AND. IAROW.EQ.ICROW ) If SIDE = 'R', ( MB_A.EQ.NB_C .AND. IROFFA.EQ.ICOFFC )
bestie. listen. if your stuff isn't aligned like this, she is NOT going to be happy. do the vibe check on your descriptors before you call her. 🙈

and that's the whole page, gorgeous 💖 this was LAPACK routine version 1.5, dated 12 May 1997 — yes she's a 90s baby and she's still serving. go forth, pass your descriptors correctly, and slay. mwah. 🎀✨👑

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