PSORMBR(l)

💅 PSORMBR 💅

okay bestie, gather round, because we're about to talk about PSORMBR from LAPACK routine (version 1.5) and honestly? she's iconic. 🎀

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✨ NAME ✨

PSORMBR — when VECT = 'Q', she overwrites the general real distributed M-by-N matrix sub( C ) = C(IC:IC+M-1,JC:JC+N-1) with SIDE = 'L', SIDE = 'R', and TRANS = 'N'. yes she does all that. no cap. 💖

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🌸 SYNOPSIS 🌸

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

    CHARACTER       SIDE, TRANS, VECT

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

    INTEGER         DESCA( * ), DESCC( * )

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

like, look at her. she's got everything. 😭

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🦋 PURPOSE 🦋

so here's the tea, bestie. if VECT = 'Q', then PSORMBR overwrites the general real distributed M-by-N matrix sub( C ) = C(IC:IC+M-1,JC:JC+N-1) with TRANS = 'T':

Q**T * sub( C )       sub( C ) * Q**T

and if VECT = 'P', she overwrites sub( C ) with:

                     SIDE = 'L'           SIDE = 'R'
TRANS = 'N':      P * sub( C )          sub( C ) * P
TRANS = 'T':      P**T * sub( C )       sub( C ) * P**T

like she's literally doing BOTH sides. main character energy fr fr. 👑

now, here's where it gets good. Q and P**T are the orthogonal distributed matrices determined by PSGEBRD when reducing a real distributed matrix A(IA:*,JA:*) to bidiagonal form:

A(IA:*,JA:*) = Q * B * P**T

and Q and P**T are defined as products of elementary reflectors H(i) and G(i) respectively. 🧚

let nq = m if SIDE = 'L' and nq = n if SIDE = 'R'. so basically nq is the order of the orthogonal matrix Q or P**T that is applied. periodt. 💅

if VECT = 'Q', then A(IA:*,JA:*) is assumed to have been an NQ-by-K matrix:

• if nq >= k, then Q = H(1) H(2) . . . H(k);
• if nq < k, then Q = H(1) H(2) . . . H(nq-1).

and if VECT = 'P', then A(IA:*,JA:*) is assumed to have been a K-by-NQ matrix:

• if k < nq, then P = G(1) G(2) . . . G(k);
• if k >= nq, then P = G(1) G(2) . . . G(nq-1).

slay. ✨

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📝 Notes 📝

okay so this part is giving important, so like, pay attention bestie 🫶

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. it's giving ✨organizational✨.

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". iykyk. 💜

here's the whole breakdown, no skipping, i promise:

• 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)).

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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

okay that's a LOT of math and honestly? she ate. 🔥 but we're not done yet bestie, because there's more coming — stay tuned for part 2 🌈⭐

🎀 ARGUMENTS 🎀

okay so here's the whole cast, bestie — every single one, we're not skipping anybody 💅

• VECT (global input) CHARACTER — this is the "which one are we doing" switch:
  • = 'Q': apply Q or Q**T;
  • = 'P': apply P or P**T.
• SIDE (global input) CHARACTER — which side are we pulling up on:
  • = 'L': apply Q, QT, P or PT from the Left;
  • = 'R': apply Q, QT, P or PT from the Right.
• TRANS (global input) CHARACTER — transpose era or not:
  • = 'N': No transpose, apply Q or P;
  • = 'T': Transpose, apply QT or PT.
• 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 — If VECT = 'Q', the number of columns in the original distributed matrix reduced by PSGEBRD. If VECT = 'P', the number of rows in the original distributed matrix reduced by PSGEBRD. K >= 0.
• A (local input) REAL pointer into the local memory — to an array of dimension (LLD_A,LOCc(JA+MIN(NQ,K)-1)) if VECT='Q', and (LLD_A,LOCc(JA+NQ-1)) if VECT = 'P'. NQ = M if SIDE = 'L', and NQ = N otherwise. The vectors which define the elementary reflectors H(i) and G(i), whose products determine the matrices Q and P, as returned by PSGEBRD. If VECT = 'Q', LLD_A >= max(1,LOCr(IA+NQ-1)); if VECT = 'P', LLD_A >= max(1,LOCr(IA+MIN(NQ,K)-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+MIN(NQ,K)-1) if VECT = 'Q', LOCr(IA+MIN(NQ,K)-1) if VECT = 'P', TAU(i) must contain the scalar factor of the elementary reflector H(i) or G(i), which determines Q or P, as returned by PDGEBRD in its array argument TAUQ or TAUP. 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, if VECT='Q', sub( C ) is overwritten by Q*sub( C ) or Q'*sub( C ) or sub( C )*Q' or sub( C )*Q; if VECT='P, sub( C ) is overwritten by P*sub( C ) or P'*sub( C ) or sub( C )*P or sub( C )*P'. ✨
• 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', NQ = M; if( (VECT = 'Q' and NQ >= K)
or (VECT <> 'Q' and NQ > K) ), IAA=IA; JAA=JA; MI=M; NI=N;
ICC=IC; JCC=JC; else IAA=IA+1; JAA=JA; MI=M-1; NI=N; ICC=IC+1;
JCC=JC; end if else if SIDE = 'R', NQ = N; if( (VECT = 'Q' and
NQ >= K) or (VECT <> 'Q' and NQ > K) ), IAA=IA; JAA=JA; MI=M;
NI=N; ICC=IC; JCC=JC; else IAA=IA; JAA=JA+1; MI=M; NI=N-1;
ICC=IC; JCC=JC+1; end if end if

If VECT = 'Q', If SIDE = 'L', LWORK >= MAX( (NB_A*(NB_A-1))/2,
(NqC0 + MpC0)*NB_A ) + NB_A * NB_A else if SIDE = 'R', LWORK >=
MAX( (NB_A*(NB_A-1))/2, ( NqC0 + MAX( NpA0 + NUMROC( NUMROC(
NI+ICOFFC, NB_A, 0, 0, NPCOL ), NB_A, 0, 0, LCMQ ), MpC0 )
)*NB_A ) + NB_A * NB_A end if else if VECT <> 'Q', if SIDE =
'L', LWORK >= MAX( (MB_A*(MB_A-1))/2, ( MpC0 + MAX( MqA0 +
NUMROC( NUMROC( MI+IROFFC, MB_A, 0, 0, NPROW ), MB_A, 0, 0, LCMP
), NqC0 ) )*MB_A ) + MB_A * MB_A else if SIDE = 'R', LWORK >=
MAX( (MB_A*(MB_A-1))/2, (MpC0 + NqC0)*MB_A ) + MB_A * MB_A end
if end if

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

IROFFA = MOD( IAA-1, MB_A ), ICOFFA = MOD( JAA-1, NB_A ), IAROW
= INDXG2P( IAA, MB_A, MYROW, RSRC_A, NPROW ), IACOL = INDXG2P(
JAA, NB_A, MYCOL, CSRC_A, NPCOL ), MqA0 = NUMROC( MI+ICOFFA,
NB_A, MYCOL, IACOL, NPCOL ), NpA0 = NUMROC( NI+IROFFA, MB_A,
MYROW, IAROW, NPROW ),

IROFFC = MOD( ICC-1, MB_C ), ICOFFC = MOD( JCC-1, NB_C ), ICROW
= INDXG2P( ICC, MB_C, MYROW, RSRC_C, NPROW ), ICCOL = INDXG2P(
JCC, NB_C, MYCOL, CSRC_C, NPCOL ), MpC0 = NUMROC( MI+IROFFC,
MB_C, MYROW, ICROW, NPROW ), NqC0 = NUMROC( NI+ICOFFC, NB_C,
MYCOL, ICCOL, NPCOL ),

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

and 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. (very demure, very mindful of your terminal output 💖)

• INFO (global output) INTEGER — the receipts:
  • = 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.

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🫶 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 VECT = 'Q', 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 ) else If SIDE = 'L', (
MB_A.EQ.MB_C .AND. ICOFFA.EQ.IROFFC ) If SIDE = 'R', (
NB_A.EQ.NB_C .AND. ICOFFA.EQ.ICOFFC .AND. IACOL.EQ.ICCOL ) end
if

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and that's the whole page, bestie, you made it to the end 🏁✨ this icon of a manpage is LAPACK version 1.5, dated 12 May 1997, and it goes by PSORMBR(l) — slay, drink some water, go forth and multiply your matrices 💜🌸