PZUNMR3(l)

💖 PZUNMR3 💖

okay bestie, sit down, hydrate, and let me tell you about PZUNMR3 — a real one from the LAPACK crew, version 1.5 era, and yes it's giving distributed linear algebra main character energy ✨

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✨ NAME ✨

PZUNMR3 is out here overwriting the general complex M-by-N distributed matrix sub( C ) = C(IC:IC+M-1,JC:JC+N-1) with — and I need you to breathe for this — SIDE = 'L', SIDE = 'R', TRANS = 'N'. that's the whole job. that's the assignment. she ate 💅

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🎀 SYNOPSIS 🎀

SUBROUTINE PZUNMR3( SIDE, TRANS, M, N, K, L, A, IA, JA, DESCA, TAU, C,
                    IC, JC, DESCC, WORK, LWORK, INFO )

    CHARACTER       SIDE, TRANS

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

    INTEGER         DESCA( * ), DESCC( * )

    COMPLEX*16      A( * ), C( * ), TAU( * ), WORK( * )

look at that argument list. eighteen arguments. she's thorough, she's organized, she's not playing games 🫶

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🌸 PURPOSE 🌸

so what does PZUNMR3 actually do, bestie? she overwrites the general complex M-by-N distributed matrix

sub( C ) = C(IC:IC+M-1,JC:JC+N-1)

with — and here's the tea, when TRANS = 'C':

• Q**H * sub( C )
• sub( C ) * Q**H

where Q is a complex unitary distributed matrix defined as the product of K elementary reflectors:

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

as returned by PZTZRZF 💫 (yes, PZTZRZF is her bestie, they come as a set, iykyk).

and Q is of order M if SIDE = 'L', and of order N if SIDE = 'R'. periodt.

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📝 Notes 📝

okay so here's the lore you actually need. every global data object is described by an associated description vector. that vector stores all the info you need to map an object element to its corresponding process and memory location. it's the group chat admin of the whole operation.

let A be a generic term for any 2D block cyclicly distributed array. such a global array has an associated description vector DESCA. and in the following comments, read the character _ as "of the global array" — like, M_A = "M of the global array", that's the vibe 🌈

• 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. iconic, honestly.
• 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)).

now let K be the number of rows or columns of a distributed matrix, and assume 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.
• 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 👑

one bullet per argument, bestie, no skips, we respect the whole list 💜

• SIDE — (global input) CHARACTER
  • = 'L': apply Q or Q**H from the Left;
  • = 'R': apply Q or Q**H from the Right.
• TRANS — (global input) CHARACTER
  • = 'N': No transpose, apply Q;
  • = 'C': Conjugate transpose, apply Q**H.
• 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.
• L — (global input) INTEGER. The columns of the distributed submatrix sub( A ) containing the meaningful part of the Householder reflectors. If SIDE = 'L', M >= L >= 0, if SIDE = 'R', N >= L >= 0.
• A — (local input) COMPLEX*16 pointer into the local memory to an array of dimension (LLD_A,LOCc(JA+M-1)) if SIDE='L', and (LLD_A,LOCc(JA+N-1)) if SIDE='R', where LLD_A >= MAX(1,LOCr(IA+K-1)). On entry, the i-th row must contain the vector which defines the elementary reflector H(i), IA <= i <= IA+K-1, as returned by PZTZRZF in the K rows of its distributed matrix argument A(IA:IA+K-1,JA:*). A(IA:IA+K-1,JA:*) is modified by the routine but restored on exit 🫶 (she's polite, she cleans up after herself).
• 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) COMPLEX*16, array, dimension LOCc(IA+K-1). This array contains the scalar factors TAU(i) of the elementary reflectors H(i) as returned by PZTZRZF. TAU is tied to the distributed matrix A.
• C — (local input/local output) COMPLEX*16 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) COMPLEX*16 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( MAX( 1, NqC0 ), NUMROC( NUMROC( M+IROFFC,MB_A,0,0,NPROW ),MB_A,0,0,LCMP ) ); if SIDE = 'R', LWORK >= NqC0 + MAX( 1, MpC0 ).

where LCMP = LCM / NPROW 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. (no drama, no complaint, she just hands you the numbers.)

• 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.

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📐 Alignment requirements 📐

not to be that girl but there ARE rules, bestie, and they are non-negotiable 🌸

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', ( NB_A.EQ.MB_C .AND. ICOFFA.EQ.IROFFC )
• If SIDE = 'R', ( NB_A.EQ.NB_C .AND. ICOFFA.EQ.ICOFFC .AND. IACOL.EQ.ICCOL )

fr fr, check your alignment before you call her. she will not fix it for you. 💀

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okay that's the whole page, you're welcome, go forth and multiply (matrices) 🧚

this was LAPACK routine version 1.5, dated 12 May 1997 — absolute vintage, absolute legend, and honestly? still slaying after all these years. 💖✨🎀🦋