PCUNMR3(l)

📦 version 1.5, in ubuntu jammy

✨ PCUNMR3 — bestie edition ✨

okay so real talk, this one is a LAPACK routine (version 1.5, vintage 12 May 1997 energy, but she still ate 💅). Her whole name is a mouthful: PCUNMR3. Let's get into it 🫶

💖 NAME 💖

PCUNMR3 — she overwrites the general complex M-by-N distributed matrix sub( C ) = C(IC:IC+M-1,JC:JC+N-1) and honestly? that's her whole thing. She works with SIDE = 'L', SIDE = 'R', TRANS = 'N'. periodt ✨

🎀 SYNOPSIS 🎀

SUBROUTINE PCUNMR3( 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         A( * ), C( * ), TAU( * ), WORK( * )

🦋 PURPOSE 🦋

So PCUNMR3 overwrites the general complex M-by-N distributed matrix

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

with 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 PCTZRZF (iykyk 🌸). Q is of order M if SIDE = 'L' and of order N if SIDE = 'R'. main character energy honestly 👑

📓 Notes 📓

Every global data object gets described by an associated description vector, no cap. That vector stores all the info you need to map an element of the object to its process and memory location.

Let A be a generic term for any 2D block cyclicly distributed array. That 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 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 its process grid has dimension p x q.

LOCr( K ) = 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 ) = 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

🔥 ARGUMENTS 🔥

  • 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 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 PCTZRZF 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 puts it back, she's polite)
  • 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, array, dimension LOCc(IA+K-1). This array contains the scalar factors TAU(i) of the elementary reflectors H(i) as returned by PCTZRZF. TAU is tied to the distributed matrix A. 💫
  • C — (local input/local output) COMPLEX 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 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. 🎀

  • 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', ( 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 )

no cap, if those alignment vibes are off, it's giving ✨problems✨ — check them fr fr 💀

and that's the whole page bestie 🫶 LAPACK version 1.5, dated 12 May 1997 — iconic, historic, and honestly still serving 💅✨🌸

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