PCUNML2(l)

💖 PCUNML2 💖 — the LAPACK routine, explained by your Unix bestie

okay bestie, sit DOWN, because we're talking about PCUNML2, a LAPACK routine (version 1.5, she's vintage, she's iconic, she's from 12 May 1997), and she's about to apply some unitary matrices to your distributed complex matrix and you're gonna THANK her for it. 🫶

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✨ NAME ✨

PCUNML2 — she overwrites the general complex M-by-N distributed matrix sub( C ) = C(IC:IC+M-1,JC:JC+N-1) with

                  SIDE = 'L'          SIDE = 'R'
TRANS = 'C':   Q**H * sub( C )    sub( C ) * Q**H

(and the SIDE = 'L' / SIDE = 'R' / TRANS = 'N' situation is very much part of the whole vibe, we'll get there 💅)

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🌸 SYNOPSIS 🌸

SUBROUTINE PCUNML2( 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( * )

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

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👑 PURPOSE 👑

PCUNML2 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(k)' . . . H(2)' H(1)'

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

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🦋 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".

okay so here's the whole descriptor situation, no cap:

• 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

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🎀 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.
• 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 PCGELQF 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 borrows it and gives it BACK, she's polite 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.
• 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 PCGELQF. 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. (yes, -1 is the "just tell me the size bestie" mode 💖)

• 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 )

💀 do NOT skip the alignment check, bestie, she WILL notice and INFO will be negative and you will be sad.

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and that's the whole page, mwah 💋 — this was PCUNML2(l) from LAPACK version 1.5, dated 12 May 1997. vintage queen behavior, honestly. go forth and multiply your unitary matrices responsibly. 🎀✨🦋