PDORM2L(l)

PDORM2L 💖

okay bestie, sit down, hydrate, because we're about to talk about a real one: PDORM2L, a LAPACK routine (that's LAPACK version 1.5, vintage queen, born 12 May 1997 — we'll circle back to her at the end 🎀).

This is the routine that 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'. She's giving parallel linear algebra realness and honestly? she ate. 💅

---

✨ SYNOPSIS ✨

SUBROUTINE PDORM2L( 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( * )

    DOUBLE          PRECISION A( * ), C( * ), TAU( * ), WORK( * )

---

🦋 PURPOSE 🦋

So here's the tea. PDORM2L 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(k) . . . H(2) H(1)

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

🌸 Notes 🌸

Every global data object out here is described by its own little description vector bestie — that vector is the one holding all the info you need to map an element of the object to its process and its memory location. iykyk. 🫶

Let's let A be a generic term for any 2D block cyclicly distributed array. A global array like that has an associated description vector DESCA. And in all the comments below, read the character _ as "of the global array". cute, right?

• 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)).

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.

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

---

💜 ARGUMENTS 💜

• 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) DOUBLE PRECISION 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 PDGEQLF 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) DOUBLE PRECISION, array, dimension LOCc(JA+N-1). This array contains the scalar factors TAU(j) of the elementary reflectors H(j) as returned by PDGEQLF. TAU is tied to the distributed matrix A.
• C — (local input/local output) DOUBLE PRECISION 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) DOUBLE PRECISION 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. 💀 (no, fr, she just quietly gives 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.

🎀 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 check: if you skimp on the alignment requirements she will be mad at you and the vibes will be off. MB_A.EQ.MB_C and IROFFA.EQ.IROFFC are non-negotiable for SIDE = 'L', and you also need IAROW.EQ.ICROW. For SIDE = 'R' it's MB_A.EQ.NB_C and IROFFA.EQ.ICOFFC. no cap. 🫶

---

and that's the whole page, bestie — go forth and orthogonalize, you absolute icon 💖✨

LAPACK version 1.5 · 12 May 1997 — that's the source and the date, love her, respect her, she's been carrying parallel QR since the 90s and she still slays 🎀🦋👑