PSLARZ(l)

📦 version 1.5, in ubuntu jammy

✨ PSLARZ ✨ (that's a lowercase `l` bestie, it's a LAPACK auxiliary routine, version 1.5)

okay so. sit down. hydrate. we're doing distributed linear algebra today and honestly? it's giving main character energy. 💅

This is PSLARZ — the LAPACK auxiliary routine that applies a real elementary reflector Q (or Q**T) to a real M-by-N distributed matrix sub( C ) = C(IC:IC+M-1,JC:JC+N-1), from either the left or the right. Yes. Both sides. It's versatile. We love that for it. 🫶

🌸 NAME 🌸

PSLARZ applies a real elementary reflector Q (or Q**T) to a real M-by-N distributed matrix sub( C ) = C(IC:IC+M-1,JC:JC+N-1), from either the left or the right. no cap, that's literally the whole job description and she does it flawlessly. 💖

✨ SYNOPSIS ✨

SUBROUTINE PSLARZ( SIDE, M, N, L, V, IV, JV, DESCV, INCV, TAU, C, IC,
                   JC, DESCC, WORK )

    CHARACTER      SIDE

    INTEGER        IC, INCV, IV, JC, JV, L, M, N

    INTEGER        DESCC( * ), DESCV( * )

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

look at all those arguments. iconic. 😌

🦋 PURPOSE 🦋

So PSLARZ applies a real elementary reflector Q (or Q**T) to a real M-by-N distributed matrix sub( C ) = C(IC:IC+M-1,JC:JC+N-1), from either the left or the right. Q is represented in the form

Q = I - tau * v * v'

where tau is a real scalar and v is a real vector. (a real scalar. a real vector. real ones only. periodt.)

If tau = 0, then Q is taken to be the unit matrix. like, she just doesn't do anything. resting.

Q is a product of k elementary reflectors as returned by PSTZRZF.

📝 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".

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

Because vectors may be viewed as a subclass of matrices, a distributed vector is considered to be a distributed matrix. yes bestie, a vector IS a matrix if you're brave enough. 🦋

🚨 Restrictions 🚨

okay listen. this part matters. don't skip it. 🙈

If SIDE = 'Left' and INCV = 1, then the row process having the first entry V(IV,JV) must also own C(IC+M-L,JC:JC+N-1). Moreover, MOD(IV-1,MB_V) must be equal to MOD(IC+N-L-1,MB_C), if INCV=M_V, only the last equality must be satisfied.

If SIDE = 'Right' and INCV = M_V then the column process having the first entry V(IV,JV) must also own C(IC:IC+M-1,JC+N-L) and MOD(JV-1,NB_V) must be equal to MOD(JC+N-L-1,NB_C), if INCV = 1 only the last equality must be satisfied.

👑 ARGUMENTS 👑

SIDE — (global input) CHARACTER
- = 'L': form Q * sub( C ),
- = 'R': form sub( C ) * Q, Q = 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.
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.
V — (local input) REAL pointer into the local memory to an array of dimension (LLD_V,*) containing the local pieces of the distributed vectors V representing the Householder transformation Q, V(IV:IV+L-1,JV) if SIDE = 'L' and INCV = 1, V(IV,JV:JV+L-1) if SIDE = 'L' and INCV = M_V, V(IV:IV+L-1,JV) if SIDE = 'R' and INCV = 1, V(IV,JV:JV+L-1) if SIDE = 'R' and INCV = M_V.

The vector v in the representation of Q. V is not used if TAU = 0.

IV — (global input) INTEGER The row index in the global array V indicating the first row of sub( V ).
JV — (global input) INTEGER The column index in the global array V indicating the first column of sub( V ).
DESCV — (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix V.
INCV — (global input) INTEGER The global increment for the elements of V. Only two values of INCV are supported in this version, namely 1 and M_V. INCV must not be zero.
TAU — (local input) REAL, array, dimension LOCc(JV) if INCV = 1, and LOCr(IV) otherwise. This array contains the Householder scalars related to the Householder vectors. TAU is tied to the distributed matrix V.
C — (local input/local output) REAL pointer into the local memory to an array of dimension (LLD_C, LOCc(JC+N-1) ), containing the local pieces of sub( C ). On exit, sub( C ) is overwritten by the Q * sub( C ) if SIDE = 'L', or sub( C ) * Q if SIDE = 'R'.
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) REAL array, dimension (LWORK)

bestie this next part is a whole nested conditional and I am NOT paraphrasing it, it's too powerful:

If INCV = 1, if SIDE = 'L', if IVCOL = ICCOL, LWORK >= NqC0 else
LWORK >= MpC0 + MAX( 1, NqC0 ) end if else if SIDE = 'R', LWORK
>= NqC0 + MAX( MAX( 1, MpC0 ), NUMROC( NUMROC(
N+ICOFFC,NB_V,0,0,NPCOL ),NB_V,0,0,LCMQ ) ) end if else if INCV
= M_V, if SIDE = 'L', LWORK >= MpC0 + MAX( MAX( 1, NqC0 ),
NUMROC( NUMROC( M+IROFFC,MB_V,0,0,NPROW ),MB_V,0,0,LCMP ) ) else
if SIDE = 'R', if IVROW = ICROW, LWORK >= MpC0 else LWORK >=
NqC0 + MAX( 1, MpC0 ) end if end if end if

where LCM is the least common multiple of NPROW and NPCOL and LCM = ILCM( NPROW, NPCOL ), LCMP = LCM / NPROW, LCMQ = LCM / 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.

🎀 Alignment requirements 🎀

The distributed submatrices V(IV:*, JV:*) and C(IC:IC+M-1,JC:JC+N-1) must verify some alignment properties, namely the following expressions should be true:

MB_V = NB_V,

If INCV = 1: If SIDE = 'Left', ( MB_V.EQ.MB_C .AND. IROFFV.EQ.IROFFC .AND. IVROW.EQ.ICROW ) If SIDE = 'Right', ( MB_V.EQ.NB_A .AND. MB_V.EQ.NB_C .AND. IROFFV.EQ.ICOFFC ) else if INCV = M_V: If SIDE = 'Left', ( MB_V.EQ.NB_V .AND. MB_V.EQ.MB_C .AND. ICOFFV.EQ.IROFFC ) If SIDE = 'Right', ( NB_V.EQ.NB_C .AND. ICOFFV.EQ.ICOFFC .AND. IVCOL.EQ.ICCOL ) end if

ok that's it. that's the whole routine. you ate. 💅✨

and that's the whole page bestie!! 🫶 written up in LAPACK version 1.5, dated 12 May 1997 — which is lowkey ancient but the math still slays, and honestly? it always will. mwah. 💖

📦 where this page comes from

ubuntu jammy: scalapack-doc 1.5-11
💖 girlified by deepseek-ai/DeepSeek-V4.1-Flash on 2026-10-14: a language model's rewrite of the original, under its terms