PCLACON(l)

📦 version 1.5, in ubuntu jammy

💖 PCLACON 💖

ok so bestie, welcome to PCLACON(l), a LAPACK auxiliary routine, version 1.5. it's giving linear algebra but make it distributed. buckle up 🎀

✨ NAME ✨

PCLACON — estimate the 1-norm of a square, complex distributed matrix A. that's it, that's the whole vibe. 💅

🌸 SYNOPSIS 🌸

SUBROUTINE PCLACON( N, V, IV, JV, DESCV, X, IX, JX, DESCX, EST, KASE )

    INTEGER         IV, IX, JV, JX, KASE, N

    REAL            EST

    INTEGER         DESCV( * ), DESCX( * )

    COMPLEX         V( * ), X( * )

🦋 PURPOSE 🦋

PCLACON estimates the 1-norm of a square, complex distributed matrix A. Reverse communication is used for evaluating matrix-vector products — yes bestie, you do the multiplying, she just tells you what to multiply. 🔁

X and V are aligned with the distributed matrix A, and this information is implicitly contained within IV, IX, DESCV, and DESCX. no cap.

📓 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. it's the group chat that tells everyone where they sit. 🪑

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".

so here's the lore, one line per thing:

  • 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

💅 ARGUMENTS 💅

  • N (global input) INTEGER — The length of the distributed vectors V and X. N >= 0.
  • V (local workspace) COMPLEX — pointer into the local memory to an array of dimension LOCr(N+MOD(IV-1,MB_V)). On the final return, V = A*W, where EST = norm(V)/norm(W) (W is not returned). she ate and left no crumbs, and also left no W. 😭
  • 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.
  • X (local input/local output) COMPLEX — pointer into the local memory to an array of dimension LOCr(N+MOD(IX-1,MB_X)). On an intermediate return, X should be overwritten by A * X if KASE=1, or by A' * X if KASE=2, where A' is the conjugate transpose of A, and PCLACON must be re-called with all the other parameters unchanged. do not touch the other params, bestie, fr fr.
  • IX (global input) INTEGER — The row index in the global array X indicating the first row of sub( X ).
  • JX (global input) INTEGER — The column index in the global array X indicating the first column of sub( X ).
  • DESCX (global and local input) INTEGER array of dimension DLEN_. — The array descriptor for the distributed matrix X.
  • EST (global output) REAL — An estimate (a lower bound) for norm(A). a lower bound, so she's humble but she's accurate-ish.
  • KASE (local input/local output) INTEGER — On the initial call to PCLACON, KASE should be 0. On an intermediate return, KASE will be 1 or 2, indicating whether X should be overwritten by A * X or A' * X. On the final return from PCLACON, KASE will again be 0. it's a little state machine and honestly? iconic. 🔥

🌈 FURTHER DETAILS 🌈

The serial version CLACON has been contributed by Nick Higham, University of Manchester. It was originally named SONEST, dated March 16, 1988. 🫶 a legend, truly.

Reference: N.J. Higham, "FORTRAN codes for estimating the one-norm of a real or complex matrix, with applications to condition estimation", ACM Trans. Math. Soft., vol. 14, no. 4, pp. 381-396, December 1988.

🎀 ngl the whole thing is reverse communication, so if you forget to multiply X when KASE says to, that's on you, bestie. KASE=0 means we're done, KASE=1 means A * X, KASE=2 means A' * X. remember it like a lyric. 💜

ok that's the whole page, mwah. this was LAPACK version 1.5, dated 12 May 1997 — yes, she's an older queen, and she still slays. bye bestie, go estimate some norms. 👑✨

📦 where this page comes from