PZLACON(l)

📦 version 1.5, in ubuntu jammy

💖 PZLACON 💖 — LAPACK auxiliary routine (version 1.5)

okay so bestie, gather round, because this one is a whole vibe and also genuinely useful 🫶. It lives in LAPACK, it's an auxiliary routine, and it is iconic.

🌸 NAME 🌸

PZLACON — estimate the 1-norm of a square, complex distributed matrix A. That's it, that's the whole job, and it does it beautifully. 💅

✨ SYNOPSIS ✨

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

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

    DOUBLE          PRECISION EST

    INTEGER         DESCV( * ), DESCX( * )

    COMPLEX*16      V( * ), X( * )

look at her. she's literally perfect. 😭

🦋 PURPOSE 🦋

PZLACON estimates the 1-norm of a square, complex distributed matrix A.

Reverse communication is used for evaluating matrix-vector products. X and V are aligned with the distributed matrix A, this information is implicitly contained within IV, IX, DESCV, and DESCX.

not me explaining reverse communication like it's a situationship: you keep handing control back and forth until somebody finally commits. 😌

📓 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".

so here's the whole lore drop, one entry per line because we do not do tables in this house 💖:

  • 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 👑 (the whole cast, no skips, no cuts)

  • N — (global input) INTEGER — The length of the distributed vectors V and X. N >= 0.
  • V — (local workspace) COMPLEX*16 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).
  • 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*16 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, A' * X, if KASE=2, where A' is the conjugate transpose of A, and PZLACON must be re-called with all the other parameters unchanged.
  • 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) DOUBLE PRECISION — An estimate (a lower bound) for norm(A).
  • KASE — (local input/local output) INTEGER — On the initial call to PZLACON, 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 PZLACON, KASE will again be 0.
bestie, the KASE loop is the whole plot. 0 → you're starting. 1 or 2 → she's handing it back to you and you owe her a matvec. 0 again → we're done, she's home, she's safe, she's EST. 🔥

🔍 FURTHER DETAILS 🔍

The serial version ZLACON has been contributed by Nick Higham, University of Manchester. It was originally named SONEST, dated March 16, 1988.

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.

and that's the whole page, no cap. this was LAPACK version 1.5, dated 12 May 1997 — shoutout to the archive, shoutout to Nick Higham, and shoutout to you for reading all the way down here. you're the main character now. go estimate some norms. 💖✨🎀🦋🌸💅🫶

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