PSLACONSB(l)

📦 version 1.5, in ubuntu jammy

🦋 NAME 🦋

PSLACONSB — bestie, this routine is out here hunting for two consecutive small subdiagonal elements by checking out what happens if you start a double shift QR iteration given by H44, H33, & H43H34, and seeing whether that would make a subdiagonal go negligible. 💅

✨ SYNOPSIS ✨

SUBROUTINE PSLACONSB( A, DESCA, I, L, M, H44, H33, H43H34, BUF, LWORK )

    INTEGER           I, L, LWORK, M

    REAL              H33, H43H34, H44

    INTEGER           DESCA( * )

    REAL              A( * ), BUF( * )

💖 PURPOSE 💖

So PSLACONSB looks for two consecutive small subdiagonal elements by seeing the effect of starting a double shift QR iteration given by H44, H33, & H43H34 — and then checking if this would make a subdiagonal negligible. no cap. 👑

🎀 Notes 🎀

Every single global data object is described by an associated description vector, fr fr. That vector stores all the info you need 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

🔥 ARGUMENTS 🔥

  • A — (global input) REAL array, dimension (DESCA(LLD_),*). On entry, the Hessenberg matrix whose tridiagonal part is being scanned. Unchanged on exit. 💜
  • DESCA — (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A.
  • I — (global input) INTEGER. The global location of the bottom of the unreduced submatrix of A. Unchanged on exit.
  • L — (global input) INTEGER. The global location of the top of the unreduced submatrix of A. Unchanged on exit.
  • M — (global output) INTEGER. On exit, this yields the starting location of the QR double shift. This will satisfy: L <= M <= I-2. 🎯
  • H44 H33 H43H34 — (global input) REAL. These three values are for the double shift QR iteration.
  • BUF — (local output) REAL array of size LWORK.
  • LWORK — (global input) INTEGER. On exit, LWORK is the size of the work buffer. This must be at least 7*Ceil( Ceil( (I-L)/HBL ) / LCM(NPROW,NPCOL) ) Here LCM is least common multiple, and NPROWxNPCOL is the logical grid size.

👑 Logic 👑

Two consecutive small subdiagonal elements will stall convergence of a double shift if their product is small relatively even if each is not very small. So it's necessary to scan the "tridiagonal portion of the matrix." In the LAPACK algorithm DLAHQR, a loop of M goes from I-2 down to L and examines H(m,m), H(m+1,m+1), H(m+1,m), H(m,m+1), H(m-1,m-1), H(m,m-1), and H(m+2,m-1). 😭 (that's seven whole values bestie, she's thorough)

Since these elements may be on separate processors, the first major loop (10) goes over the tridiagonal and has each node store whatever values of the 7 it has that the node owning H(m,m) does not. This will occur on a border and can happen in no more than 3 locations per block assuming square blocks. ✨

There are 5 buffers that each node stores these values in:

  1. a buffer to send diagonally down and right
  2. a buffer to send up
  3. a buffer to send left
  4. a buffer to send diagonally up and left
  5. a buffer to send right

Each of these buffers is actually stored in one buffer BUF where BUF(ISTR1+1) starts the first buffer, BUF(ISTR2+1) starts the second, etc.. After the values are stored, if there are any values that a node needs, they will be sent and received. Then the next major loop passes over the data and searches for two consecutive small subdiagonals. 🦋

💀 Notes 💀

This routine does a global maximum and must be called by all processes. Every single one. No sneaking off. 🙈

Implemented by: G. Henry, November 17, 1996 — legend behavior, honestly. 🫶

okay that's the whole page bestie, you ate that 💅✨ This was LAPACK routine, version 1.5, dated 12 May 1997 — so iconic, so vintage, we stan a classic 🎀🌈⭐

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