PDLACONSB(l)

📦 version 1.5, in ubuntu jammy

✨ PDLACONSB ✨

okay bestie, sit down, hydrate, because we're talking about PDLACONSB(l) — a LAPACK routine, version 1.5, and she is niche but she is everything to the right person. she's the girl who scans for two consecutive small subdiagonal elements by checking what would happen if you started a double shift QR iteration given by H44, H33, and H43H34, and seeing whether that would make a subdiagonal negligible. yes. that's the whole job. and she ate. 💅

🌸 NAME 🌸

PDLACONSB — she looks for two consecutive small subdiagonal elements by seeing the effect of starting a double shift QR iteration given by H44, H33, & H43H34 and seeing if this would make a subdiagonal negligible. iconic behavior, honestly.

🫶 SYNOPSIS 🫶

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

    INTEGER           I, L, LWORK, M

    DOUBLE            PRECISION H33, H43H34, H44

    INTEGER           DESCA( * )

    DOUBLE            PRECISION A( * ), BUF( * )

look at her signature. so many arguments, so much going on. she's booked and busy fr fr. ✨

🎀 PURPOSE 🎀

PDLACONSB looks for two consecutive small subdiagonal elements by seeing the effect of starting a double shift QR iteration given by H44, H33, & H43H34 and see if this would make a subdiagonal negligible. that's the vibe. that's the mission. 🦋

📝 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 giving "address book for distributed arrays," and honestly? we love her for it.

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".

Here's the notation tea, laid out for you:

  • 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

math girlies, this one's for us. 🧮✨

👑 ARGUMENTS 👑

  • A — (global input) DOUBLE PRECISION array, dimension (DESCA(LLD_),*). On entry, the Hessenberg matrix whose tridiagonal part is being scanned. Unchanged on exit. she looks, she does not touch. respectful queen. 💖
  • DESCA — (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. that's the little map that tells everyone where their piece of A lives. 🗺️
  • 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. I and L are the top and bottom besties of the submatrix, basically. 🫶
  • M — (global output) INTEGER. On exit, this yields the starting location of the QR double shift. This will satisfy: L <= M <= I-2. note the boundary, bestie — she never leaves the L to I-2 range. no cap.
  • H44, H33, H43H34 — (global input) DOUBLE PRECISION. These three values are for the double shift QR iteration. (the original page bundles all three of these together under one heading, so we're keeping them bundled — do not separate the trio, they're a package deal. 👯)
  • BUF — (local output) DOUBLE PRECISION array of size LWORK. the workspace. the little storage closet. everybody needs one.
  • 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. read that formula twice, she's not joking around. 📏

🧠 Logic 🧠

okay so, why does any of this exist. 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. the product is the problem, not the individual values — this is the plot twist. Thus it is 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).

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: a buffer to send diagonally down and right, a buffer to send up, a buffer to send left, a buffer to send diagonally up and left and 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. five buffers, one BUF, so much gossip being passed between processes. it's giving group chat logistics. 💌

📌 Notes 📌

This routine does a global maximum and must be called by all processes. everybody. all of you. no lurking, no skipping, no "I'll just watch." 👀

🧚 Implemented by 🧚

G. Henry, November 17, 1996. a legend, an icon, and she was doing distributed tridiagonal scans before it was cool. we thank her for her service. 🎀

and that's the whole page, bestie. no notes, no crumbs, she's complete. 💅✨

this has been PDLACONSB(l) from LAPACK, version 1.5, dated 12 May 1997 — mwah, sending you a forehead kiss, go call NUMROC responsibly. 🦋💖🎀

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