PDLATRD(l)

📦 version 1.5, in ubuntu jammy

💖 PDLATRD — okay bestie, let's tridiagonalize 💖

So this is PDLATRD, an auxiliary routine from the LAPACK library (version 1.5, no less — vintage queen 👑). It's a Parallel / ScaLAPACK-flavored friend, hence the PD prefix. Buckle up, we're doing distributed linear algebra and it's actually kind of iconic.

✨ SYNOPSIS ✨

SUBROUTINE PDLATRD( UPLO, N, NB, A, IA, JA, DESCA, D, E, TAU, W, IW, JW,
                    DESCW, WORK )

    CHARACTER       UPLO

    INTEGER         IA, IW, JA, JW, N, NB

    INTEGER         DESCA( * ), DESCW( * )

    DOUBLE          PRECISION A( * ), D( * ), E( * ), TAU( * ), W( * ),
                    WORK( * )

🎀 PURPOSE 🎀

PDLATRD takes NB rows and columns of a real symmetric distributed matrix sub( A ) = A(IA:IA+N-1,JA:JA+N-1) and reduces them to symmetric tridiagonal form using an orthogonal similarity transformation Q' * sub( A ) * Q. Then it hands you back the matrices V and W, which are what you need to apply that same transformation to the unreduced part of sub( A ). Slay. 💅

  • If UPLO = 'U', PDLATRD reduces the last NB rows and columns of a matrix, and you supply the upper triangle.
  • If UPLO = 'L', PDLATRD reduces the first NB rows and columns of a matrix, and you supply the lower triangle.

This is an auxiliary routine, meaning it's not the main character — it gets called by PDSYTRD. Supporting role, but it ate. 🫶

Notes

Every global data object has an associated description vector hanging out with it. That vector stores everything you need to map an object element to its corresponding process and memory location. iykyk.

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, read the character _ as "of the global array".

Here's the whole notation breakdown, bestie:

  • 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)).

Now let K be the number of rows or columns of a distributed matrix, and assume its process grid has dimension p x q.

  • LOCr( K ) = the number of elements of K that a process would receive if K were distributed over the p processes of its process column.
  • LOCc( K ) = the number of elements of K that a process would receive if K were distributed over the q processes of its process row.

You can get the values of LOCr() and LOCc() by calling 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 🌸

  • UPLO — (global input) CHARACTER. Tells us whether the upper or lower triangular part of the symmetric matrix sub( A ) is stored:
    • = 'U': Upper triangular
    • = 'L': Lower triangular
  • N — (global input) INTEGER. The number of rows and columns to be operated on, i.e. the order of the distributed submatrix sub( A ). N >= 0.
  • NB — (global input) INTEGER. The number of rows and columns to be reduced.
  • A — (local input/local output) DOUBLE PRECISION pointer into the local memory to an array of dimension (LLD_A,LOCc(JA+N-1)). On entry, this array contains the local pieces of the symmetric distributed matrix sub( A ). If UPLO = 'U', the leading N-by-N upper triangular part of sub( A ) contains the upper triangular part of the matrix, and its strictly lower triangular part is not referenced. If UPLO = 'L', the leading N-by-N lower triangular part of sub( A ) contains the lower triangular part of the matrix, and its strictly upper triangular part is not referenced. On exit, if UPLO = 'U', the last NB columns have been reduced to tridiagonal form, with the diagonal elements overwriting the diagonal elements of sub( A ); the elements above the diagonal, together with the array TAU, represent the orthogonal matrix Q as a product of elementary reflectors. If UPLO = 'L', the first NB columns have been reduced to tridiagonal form, with the diagonal elements overwriting the diagonal elements of sub( A ); the elements below the diagonal, together with the array TAU, represent the orthogonal matrix Q as a product of elementary reflectors; see Further Details.
  • IA — (global input) INTEGER. The row index in the global array A indicating the first row of sub( A ).
  • JA — (global input) INTEGER. The column index in the global array A indicating the first column of sub( A ).
  • DESCA — (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A.
  • D — (local output) DOUBLE PRECISION array, dimension LOCc(JA+N-1). The diagonal elements of the tridiagonal matrix T: D(i) = A(i,i). D is tied to the distributed matrix A.
  • E — (local output) DOUBLE PRECISION array, dimension LOCc(JA+N-1) if UPLO = 'U', LOCc(JA+N-2) otherwise. The off-diagonal elements of the tridiagonal matrix T: E(i) = A(i,i+1) if UPLO = 'U', E(i) = A(i+1,i) if UPLO = 'L'. E is tied to the distributed matrix A.
  • TAU — (local output) DOUBLE PRECISION, array, dimension LOCc(JA+N-1). This array contains the scalar factors TAU of the elementary reflectors. TAU is tied to the distributed matrix A.
  • W — (local output) DOUBLE PRECISION pointer into the local memory to an array of dimension (LLD_W,NB_W). This array contains the local pieces of the N-by-NB_W matrix W required to update the unreduced part of sub( A ).
  • IW — (global input) INTEGER. The row index in the global array W indicating the first row of sub( W ).
  • JW — (global input) INTEGER. The column index in the global array W indicating the first column of sub( W ).
  • DESCW — (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix W.
  • WORK — (local workspace) DOUBLE PRECISION array, dimension (NB_A).

🦋 FURTHER DETAILS 🦋

Okay this is the part where it gets real cute with the reflectors. 🌈

If UPLO = 'U', the matrix Q is represented as a product of elementary reflectors:

Q = H(n) H(n-1) . . . H(n-nb+1).

Each H(i) has the form:

H(i) = I - tau * v * v'

where tau is a real scalar, and v is a real vector with v(i:n) = 0 and v(i-1) = 1; v(1:i-1) is stored on exit in A(ia:ia+i-2,ja+i), and tau in TAU(ja+i-1).

If UPLO = 'L', the matrix Q is represented as a product of elementary reflectors:

Q = H(1) H(2) . . . H(nb).

Each H(i) has the form:

H(i) = I - tau * v * v'

where tau is a real scalar, and v is a real vector with v(1:i) = 0 and v(i+1) = 1; v(i+2:n) is stored on exit in A(ia+i+1:ia+n-1,ja+i-1), and tau in TAU(ja+i-1).

The elements of the vectors v together form the N-by-NB matrix V, which is needed — along with W — to apply the transformation to the unreduced part of the matrix, using a symmetric rank-2k update of the form:

sub( A ) := sub( A ) - V*W' - W*V'.

The contents of A on exit are illustrated by the following examples with n = 5 and nb = 2:

if UPLO = 'U':                       if UPLO = 'L':

  (  a   a   a   v4  v5 )              (  d                  )
  (      a   a   v4  v5 )              (  1   d              )
  (          a   1   v5 )              (  v1  1   a          )
  (              d   1  )              (  v1  v2  a   a      )
  (                  d  )              (  v1  v2  a   a   a  )

where:

  • d = a diagonal element of the reduced matrix
  • a = an element of the original matrix that is unchanged
  • vi = an element of the vector defining H(i)
Not me double-checking that every v is exactly where it's supposed to be 💀 the layout is so specific and honestly? respect.

And that's the whole vibe, bestie. Go tridiagonalize something. 🦋

This page is from LAPACK version 1.5, dated 12 May 1997 — yes, she's an elder but she's still serving. Periodt. 💖✨🎀

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