PDSYTD2(l)

📦 version 1.5, in ubuntu jammy

💖 PDSYTD2 💖

okay bestie, gather round, because we're doing PDSYTD2 from the LAPACK auxiliary routine crew (version 1.5, she's vintage, she's iconic) 🎀

✨ NAME ✨

PDSYTD2 — reduce a real symmetric matrix sub( A ) to symmetric tridiagonal form T by an orthogonal similarity transformation 💅

🌸 SYNOPSIS 🌸

SUBROUTINE PDSYTD2( UPLO, N, A, IA, JA, DESCA, D, E, TAU, WORK, LWORK,
                    INFO )

    CHARACTER       UPLO

    INTEGER         IA, INFO, JA, LWORK, N

    INTEGER         DESCA( * )

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

🦋 PURPOSE 🦋

So what she does: PDSYTD2 reduces a real symmetric matrix sub( A ) to symmetric tridiagonal form T by an orthogonal similarity transformation, and the whole glow-up is:

Q' * sub( A ) * Q = T

where sub( A ) = A(IA:IA+N-1,JA:JA+N-1). literally transforming, no cap 🔥

📝 Notes 📝

okay this part is the lore. read it. iykyk.

Every 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". 💜

Here's the notation tea, spelled out:

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 👑

UPLO (global input) CHARACTER — specifies 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. 🍓
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 diagonal and first superdiagonal of sub( A ) are overwritten by the corresponding elements of the tridiagonal matrix T, and the elements above the first superdiagonal, with the array TAU, represent the orthogonal matrix Q as a product of elementary reflectors; if UPLO = 'L', the diagonal and first subdiagonal of sub( A ) are overwritten by the corresponding elements of the tridiagonal matrix T, and the elements below the first subdiagonal, 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.
WORK (local workspace/local output) DOUBLE PRECISION array, dimension (LWORK) — on exit, WORK( 1 ) returns the minimal and optimal LWORK. ✨
LWORK (local or global input) INTEGER — the dimension of the array WORK. LWORK is local input and must be at least LWORK >= 3*N.

If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. 🫢

INFO (local output) INTEGER —
- = 0: successful exit 🎉
- < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i.

🌈 FURTHER DETAILS 🌈

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

Q = H(n-1) . . . H(2) H(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+1:n) = 0 and v(i) = 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(n-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(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 contents of sub( A ) on exit are illustrated by the following examples with n = 5:

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

  (  d   e   v2  v3  v4 )              (  d                  )
  (      d   e   v3  v4 )              (  e   d              )
  (          d   e   v4 )              (  v1  e   d          )
  (              d   e  )              (  v1  v2  e   d      )
  (                  d  )              (  v1  v2  v3  e   d  )

where d and e denote diagonal and off-diagonal elements of T, and vi denotes an element of the vector defining H(i).

🧚 Alignment requirements 🧚

The distributed submatrix sub( A ) must verify some alignment properties, namely the following expression should be true:

( MB_A.EQ.NB_A .AND. IROFFA.EQ.ICOFFA )

with

IROFFA = MOD( IA-1, MB_A )

and

ICOFFA = MOD( JA-1, NB_A ).

⚠️ bestie if that alignment check isn't giving what it needs to give, that's a whole situation. MB_A.EQ.NB_A AND IROFFA.EQ.ICOFFA. both. periodt. 💀

and that's the whole page!! 🎀 she's LAPACK version 1.5, dated 12 May 1997, and she's still serving tridiagonal realness to this day. mwah, love u, go reduce something symmetric 💖✨🦋

📦 where this page comes from

ubuntu jammy: scalapack-doc 1.5-11
💖 girlified by deepseek-ai/DeepSeek-V4.1-Flash on 2026-10-14: a language model's rewrite of the original, under its terms