PSSYTRD(l)

💖 PSSYTRD 💖

okay bestie, sit DOWN, because we're about to talk about a LAPACK routine and it's giving main character energy. this is PSSYTRD from the LAPACK routine collection, version 1.5, and yes she's an icon, yes she's been around since before you were born, and yes she still slays. 💅

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✨ NAME ✨

PSSYTRD — reduce a real symmetric matrix sub( A ) to symmetric tridiagonal form T by an orthogonal similarity transformation. that's the whole job, that's the whole vibe. 🦋

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🌸 SYNOPSIS 🌸

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

    CHARACTER       UPLO

    INTEGER         IA, INFO, JA, LWORK, N

    INTEGER         DESCA( * )

    REAL            A( * ), D( * ), E( * ), TAU( * ), WORK( * )

look at all those arguments. so many little guys. each one matters, no cap. 👑

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💜 PURPOSE 💜

PSSYTRD reduces a real symmetric matrix sub( A ) to symmetric tridiagonal form T by an orthogonal similarity transformation:

Q' * sub( A ) * Q = T

where sub( A ) = A(IA:IA+N-1,JA:JA+N-1).

like. she takes your big messy symmetric situation and turns it into something tridiagonal. that's a glow-up fr fr. ✨

📝 Notes

every global data object is described by an associated description vector. this vector stores 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".

here's the whole notation breakdown, and yes you need all of it, 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)).

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 rise UP 🧚

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🎀 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) REAL 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. 😭 (yes it's a lot, yes you'll survive)
• 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) REAL 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) REAL 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) REAL, 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) REAL 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 >= MAX( NB * ( NP +1 ), 3 * NB )

where

NB = MB_A = NB_A,
NP = NUMROC( N, NB, MYROW, IAROW, NPROW ),
IAROW = INDXG2P( IA, NB, MYROW, RSRC_A, NPROW ).

INDXG2P and NUMROC are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO.

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. 💀 (workspace query my beloved)

• INFO — (global 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.

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🔥 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 .AND. IROFFA.EQ.0 )

with

IROFFA = MOD( IA-1, MB_A )
ICOFFA = MOD( JA-1, NB_A )

bestie if you don't satisfy the alignment requirements, do NOT come crying to me. it literally says "should be true". that's a requirement, not a suggestion. 💅

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okay that's the whole page! she's from LAPACK version 1.5, dated 12 May 1997 — a true vintage queen who has been serving tridiagonal realness since the 90s and honestly? still iconic. go forth and reduce. 🫶✨🌈💖