PSSYEV(l)

✨ PSSYEV(l) — LAPACK routine (version 1.3) ✨

okay bestie, gather round 🫶 this is the tea on PSSYEV(l) from the LAPACK routine (version 1.3) era. It's giving linear algebra main character energy and we're gonna read it together, no cap.

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💖 NAME 💖

PSSYEV computes all eigenvalues and, optionally, eigenvectors of a real symmetric matrix A by calling the recommended sequence of ScaLAPACK routines. Periodt. 💅

📜 SYNOPSIS 📜

       SUBROUTINE PSSYEV( JOBZ, UPLO, N, A, IA, JA, DESCA, W, Z, IZ, JZ, DESCZ,
                          WORK, LWORK, INFO )

           CHARACTER      JOBZ, UPLO

           INTEGER        IA, INFO, IZ, JA, JZ, LWORK, N

           INTEGER        DESCA( * ), DESCZ( * )

           REAL           A( * ), W( * ), WORK( * ), Z( * )

           INTEGER        BLOCK_CYCLIC_2D, DLEN_, DTYPE_, CTXT_, M_, N_, MB_,
                          NB_, RSRC_, CSRC_, LLD_

           PARAMETER      ( BLOCK_CYCLIC_2D = 1, DLEN_ = 9, DTYPE_ = 1, CTXT_ =
                          2, M_ = 3, N_ = 4, MB_ = 5, NB_ = 6, RSRC_ = 7, CSRC_
                          = 8, LLD_ = 9 )

           REAL           ZERO, ONE

           PARAMETER      ( ZERO = 0.0E+0, ONE = 1.0E+0 )

           INTEGER        ITHVAL

           PARAMETER      ( ITHVAL = 10 )

           LOGICAL        LOWER, WANTZ

           INTEGER        CONTEXTC, CSRC_A, I, IACOL, IAROW, ICOFFA, IINFO,
                          INDD, INDD2, INDE, INDE2, INDTAU, INDWORK, INDWORK2,
                          IROFFA, IROFFZ, ISCALE, IZROW, J, K, LCM, LCMQ, LDC,
                          LLWORK, LWMIN, MB_A, MB_Z, MYCOL, MYPCOLC, MYPROWC,
                          MYROW, NB, NB_A, NB_Z, NN, NP, NPCOL, NPCOLC, NPROCS,
                          NPROW, NPROWC, NQ, NRC, QRMEM, RSRC_A, RSRC_Z,
                          SIZEMQRLEFT, SIZEMQRRIGHT

           REAL           ANRM, BIGNUM, EPS, RMAX, RMIN, SAFMIN, SIGMA, SMLNUM

           INTEGER        DESCQR( 10 ), IDUM1( 3 ), IDUM2( 3 )

           LOGICAL        LSAME

           INTEGER        ILCM, INDXG2P, NUMROC, SL_GRIDRESHAPE

           REAL           PSLAMCH, PSLANSY

           EXTERNAL       LSAME, ILCM, INDXG2P, NUMROC, SL_GRIDRESHAPE,
                          PSLAMCH, PSLANSY

           EXTERNAL       BLACS_GRIDEXIT, BLACS_GRIDINFO, CHK1MAT, DESCINIT,
                          PCHK2MAT, PSELGET, PSGEMR2D, PSLASCL, PSLASET,
                          PSORMTR, PSSYTRD, PXERBLA, SCOPY, SGAMN2D, SGAMX2D,
                          SSCAL, SSTEQR2

           INTRINSIC      ICHAR, MAX, MIN, MOD, REAL, SQRT

🎯 PURPOSE 🎯

PSSYEV computes all eigenvalues and, optionally, eigenvectors of a real symmetric matrix A by calling the recommended sequence of ScaLAPACK routines. Slay. ✨

But heads up, bestie 🚨 in its present form, PSSYEV assumes a homogeneous system and makes no checks for consistency of the eigenvalues or eigenvectors across the different processes. Because of this, it is possible that a heterogeneous system may return incorrect results without any error messages. Not me side-eyeing that caveat. 💀

📝 NOTES 📝

A description vector is associated with each 2D block-cyclicly distributed matrix. This vector stores the information required to establish the mapping between a matrix entry and its corresponding process and memory location. It's giving the blueprint, fr fr.

In the following comments, the character _ should be read as "of the distributed matrix". Let A be a generic term for any 2D block cyclicly distributed matrix. Its description vector is DESCA:

• DTYPE_A(global) — stored in DESCA( DTYPE_): the descriptor type.
• 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 distributed matrix A.
• N_A (global) — stored in DESCA( N_ ): the number of columns in the distributed matrix A.
• MB_A (global) — stored in DESCA( MB_ ): the blocking factor used to distribute the rows of A.
• NB_A (global) — stored in DESCA( NB_ ): the blocking factor used to distribute the columns of A.
• RSRC_A (global) — stored in DESCA( RSRC_ ): the process row over which the first row of the matrix A is distributed.
• CSRC_A (global) — stored in DESCA( CSRC_ ): the process column over which the first column of A is distributed.
• LLD_A (local) — stored in DESCA( LLD_ ): the leading dimension of the local array storing the local blocks of the distributed matrix A. 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.S

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 ).

✨ ARGUMENTS ✨

okay bestie, here's the full cast of characters for PSSYEV — every single one matters, no skips. 💖

NP, NQ

• NP — the number of rows local to a given process.
• NQ — the number of columns local to a given process.

JOBZ (global input) CHARACTER*1

Specifies whether or not to compute the eigenvectors:

• = 'N': Compute eigenvalues only.
• = 'V': Compute eigenvalues and eigenvectors.

UPLO (global input) CHARACTER*1

Specifies whether the upper or lower triangular part of the symmetric matrix A is stored:

• = 'U': Upper triangular
• = 'L': Lower triangular

N (global input) INTEGER

The number of rows and columns of the matrix A. N >= 0.

A (local input/workspace) block cyclic DOUBLE PRECISION array,

global dimension (N, N), local dimension ( LLD_A, LOCc(JA+N-1) )

On entry, the symmetric matrix A. If UPLO = 'U', only the upper triangular part of A is used to define the elements of the symmetric matrix. If UPLO = 'L', only the lower triangular part of A is used to define the elements of the symmetric matrix.

On exit, the lower triangle (if UPLO='L') or the upper triangle (if UPLO='U') of A, including the diagonal, is destroyed. 😭 she's gone, periodt.

IA (global input) INTEGER

A's global row index, which points to the beginning of the submatrix which is to be operated on.

JA (global input) INTEGER

A's global column index, which points to the beginning of the submatrix which is to be operated on.

DESCA (global and local input) INTEGER array of dimension DLEN_.

The array descriptor for the distributed matrix A. If DESCA( CTXT_ ) is incorrect, PSSYEV cannot guarantee correct error reporting.

not me trusting a bad context 💀

W (global output) REAL array, dimension (N)

On normal exit, the first M entries contain the selected eigenvalues in ascending order.

Z (local output) REAL array,

global dimension (N, N), local dimension ( LLD_Z, LOCc(JZ+N-1) ) If JOBZ = 'V', then on normal exit the first M columns of Z contain the orthonormal eigenvectors of the matrix corresponding to the selected eigenvalues. If JOBZ = 'N', then Z is not referenced.

IZ (global input) INTEGER

Z's global row index, which points to the beginning of the submatrix which is to be operated on.

JZ (global input) INTEGER

Z's global column index, which points to the beginning of the submatrix which is to be operated on.

DESCZ (global and local input) INTEGER array of dimension DLEN_.

The array descriptor for the distributed matrix Z. DESCZ( CTXT_ ) must equal DESCA( CTXT_ )

WORK (local workspace/output) REAL array,

dimension (LWORK) Version 1.0: on output, WORK(1) returns the workspace needed to guarantee completion. If the input parameters are incorrect, WORK(1) may also be incorrect.

If JOBZ='N' WORK(1) = minimal=optimal amount of workspace If JOBZ='V' WORK(1) = minimal workspace required to generate all the eigenvectors.

LWORK (local input) INTEGER

See below for definitions of variables used to define LWORK.

If no eigenvectors are requested (JOBZ = 'N') then

LWORK >= 5*N + SIZESYTRD + 1

where

SIZESYTRD = The workspace requirement for PSSYTRD
            and is MAX( NB * ( NP +1 ), 3 * NB )

If eigenvectors are requested (JOBZ = 'V' ) then the amount of workspace required to guarantee that all eigenvectors are computed is:

QRMEM = 2*N-2
LWMIN = 5*N + N*LDC + MAX( SIZEMQRLEFT, QRMEM ) + 1

Variable definitions:

NB = DESCA( MB_ ) = DESCA( NB_ ) =
     DESCZ( MB_ ) = DESCZ( NB_ )
NN = MAX( N, NB, 2 )
DESCA( RSRC_ ) = DESCA( RSRC_ ) = DESCZ( RSRC_ ) =
                 DESCZ( CSRC_ ) = 0
NP = NUMROC( NN, NB, 0, 0, NPROW )
NQ = NUMROC( MAX( N, NB, 2 ), NB, 0, 0, NPCOL )
NRC = NUMROC( N, NB, MYPROWC, 0, NPROCS)
LDC = MAX( 1, NRC )
SIZEMQRLEFT = The workspace requirement for PSORMTR
              when it's SIDE argument is 'L'.

With MYPROWC defined when a new context is created as:

CALL BLACS_GET( DESCA( CTXT_ ), 0, CONTEXTC )
CALL BLACS_GRIDINIT( CONTEXTC, 'R', NPROCS, 1 )
CALL BLACS_GRIDINFO( CONTEXTC, NPROWC, NPCOLC, MYPROWC,
                     MYPCOLC )

If LWORK = -1, the LWORK is global input and a workspace query is assumed; the routine only calculates the minimum size for the WORK array. The required workspace is returned as the first element of WORK and no error message is issued by PXERBLA.

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.
• > 0: If INFO = 1 through N, the i(th) eigenvalue did not converge in SSTEQR2 after a total of 30*N iterations. If INFO = N+1, then PSSYEV has detected heterogeneity by finding that eigenvalues were not identical across the process grid. In this case, the accuracy of the results from PSSYEV cannot be guaranteed.

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and that's the whole page, bestie! this iconic manual page is from LAPACK version 1.3, dated 12 May 1997, and its manpage name is PSSYEV(l). periodt 💅✨