PCHEGVX(l)

📦 version 1.5, in ubuntu jammy

💖 NAME 💖

Welcome to the PCHEGVX(l) glow-up, bestie — this page comes to you straight from the LAPACK routine family, version 1.5. So what does it do? It computes all the eigenvalues, and optionally the eigenvectors, of a complex generalized Hermitian-definite eigenproblem. That's the whole vibe. 👑

✨ SYNOPSIS ✨

Okay so here's the literal shape of the call, no cap, copy it exactly:

SUBROUTINE PCHEGVX( IBTYPE, JOBZ, RANGE, UPLO, N, A, IA, JA, DESCA, B,
                    IB, JB, DESCB, VL, VU, IL, IU, ABSTOL, M, NZ, W,
                    ORFAC, Z, IZ, JZ, DESCZ, WORK, LWORK, RWORK, LRWORK,
                    IWORK, LIWORK, IFAIL, ICLUSTR, GAP, INFO )

    CHARACTER       JOBZ, RANGE, UPLO

    INTEGER         IA, IB, IBTYPE, IL, INFO, IU, IZ, JA, JB, JZ,
                    LIWORK, LRWORK, LWORK, M, N, NZ

    REAL            ABSTOL, ORFAC, VL, VU

    INTEGER         DESCA( * ), DESCB( * ), DESCZ( * ), ICLUSTR( * ),
                    IFAIL( * ), IWORK( * )

    REAL            GAP( * ), RWORK( * ), W( * )

    COMPLEX         A( * ), B( * ), 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 )

    COMPLEX         ONE

    PARAMETER       ( ONE = 1.0E+0 )

    REAL            FIVE, ZERO

    PARAMETER       ( FIVE = 5.0E+0, ZERO = 0.0E+0 )

    INTEGER         IERRNPD

    PARAMETER       ( IERRNPD = 16 )

    LOGICAL         ALLEIG, INDEIG, LQUERY, UPPER, VALEIG, WANTZ

    CHARACTER       TRANS

    INTEGER         IACOL, IAROW, IBCOL, IBROW, ICOFFA, ICOFFB, ICTXT,
                    IROFFA, IROFFB, LIWMIN, LRWMIN, LWMIN, MQ0, MYCOL,
                    MYROW, NB, NEIG, NN, NP0, NPCOL, NPROW

    REAL            EPS, SCALE

    INTEGER         IDUM1( 5 ), IDUM2( 5 )

    LOGICAL         LSAME

    INTEGER         ICEIL, INDXG2P, NUMROC

    REAL            PSLAMCH

    EXTERNAL        LSAME, ICEIL, INDXG2P, NUMROC, PSLAMCH

    EXTERNAL        BLACS_GRIDINFO, CHK1MAT, PCHEEVX, PCHEGST, PCHK1MAT,
                    PCHK2MAT, PCPOTRF, PCTRMM, PCTRSM, PXERBLA, SGEBR2D,
                    SGEBS2D, SSCAL

    INTRINSIC       ABS, CMPLX, ICHAR, MAX, MIN, MOD, REAL

🦋 PURPOSE 🦋

So PCHEGVX computes all the eigenvalues, and optionally the eigenvectors, of a complex generalized Hermitian-definite eigenproblem. The three forms it handles are:

  • sub( A )*x=(lambda)*sub( B )*x
  • sub( A )*sub( B )x=(lambda)*x
  • sub( B )*sub( A )*x=(lambda)*x

And here's the tea on notation: sub( A ) means A( IA:IA+N-1, JA:JA+N-1 ) and it's assumed to be Hermitian. Meanwhile sub( B ) means B( IB:IB+N-1, JB:JB+N-1 ) and it's assumed to be Hermitian positive definite. Slay. 💅

🌈 NOTES 🌈

Every global data object gets described by an associated description vector, and that vector is storing all the info you need to map an object element to its matching process and memory location. It's giving admin energy.

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

  • 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 ) is 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 ) is 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 🎀

  • IBTYPE (global input) INTEGER — specifies the problem type to be solved:
    • = 1: sub( A )*x = (lambda)*sub( B )*x
    • = 2: sub( A )*sub( B )*x = (lambda)*x
    • = 3: sub( B )*sub( A )*x = (lambda)*x
  • JOBZ (global input) CHARACTER*1 — the "how much do you want" switch:
    • = 'N': compute eigenvalues only;
    • = 'V': compute eigenvalues and eigenvectors.
  • RANGE (global input) CHARACTER*1 — which eigenvalues are we hunting:
    • = 'A': all eigenvalues will be found.
    • = 'V': all eigenvalues in the interval [VL,VU] will be found.
    • = 'I': the IL-th through IU-th eigenvalues will be found.
  • UPLO (global input) CHARACTER*1 — which triangle got stored:
    • = 'U': upper triangles of sub( A ) and sub( B ) are stored;
    • = 'L': lower triangles of sub( A ) and sub( B ) are stored.
  • N (global input) INTEGER — the order of the matrices sub( A ) and sub( B ). N >= 0.
  • A (local input/local output) COMPLEX — a 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 N-by-N Hermitian 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.
    • If UPLO = 'L', the leading N-by-N lower triangular part of sub( A ) contains the lower triangular part of the matrix.
    • On exit, if JOBZ = 'V', then if INFO = 0, sub( A ) contains the distributed matrix Z of eigenvectors. The eigenvectors are normalized as follows:
      • if IBTYPE = 1 or 2, Z**H*sub( B )*Z = I;
      • if IBTYPE = 3, Z**H*inv( sub( B ) )*Z = I.
    • If JOBZ = 'N', then on exit the upper triangle (if UPLO='U') or the lower triangle (if UPLO='L') of sub( A ), including the diagonal, is destroyed. 😭 rip.
  • 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. If DESCA( CTXT_ ) is incorrect, PCHEGVX cannot guarantee correct error reporting.
  • B (local input/local output) COMPLEX — a pointer into the local memory to an array of dimension (LLD_B, LOCc(JB+N-1)). On entry, this array contains the local pieces of the N-by-N Hermitian distributed matrix sub( B ).
    • If UPLO = 'U', the leading N-by-N upper triangular part of sub( B ) contains the upper triangular part of the matrix.
    • If UPLO = 'L', the leading N-by-N lower triangular part of sub( B ) contains the lower triangular part of the matrix.
    • On exit, if INFO <= N, the part of sub( B ) containing the matrix is overwritten by the triangular factor U or L from the Cholesky factorization sub( B ) = U**H*U or sub( B ) = L*L**H.
  • IB (global input) INTEGER — the row index in the global array B indicating the first row of sub( B ).
  • JB (global input) INTEGER — the column index in the global array B indicating the first column of sub( B ).
  • DESCB (global and local input) INTEGER array of dimension DLEN_ — the array descriptor for the distributed matrix B. DESCB( CTXT_ ) must equal DESCA( CTXT_ ).
  • VL (global input) REAL — if RANGE='V', the lower bound of the interval to be searched for eigenvalues. Not referenced if RANGE = 'A' or 'I'.
  • VU (global input) REAL — if RANGE='V', the upper bound of the interval to be searched for eigenvalues. Not referenced if RANGE = 'A' or 'I'.
  • IL (global input) INTEGER — if RANGE='I', the index (from smallest to largest) of the smallest eigenvalue to be returned. IL >= 1. Not referenced if RANGE = 'A' or 'V'.
  • IU (global input) INTEGER — if RANGE='I', the index (from smallest to largest) of the largest eigenvalue to be returned. min(IL,N) <= IU <= N. Not referenced if RANGE = 'A' or 'V'.
  • ABSTOL (global input) REAL — bestie, listen. If JOBZ='V', setting ABSTOL to PSLAMCH( CONTEXT, 'U') yields the most orthogonal eigenvectors. ✨
    • The absolute error tolerance for the eigenvalues. An approximate eigenvalue is accepted as converged when it is determined to lie in an interval [a,b] of width less than or equal to
ABSTOL + EPS *   max( |a|,|b| ) ,
  • where EPS is the machine precision. If ABSTOL is less than or equal to zero, then EPS*norm(T) will be used in its place, where norm(T) is the 1-norm of the tridiagonal matrix obtained by reducing A to tridiagonal form.
  • Eigenvalues will be computed most accurately when ABSTOL is set to twice the underflow threshold 2*PSLAMCH('S'), not zero. If this routine returns with ((MOD(INFO,2).NE.0) .OR. (MOD(INFO/8,2).NE.0)), indicating that some eigenvalues or eigenvectors did not converge, try setting ABSTOL to 2*PSLAMCH('S').
Not the ABSTOL slander, it really said "set me to zero and watch everything fall apart." Set it right, girlies. 🫶

omg okay bestie, before we get into the arg list, here's some required reading if you wanna be a real one 📚✨

"Computing Small Singular Values of Bidiagonal Matrices with Guaranteed High Relative Accuracy," by Demmel and Kahan, LAPACK Working Note #3. "On the correctness of Parallel Bisection in Floating Point" by Demmel, Dhillon and Ren, LAPACK Working Note #70

alright, now the arguments. there are so many, but we're gonna slay this list together 💅

  • M (global output) INTEGER — Total number of eigenvalues found. 0 <= M <= N.
  • NZ (global output) INTEGER — Total number of eigenvectors computed. 0 <= NZ <= M. The number of columns of Z that are filled. If JOBZ .NE. 'V', NZ is not referenced. If JOBZ .EQ. 'V', NZ = M unless the user supplies insufficient space and PCHEGVX is not able to detect this before beginning computation. To get all the eigenvectors requested, the user must supply both sufficient space to hold the eigenvectors in Z (M .LE. DESCZ(N_)) and sufficient workspace to compute them. (See LWORK below.) PCHEGVX is always able to detect insufficient space without computation unless RANGE .EQ. 'V'. periodt.
  • W (global output) REAL array, dimension (N) — On normal exit, the first M entries contain the selected eigenvalues in ascending order.
  • ORFAC (global input) REAL — Specifies which eigenvectors should be reorthogonalized. Eigenvectors that correspond to eigenvalues which are within tol=ORFAC*norm(A) of each other are to be reorthogonalized. However, if the workspace is insufficient (see LWORK), tol may be decreased until all eigenvectors to be reorthogonalized can be stored in one process. No reorthogonalization will be done if ORFAC equals zero. A default value of 10^-3 is used if ORFAC is negative. ORFAC should be identical on all processes. lowkey important that last bit, bestie.
  • Z (local output) COMPLEX 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 an eigenvector fails to converge, then that column of Z contains the latest approximation to the eigenvector, and the index of the eigenvector is returned in IFAIL. If JOBZ = 'N', then Z is not referenced.
  • IZ (global input) INTEGER — The row index in the global array Z indicating the first row of sub( Z ).
  • JZ (global input) INTEGER — The column index in the global array Z indicating the first column of sub( Z ).
  • DESCZ (global and local input) INTEGER array of dimension DLEN_. — The array descriptor for the distributed matrix Z. DESCZ( CTXT_ ) must equal DESCA( CTXT_ ) no cap, they must match.
  • WORK (local workspace/output) COMPLEX array, dimension (LWORK) — WORK(1) returns the optimal workspace.
  • LWORK (local input) INTEGER — Size of WORK array. If only eigenvalues are requested:
LWORK >= N + MAX( NB * ( NP0 + 1 ), 3 )

If eigenvectors are requested:

LWORK >= N + ( NP0 + MQ0 + NB ) * NB

with NQ0 = NUMROC( NN, NB, 0, 0, NPCOL ).

For optimal performance, greater workspace is needed, i.e.

LWORK >= MAX( LWORK, N + NHETRD_LWOPT,
  NHEGST_LWOPT )

Where LWORK is as defined above, and

NHETRD_LWORK = 2*( ANB+1 )*( 4*NPS+2 ) +
  ( NPS + 1 ) * NPS
NHEGST_LWOPT =  2*NP0*NB + NQ0*NB + NB*NB
NB = DESCA( MB_ )
NP0 = NUMROC( N, NB, 0, 0, NPROW )
NQ0 = NUMROC( N, NB, 0, 0, NPCOL )
ICTXT = DESCA( CTXT_ )
ANB = PJLAENV( ICTXT, 3, 'PCHETTRD', 'L', 0, 0, 0, 0 )
SQNPC = SQRT( DBLE( NPROW * NPCOL ) )
NPS = MAX( NUMROC( N, 1, 0, 0, SQNPC ), 2*ANB )

NUMROC is a ScaLAPACK tool functions; PJLAENV is a ScaLAPACK envionmental inquiry function 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 optimal size for all work arrays. Each of these values is returned in the first entry of the correspondingwork array, and no error message is issued by PXERBLA. iykyk, workspace query is a whole vibe.

  • RWORK (local workspace/output) REAL array, dimension (LRWORK) — On return, RWORK(1) contains the amount of workspace required for optimal efficiency. If JOBZ='N' RWORK(1) = optimal amount of workspace required to compute eigenvalues efficiently. If JOBZ='V' RWORK(1) = optimal amount of workspace required to compute eigenvalues and eigenvectors efficiently with no guarantee on orthogonality. If RANGE='V', it is assumed that all eigenvectors may be required when computing optimal workspace.
  • LRWORK (local input) INTEGER — Size of RWORK. See below for definitions of variables used to define LRWORK. If no eigenvectors are requested (JOBZ = 'N') then
LRWORK >= 5 * NN + 4 * N

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

LRWORK >= 4*N + MAX( 5*NN, NP0 * MQ0 ) +
  ICEIL( NEIG, NPROW*NPCOL)*NN

The computed eigenvectors may not be orthogonal if the minimal workspace is supplied and ORFAC is too small. If you want to guarantee orthogonality (at the cost of potentially poor performance) you should add the following to LRWORK:

(CLUSTERSIZE-1)*N

where CLUSTERSIZE is the number of eigenvalues in the largest cluster, where a cluster is defined as a set of close eigenvalues: { W(K),...,W(K+CLUSTERSIZE-1) | W(J+1) <= W(J) + ORFAC*2*norm(A) }

Variable definitions:

NEIG = number of eigenvectors requested
NB = DESCA( MB_ ) = DESCA( NB_ ) = DESCZ( MB_ ) =
     DESCZ( NB_ )
NN = MAX( N, NB, 2 )
DESCA( RSRC_ ) = DESCA( NB_ ) = DESCZ( RSRC_ ) =
                 DESCZ( CSRC_ ) = 0
NP0 = NUMROC( NN, NB, 0, 0, NPROW )
MQ0 = NUMROC( MAX( NEIG, NB, 2 ), NB, 0, 0, NPCOL )
ICEIL( X, Y ) is a ScaLAPACK function returning
ceiling(X/Y)

When LRWORK is too small: If LRWORK is too small to guarantee orthogonality, PCHEGVX attempts to maintain orthogonality in the clusters with the smallest spacing between the eigenvalues. If LRWORK is too small to compute all the eigenvectors requested, no computation is performed and INFO=-25 is returned. Note that when RANGE='V', PCHEGVX does not know how many eigenvectors are requested until the eigenvalues are computed. Therefore, when RANGE='V' and as long as LRWORK is large enough to allow PCHEGVX to compute the eigenvalues, PCHEGVX will compute the eigenvalues and as many eigenvectors as it can. it's giving "we tried our best" energy.

Relationship between workspace, orthogonality & performance: If CLUSTERSIZE >= N/SQRT(NPROW*NPCOL), then providing enough space to compute all the eigenvectors orthogonally will cause serious degradation in performance. In the limit (i.e. CLUSTERSIZE = N-1) PCSTEIN will perform no better than CSTEIN on 1 processor. For CLUSTERSIZE = N/SQRT(NPROW*NPCOL) reorthogonalizing all eigenvectors will increase the total execution time by a factor of 2 or more. For CLUSTERSIZE > N/SQRT(NPROW*NPCOL) execution time will grow as the square of the cluster size, all other factors remaining equal and assuming enough workspace. Less workspace means less reorthogonalization but faster execution. trade-offs, bestie. it's the way that it is.

If LRWORK = -1, then LRWORK 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.

  • IWORK (local workspace) INTEGER array — On return, IWORK(1) contains the amount of integer workspace required.
  • LIWORK (local input) INTEGER — size of IWORK LIWORK >= 6 * NNP Where:
NNP = MAX( N, NPROW*NPCOL + 1, 4 )

If LIWORK = -1, then LIWORK 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.

  • IFAIL (output) INTEGER array, dimension (N) — IFAIL provides additional information when INFO .NE. 0 If (MOD(INFO/16,2).NE.0) then IFAIL(1) indicates the order of the smallest minor which is not positive definite. If (MOD(INFO,2).NE.0) on exit, then IFAIL contains the indices of the eigenvectors that failed to converge. If neither of the above error conditions hold and JOBZ = 'V', then the first M elements of IFAIL are set to zero.
  • ICLUSTR (global output) integer array, dimension (2*NPROW*NPCOL) — This array contains indices of eigenvectors corresponding to a cluster of eigenvalues that could not be reorthogonalized due to insufficient workspace (see LWORK, ORFAC and INFO). Eigenvectors corresponding to clusters of eigenvalues indexed ICLUSTR(2*I-1) to ICLUSTR(2*I), could not be reorthogonalized due to lack of workspace. Hence the eigenvectors corresponding to these clusters may not be orthogonal. ICLUSTR() is a zero terminated array. (ICLUSTR(2*K).NE.0 .AND. ICLUSTR(2*K+1).EQ.0) if and only if K is the number of clusters ICLUSTR is not referenced if JOBZ = 'N'.
  • GAP (global output) REAL array, dimension (NPROW*NPCOL) — bestie, this one holds the gap between eigenvalues whose eigenvectors could not be reorthogonalized. The output values in this array line up with the clusters that ICLUSTR points at. So lowkey, the dot product between eigenvectors corresponding to the I-th cluster can be as big as ( C * n ) / GAP(I), where C is a tiny lil constant. ✨
  • INFO (global output) INTEGER — the vibe check of the whole call, fr fr:
    • = 0: successful exit, we ate 💅
    • < 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. Not iconic. 😭
    • > 0: this is where it gets messy:
      • if (MOD(INFO,2).NE.0), then one or more eigenvectors failed to converge. Their indices are stored in IFAIL. Send e-mail to scalapack@cs.utk.edu 📧
      • if (MOD(INFO/2,2).NE.0), then eigenvectors corresponding to one or more clusters of eigenvalues could not be reorthogonalized because of insufficient workspace. The indices of the clusters are stored in the array ICLUSTR.
      • if (MOD(INFO/4,2).NE.0), then space limit prevented PCHEGVX from computing all of the eigenvectors between VL and VU. The number of eigenvectors computed is returned in NZ.
      • if (MOD(INFO/8,2).NE.0), then PCSTEBZ failed to compute eigenvalues. Send e-mail to scalapack@cs.utk.edu 📧
      • if (MOD(INFO/16,2).NE.0), then B was not positive definite. IFAIL(1) indicates the order of the smallest minor which is not positive definite.

okay bye bestie, this was LAPACK version 1.5, dated 12 May 1997, and the page is PCHEGVX(l) — periodt. 🎀✨

📦 where this page comes from