PZPOCON(l)

📦 version 1.5, in ubuntu jammy

💖 PZPOCON — the vibe check for your Hermitian positive definite matrix 💖

okay bestie, so this is PZPOCON from the LAPACK routine crew (version 1.5, she's a classic), and she's here to estimate the reciprocal of the condition number (in the 1-norm) of a complex Hermitian positive definite distributed matrix, using the Cholesky factorization A = U**H*U or A = L*L**H that PZPOTRF already computed for you. literally her whole job. 🫶

✨ SYNOPSIS ✨

SUBROUTINE PZPOCON( UPLO, N, A, IA, JA, DESCA, ANORM, RCOND, WORK,
                    LWORK, RWORK, LRWORK, INFO )

    CHARACTER       UPLO

    INTEGER         IA, INFO, JA, LRWORK, LWORK, N

    DOUBLE          PRECISION ANORM, RCOND

    INTEGER         DESCA( * )

    DOUBLE          PRECISION RWORK( * )

    COMPLEX*16      A( * ), WORK( * )

🌸 PURPOSE 🌸

PZPOCON estimates the reciprocal of the condition number (in the 1-norm) of a complex Hermitian positive definite distributed matrix using the Cholesky factorization A = U**H*U or A = L*L**H computed by PZPOTRF.

An estimate is obtained for norm(inv(A(IA:IA+N-1,JA:JA+N-1))), and the reciprocal of the condition number is computed as

RCOND = 1 / ( norm( A(IA:IA+N-1,JA:JA+N-1)      ) *
              norm( inv(A(IA:IA+N-1,JA:JA+N-1)) ) ).

basically: how close is your matrix to being a hot mess (singular)? this tells you. 👑

🎀 Notes 🎀

Each 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".

  • 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 factor stored in A(IA:IA+N-1,JA:JA+N-1) is upper or lower triangular.
    • = 'U': Upper triangular
    • = 'L': Lower triangular
  • N — (global input) INTEGER. The order of the distributed matrix A(IA:IA+N-1,JA:JA+N-1). N >= 0.
  • A — (local input) COMPLEX*16 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 factors L or U from the Cholesky factorization A(IA:IA+N-1,JA:JA+N-1) = U'*U or L*L', as computed by PZPOTRF.
  • 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.
  • ANORM — (global input) DOUBLE PRECISION. The 1-norm (or infinity-norm) of the hermitian distributed matrix A(IA:IA+N-1,JA:JA+N-1).
  • RCOND — (global output) DOUBLE PRECISION. The reciprocal of the condition number of the distributed matrix A(IA:IA+N-1,JA:JA+N-1), computed as
RCOND = 1 / ( norm( A(IA:IA+N-1,JA:JA+N-1)      ) *
norm( inv(A(IA:IA+N-1,JA:JA+N-1)) ) ).
  • WORK — (local workspace/local output) COMPLEX*16 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 >= 2*LOCr(N+MOD(IA-1,MB_A)) + MAX( 2,
MAX(NB_A*MAX(1,CEIL(P-1,Q)),LOCc(N+MOD(JA-1,NB_A)) +
NB_A*MAX(1,CEIL(Q-1,P))) ).

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.

  • RWORK — (local workspace/local output) DOUBLE PRECISION array, dimension (LRWORK). On exit, RWORK(1) returns the minimal and optimal LRWORK.
  • LRWORK — (local or global input) INTEGER. The dimension of the array RWORK. LRWORK is local input and must be at least
LRWORK >= 2*LOCc(N+MOD(JA-1,NB_A)).

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.

  • 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.
💀 heads up bestie: if you see a negative INFO, something you passed in was not it. the encoding is -(i*100+j) for arrays and -i for scalars, so decode it before you spiral.

LAPACK version 1.5, dated 12 May 1997 — that's the lore, that's the source, she's been iconic since '97 and she's still serving 💅✨

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