PCPOCON(l)

📦 version 1.5, in ubuntu jammy

🌸 PCPOCON 🌸 (the LAPACK routine, bestie, and yes it's giving linear algebra royalty)

okay so like. hi. this is PCPOCON(l), straight out of LAPACK, version 1.5, and it is lowkey iconic. no cap. it's a distributed-matrix girlie and she's here to tell you how well-conditioned your Hermitian positive definite matrix is. 💅✨

✨ NAME ✨

PCPOCON — bestie, she 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 PCPOTRF. 🫶

🦋 SYNOPSIS 🦋

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

    CHARACTER       UPLO

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

    REAL            ANORM, RCOND

    INTEGER         DESCA( * )

    REAL            RWORK( * )

    COMPLEX         A( * ), WORK( * )

👑 PURPOSE 👑

PCPOCON 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 PCPOTRF.

An estimate is obtained for norm(inv(A(IA:IA+N-1,JA:JA+N-1))), and then 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)) ) ).

like. she's dividing one by a product of two norms. math ate. 🔥

💜 Notes 💜

okay so here's the lore you actually need. every global data object has an associated description vector. that vector is the one holding all the info you need to map an element of the object onto its process and its memory location. it's the group chat between your array and the grid. 🧚

let A be a generic term for any 2D block cyclicly distributed array. such a global array has an associated description vector DESCA. and in the comments that follow, bestie, the character _ should be read as "of the global array". periodt.

so, the notation, stored in, and explanation — served three ways because we don't do tables here:

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 ) 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 ).

and 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

you got this, bestie. 🦋

🎀 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, 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 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 PCPOTRF. 💖
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) REAL. the 1-norm (or infinity-norm) of the hermitian distributed matrix A(IA:IA+N-1,JA:JA+N-1).
RCOND — (global output) REAL. 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 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) REAL 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: a negative INFO means somebody handed over an illegal value, and the formula tells you exactly who and which entry. read the code, not the vibes. 🫶

and that's the whole page, fr fr. 💅

this whole thing came out of LAPACK version 1.5, dated 12 May 1997, so like. vintage. archival. main character energy from another era. 🎀 if you ever forget a detail, the original is out there living its truth, but honestly? you've got this. slay. bye bestie!! 💖✨🦋🌸👑🌈⭐🧚💜

📦 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