PZTRCON(l)

📦 version 1.5, in ubuntu jammy

💖 PZTRCON 💖 — the NAME section, but make it cute

okay so bestie, welcome to PZTRCON(l), a whole LAPACK routine, and she is serving linear algebra realness. This one estimates the reciprocal of the condition number of a triangular distributed matrix A(IA:IA+N-1,JA:JA+N-1), in either the 1-norm or the infinity-norm. That's the whole assignment. No cap. 🫶

✨ SYNOPSIS ✨

SUBROUTINE PZTRCON( NORM, UPLO, DIAG, N, A, IA, JA, DESCA, RCOND, WORK,
                    LWORK, RWORK, LRWORK, INFO )

    CHARACTER       DIAG, NORM, UPLO

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

    DOUBLE          PRECISION RCOND

    INTEGER         DESCA( * )

    DOUBLE          PRECISION RWORK( * )

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

🎀 PURPOSE 🎀

PZTRCON estimates the reciprocal of the condition number of a triangular distributed matrix A(IA:IA+N-1,JA:JA+N-1), in either the 1-norm or the infinity-norm. Like, it's the vibe check for your matrix: how close is this thing to being a total mess? 🌸

The norm of A(IA:IA+N-1,JA:JA+N-1) is computed and an estimate is obtained for norm(inv(A(IA:IA+N-1,JA:JA+N-1))), 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)) ) ).

📝 Notes (the lore) 📝

Every 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. It's giving "your data lives here, on this process, mind your business". 💅

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 👑 (every single one, we don't skip, periodt)

NORM (global input) CHARACTER — specifies whether the 1-norm condition number or the infinity-norm condition number is required:
- = '1' or 'O': 1-norm;
- = 'I': Infinity-norm.
UPLO (global input) CHARACTER —
- = 'U': A(IA:IA+N-1,JA:JA+N-1) is upper triangular;
- = 'L': A(IA:IA+N-1,JA:JA+N-1) is lower triangular.
DIAG (global input) CHARACTER —
- = 'N': A(IA:IA+N-1,JA:JA+N-1) is non-unit triangular;
- = 'U': A(IA:IA+N-1,JA:JA+N-1) is unit triangular.
N (global input) INTEGER — the order of the distributed matrix A(IA:IA+N-1,JA:JA+N-1). N >= 0. (Yes bestie, zero is allowed, she's flexible. 🦋)
A (local input) COMPLEX*16 pointer into the local memory to an array of dimension ( LLD_A, LOCc(JA+N-1) ). This array contains the local pieces of the triangular distributed matrix A(IA:IA+N-1,JA:JA+N-1). If UPLO = 'U', the leading N-by-N upper triangular part of this distributed matrix contains the upper triangular matrix, and its strictly lower triangular part is not referenced. If UPLO = 'L', the leading N-by-N lower triangular part of this distributed matrix contains the lower triangular matrix, and the strictly upper triangular part is not referenced. If DIAG = 'U', the diagonal elements of A(IA:IA+N-1,JA:JA+N-1) are also not referenced and are assumed to be 1.
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.
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*CEIL(P-1,Q),LOCc(N+MOD(JA-1,NB_A)) + NB_A*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 >= 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.

⚠️ friendly reminder bestie: < 0 means somebody passed something illegal and INFO is out here snitching with the exact coordinates. Read the formula, it's literally giving you the address. 💀

and that's the whole page, no notes, you ate this 🎀 — this was LAPACK version 1.5, dated 12 May 1997, and she's still iconic fr fr. bye bestie, go compute that condition number 💅✨🦋

📦 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