PDPOSVX(l)

📦 version 1.5, in ubuntu jammy

PDPOSVX 💖 (that's a LAPACK routine, version 1.5, bestie)

okay so. sit down, grab a strawberry 🍓, because we're about to solve some linear systems and it's gonna be so good.

✨ NAME ✨

PDPOSVX uses the Cholesky factorization A = U**T*U or A = L*L**T to compute the solution to a real system of linear equations

A(IA:IA+N-1,JA:JA+N-1) * X = B(IB:IB+N-1,JB:JB+NRHS-1),

periodt. that's the whole job description 💅

🌸 SYNOPSIS 🌸

SUBROUTINE PDPOSVX( FACT, UPLO, N, NRHS, A, IA, JA, DESCA, AF, IAF, JAF,
                    DESCAF, EQUED, SR, SC, B, IB, JB, DESCB, X, IX, JX,
                    DESCX, RCOND, FERR, BERR, WORK, LWORK, IWORK,
                    LIWORK, INFO )

    CHARACTER       EQUED, FACT, UPLO

    INTEGER         IA, IAF, IB, INFO, IX, JA, JAF, JB, JX, LIWORK,
                    LWORK, N, NRHS

    DOUBLE          PRECISION RCOND

    INTEGER         DESCA( * ), DESCAF( * ), DESCB( * ), DESCX( * ),
                    IWORK( * )

    DOUBLE          PRECISION A( * ), AF( * ), B( * ), BERR( * ), FERR(
                    * ), SC( * ), SR( * ), WORK( * ), X( * )

like. look at all those arguments. main character energy fr fr 👑

🦋 PURPOSE 🦋

PDPOSVX uses the Cholesky factorization A = U**T*U or A = L*L**T to compute the solution to a real system of linear equations

where A(IA:IA+N-1,JA:JA+N-1) is an N-by-N matrix and X and B(IB:IB+N-1,JB:JB+NRHS-1) are N-by-NRHS matrices.

Error bounds on the solution and a condition estimate are also provided. In the following comments Y denotes Y(IY:IY+M-1,JY:JY+K-1) a M-by-K matrix where Y can be A, AF, B and X.

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".

so here's the whole breakdown, no cap:

  • 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

💜 DESCRIPTION 💜

The following steps are performed:

  1. If FACT = 'E', real scaling factors are computed to equilibrate the system:
diag(SR) * A * diag(SC) * inv(diag(SC)) * X = diag(SR) * B

Whether or not the system will be equilibrated depends on the scaling of the matrix A, but if equilibration is used, A is overwritten by diag(SR)*A*diag(SC) and B by diag(SR)*B.

  1. If FACT = 'N' or 'E', the Cholesky decomposition is used to factor the matrix A (after equilibration if FACT = 'E') as
A = U**T* U,  if UPLO = 'U', or
A = L * L**T,  if UPLO = 'L',

where U is an upper triangular matrix and L is a lower triangular matrix.

  1. The factored form of A is used to estimate the condition number of the matrix A. If the reciprocal of the condition number is less than machine precision, steps 4-6 are skipped.
  2. The system of equations is solved for X using the factored form of A.
  3. Iterative refinement is applied to improve the computed solution matrix and calculate error bounds and backward error estimates for it.
  4. If equilibration was used, the matrix X is premultiplied by diag(SR) so that it solves the original system before equilibration.

and that's literally the whole pipeline bestie, six little steps and you're done 🧚✨

(more coming, don't go anywhere 💖)

✨ ARGUMENTS ✨

okay bestie this is the part where we meet the whole cast 💖 every single one of these matters, so don't skim, i'm watching 👀

  • FACT (global input) CHARACTER — tells us whether the factored form of the matrix A is handed to you on entry, and if it's not, whether A should get equilibrated before it's factored. the options are:
    • 'F' — on entry, AF contains the factored form of A. if EQUED = 'Y', the matrix A has been equilibrated with scaling factors given by S. A and AF will not be modified. (iconic, hands off 🫶)
    • 'N' — the matrix A will be copied to AF and factored.
    • 'E' — the matrix A will be equilibrated if necessary, then copied to AF and factored.
  • UPLO (global input) CHARACTER — which triangle of A are we even storing, bestie:
    • 'U' — upper triangle of A is stored;
    • 'L' — lower triangle of A is stored.
  • N (global input) INTEGER — the number of rows and columns to be operated on, i.e. the order of the distributed submatrix A(IA:IA+N-1,JA:JA+N-1). N >= 0. no negatives, we don't do that here 💅
  • NRHS (global input) INTEGER — the number of right hand sides, i.e. the number of columns of the distributed submatrices B and X. NRHS >= 0.
  • A (local input/local output) DOUBLE PRECISION pointer into the local memory to an array of local dimension ( LLD_A, LOCc(JA+N-1) ). on entry, the symmetric matrix A — except if FACT = 'F' and EQUED = 'Y', in which case A must contain the equilibrated matrix diag(SR)*A*diag(SC). if UPLO = 'U', the leading N-by-N upper triangular part of A contains the upper triangular part of the matrix A, and the strictly lower triangular part of A is not referenced. if UPLO = 'L', the leading N-by-N lower triangular part of A contains the lower triangular part of the matrix A, and the strictly upper triangular part of A is not referenced. A is not modified if FACT = 'F' or 'N', or if FACT = 'E' and EQUED = 'N' on exit.

on exit, if FACT = 'E' and EQUED = 'Y', A is overwritten by diag(SR)*A*diag(SC). bestie she gets a glow-up ✨

  • 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. it's giving backstage pass 🎀
  • AF (local input or local output) DOUBLE PRECISION pointer into the local memory to an array of local dimension ( LLD_AF, LOCc(JA+N-1)). depends on the vibe:
    • if FACT = 'F', then AF is an input argument and on entry contains the triangular factor U or L from the Cholesky factorization A = U**T*U or A = L*L**T, in the same storage format as A. if EQUED .ne. 'N', then AF is the factored form of the equilibrated matrix diag(SR)*A*diag(SC).
    • if FACT = 'N', then AF is an output argument and on exit returns the triangular factor U or L from the Cholesky factorization A = U**T*U or A = L*L**T of the original matrix A.
    • if FACT = 'E', then AF is an output argument and on exit returns the triangular factor U or L from the Cholesky factorization A = U**T*U or A = L*L**T of the equilibrated matrix A (peep the description of A for the form of the equilibrated matrix).
  • IAF (global input) INTEGER — the row index in the global array AF indicating the first row of sub( AF ).
  • JAF (global input) INTEGER — the column index in the global array AF indicating the first column of sub( AF ).
  • DESCAF (global and local input) INTEGER array of dimension DLEN_. the array descriptor for the distributed matrix AF.
  • EQUED (global input/global output) CHARACTER — specifies the form of equilibration that was done. yes she's both input AND output, main character energy fr 👑
    • 'N' — no equilibration (always true if FACT = 'N').
    • 'Y' — equilibration was done, i.e., A has been replaced by diag(SR) * A * diag(SC).

EQUED is an input variable if FACT = 'F'; otherwise, it is an output variable.

  • SR (local input/local output) DOUBLE PRECISION array, dimension (LLD_A) — the scale factors for A distributed across process rows; not accessed if EQUED = 'N'. SR is an input variable if FACT = 'F'; otherwise, SR is an output variable. if FACT = 'F' and EQUED = 'Y', each element of SR must be positive. no zeroes, no negatives, we're keeping it positive 💖
  • SC (local input/local output) DOUBLE PRECISION array, dimension (LOC(N_A)) — the scale factors for A distributed across process columns; not accessed if EQUED = 'N'. SC is an input variable if FACT = 'F'; otherwise, SC is an output variable. if FACT = 'F' and EQUED = 'Y', each element of SC must be positive.
  • B (local input/local output) DOUBLE PRECISION pointer into the local memory to an array of local dimension ( LLD_B, LOCc(JB+NRHS-1) ). on entry, the N-by-NRHS right-hand side matrix B. on exit: if EQUED = 'N', B is not modified; if TRANS = 'N' and EQUED = 'R' or 'B', B is overwritten by diag(R)*B; if TRANS = 'T' or 'C' and EQUED = 'C' or 'B', B is overwritten by diag(C)*B. she's been through a lot okay 😭
  • 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.
  • X (local input/local output) DOUBLE PRECISION pointer into the local memory to an array of local dimension ( LLD_X, LOCc(JX+NRHS-1) ). if INFO = 0, the N-by-NRHS solution matrix X to the original system of equations. note that A and B are modified on exit if EQUED .ne. 'N', and the solution to the equilibrated system is inv(diag(SC))*X if TRANS = 'N' and EQUED = 'C' or or 'B'. (yes, that double "or or" is really in there, we don't edit the original, we simply respect her quirks 🙈)
  • IX (global input) INTEGER — the row index in the global array X indicating the first row of sub( X ).
  • JX (global input) INTEGER — the column index in the global array X indicating the first column of sub( X ).
  • DESCX (global and local input) INTEGER array of dimension DLEN_. the array descriptor for the distributed matrix X.
  • RCOND (global output) DOUBLE PRECISION — the estimate of the reciprocal condition number of the matrix A after equilibration (if done). if RCOND is less than the machine precision (in particular, if RCOND = 0), the matrix is singular to working precision. this condition is indicated by a return code of INFO > 0, and the solution and error bounds are not computed. so like… she's gone, no solution bestie 💀
  • FERR (local output) DOUBLE PRECISION array, dimension (LOC(N_B)) — the estimated forward error bounds for each solution vector X(j) (the j-th column of the solution matrix X). if XTRUE is the true solution, FERR(j) bounds the magnitude of the largest entry in (X(j) - XTRUE) divided by the magnitude of the largest entry in X(j). the quality of the error bound depends on the quality of the estimate of norm(inv(A)) computed in the code; if the estimate of norm(inv(A)) is accurate, the error bound is guaranteed. no cap 🫶
  • BERR (local output) DOUBLE PRECISION array, dimension (LOC(N_B)) — the componentwise relative backward error of each solution vector X(j) (i.e., the smallest relative change in any entry of A or B that makes X(j) an exact solution).
  • WORK (local workspace/local output) DOUBLE PRECISION 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 = MAX( PDPOCON( LWORK ), PDPORFS( LWORK ) ) + LOCr( N_A ).

also

LWORK = 3*DESCA( LLD_ )

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.

  • IWORK (local workspace/local output) INTEGER array, dimension (LIWORK) — on exit, IWORK(1) returns the minimal and optimal LIWORK.
  • LIWORK (local or global input) INTEGER — the dimension of the array IWORK. LIWORK is local input and must be at least
LIWORK = DESCA( LLD_ )

and also

LIWORK = LOCr(N_A).

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.

  • INFO (global output) INTEGER — the receipts, the final word:
    • = 0 — successful exit 🎉 slay.
    • < 0 — if INFO = -i, the i-th argument had an illegal value. that's on you bestie, fix your args.
    • > 0 — if INFO = i, and i is:
      • <= N — if INFO = i, the leading minor of order i of A is not positive definite, so the factorization could not be completed, and the solution and error bounds could not be computed.
      • = N+1 — RCOND is less than machine precision. the factorization has been completed, but the matrix is singular to working precision, and the solution and error bounds have not been computed.
heads up: this whole thing is LAPACK, so if any of these arguments misbehave, that's on your inputs, not on her 💅

and that's the whole page bestie, we ate, we devoured, no crumbs left 🍓 this one's LAPACK version 1.5, dated 12 May 1997 — a true vintage icon, still serving 💖✨🦋

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