PZGESVX(l)

📦 version 1.5, in ubuntu jammy

💖 PZGESVX — the LAPACK routine that solves your complex linear system and also tells you how confident it is 💖

okay bestie, listen. we're doing PZGESVX from LAPACK routine (version 1.5), and this one is a whole journey. it's the parallel, distributed-memory girlie who takes your complex system A * X = B, crunches it, and then hands you back error bounds and a condition estimate like a little overachiever. main character energy fr fr. ✨

🌸 NAME 🌸

PZGESVX — she uses the LU factorization to compute the solution to a complex 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 PZGESVX( FACT, TRANS, N, NRHS, A, IA, JA, DESCA, AF, IAF,
                    JAF, DESCAF, IPIV, EQUED, R, C, B, IB, JB, DESCB, X,
                    IX, JX, DESCX, RCOND, FERR, BERR, WORK, LWORK,
                    RWORK, LRWORK, INFO )

    CHARACTER       EQUED, FACT, TRANS

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

    DOUBLE          PRECISION RCOND

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

    DOUBLE          PRECISION BERR( * ), C( * ), FERR( * ), R( * ),
                    RWORK( * )

    COMPLEX*16      A( * ), AF( * ), B( * ), WORK( * ), X( * )

look at all those arguments. she's booked and busy. 🫶

🦋 PURPOSE 🦋

PZGESVX uses the LU factorization to compute the solution to a complex 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. like, she doesn't just solve it — she rates her own work. iconic behavior honestly. 😭👑

💜 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 little cheat sheet, rewritten with love:

  • 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

math girlies rise up, this is your moment 🧚‍♀️

🌈 DESCRIPTION 🌈

In the following description, A denotes A(IA:IA+N-1,JA:JA+N-1), B denotes B(IB:IB+N-1,JB:JB+NRHS-1) and X denotes X(IX:IX+N-1,JX:JX+NRHS-1).

The following steps are performed:

  1. If FACT = 'E', real scaling factors are computed to equilibrate the system:
TRANS = 'N':  diag(R)*A*diag(C)     *inv(diag(C))*X = diag(R)*B
TRANS = 'T': (diag(R)*A*diag(C))**T *inv(diag(R))*X = diag(C)*B
TRANS = 'C': (diag(R)*A*diag(C))**H *inv(diag(R))*X = diag(C)*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(R)*A*diag(C) and B by diag(R)*B (if TRANS='N') or diag(C)*B (if TRANS = 'T' or 'C').

  1. If FACT = 'N' or 'E', the LU decomposition is used to factor the matrix A (after equilibration if FACT = 'E') as
A = P * L * U,

where P is a permutation matrix, L is a unit lower triangular matrix, and U is upper triangular.

  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 FACT = 'E' and equilibration was used, the matrix X is premultiplied by diag(C) (if TRANS = 'N') or diag(R) (if TRANS = 'T' or 'C') so that it solves the original system before equilibration.

like, she undoes her own glow-up at the end so the answer actually matches your original problem. self-aware queen. 🎀

👑 ARGUMENTS 👑

  • FACT — (global input) CHARACTER

Specifies whether or not the factored form of the matrix A(IA:IA+N-1,JA:JA+N-1) is supplied on entry, and if not, whether the matrix A(IA:IA+N-1,JA:JA+N-1) should be equilibrated before it is factored.

  • = 'F': On entry, AF(IAF:IAF+N-1,JAF:JAF+N-1) and IPIV contain the factored form of A(IA:IA+N-1,JA:JA+N-1). If EQUED is not 'N', the matrix A(IA:IA+N-1,JA:JA+N-1) has been equilibrated with scaling factors given by R and C. A(IA:IA+N-1,JA:JA+N-1), AF(IAF:IAF+N-1,JAF:JAF+N-1), and IPIV are not modified.
  • = 'N': The matrix A(IA:IA+N-1,JA:JA+N-1) will be copied to AF(IAF:IAF+N-1,JAF:JAF+N-1) and factored.
  • = 'E': The matrix A(IA:IA+N-1,JA:JA+N-1) will be equilibrated if necessary, then copied to AF(IAF:IAF+N-1,JAF:JAF+N-1) and factored.
  • TRANS — (global input) CHARACTER

Specifies the form of the system of equations:

  • = 'N': A(IA:IA+N-1,JA:JA+N-1) * X(IX:IX+N-1,JX:JX+NRHS-1) = B(IB:IB+N-1,JB:JB+NRHS-1) (No transpose)
  • = 'T': A(IA:IA+N-1,JA:JA+N-1)**T * X(IX:IX+N-1,JX:JX+NRHS-1) = B(IB:IB+N-1,JB:JB+NRHS-1) (Transpose)
  • = 'C': A(IA:IA+N-1,JA:JA+N-1)**H * X(IX:IX+N-1,JX:JX+NRHS-1) = B(IB:IB+N-1,JB:JB+NRHS-1) (Conjugate transpose)
  • 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.

  • NRHS — (global input) INTEGER

The number of right-hand sides, i.e., the number of columns of the distributed submatrices B(IB:IB+N-1,JB:JB+NRHS-1) and X(IX:IX+N-1,JX:JX+NRHS-1). NRHS >= 0.

  • A — (local input/local output) COMPLEX*16 pointer into the local memory to an array of local dimension (LLD_A,LOCc(JA+N-1)).

On entry, the N-by-N matrix A(IA:IA+N-1,JA:JA+N-1). If FACT = 'F' and EQUED is not 'N', then A(IA:IA+N-1,JA:JA+N-1) must have been equilibrated by the scaling factors in R and/or C. A(IA:IA+N-1,JA:JA+N-1) is not modified if FACT = 'F' or 'N', or if FACT = 'E' and EQUED = 'N' on exit.

On exit, if EQUED .ne. 'N', A(IA:IA+N-1,JA:JA+N-1) is scaled as follows:

  • EQUED = 'R': A(IA:IA+N-1,JA:JA+N-1) := diag(R) * A(IA:IA+N-1,JA:JA+N-1)
  • EQUED = 'C': A(IA:IA+N-1,JA:JA+N-1) := A(IA:IA+N-1,JA:JA+N-1) * diag(C)
  • EQUED = 'B': A(IA:IA+N-1,JA:JA+N-1) := diag(R) * A(IA:IA+N-1,JA:JA+N-1) * diag(C)
  • 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.

  • AF — (local input or local output) COMPLEX*16 pointer into the local memory to an array of local dimension (LLD_AF,LOCc(JA+N-1)).

If FACT = 'F', then AF(IAF:IAF+N-1,JAF:JAF+N-1) is an input argument and on entry contains the factors L and U from the factorization A(IA:IA+N-1,JA:JA+N-1) = P*L*U as computed by PZGETRF. If EQUED .ne. 'N', then AF is the factored form of the equilibrated matrix A(IA:IA+N-1,JA:JA+N-1).

If FACT = 'N', then AF(IAF:IAF+N-1,JAF:JAF+N-1) is an output argument and on exit returns the factors L and U from the factorization A(IA:IA+N-1,JA:JA+N-1) = P*L*U of the original matrix A(IA:IA+N-1,JA:JA+N-1).

If FACT = 'E', then AF(IAF:IAF+N-1,JAF:JAF+N-1) is an output argument and on exit returns the factors L and U from the factorization A(IA:IA+N-1,JA:JA+N-1) = P*L*U of the equilibrated matrix A(IA:IA+N-1,JA:JA+N-1) (see the description of A(IA:IA+N-1,JA:JA+N-1) 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.

  • IPIV — (local input or local output) INTEGER array, dimension LOCr(M_A)+MB_A.

If FACT = 'F', then IPIV is an input argument and on entry contains the pivot indices from the factorization A(IA:IA+N-1,JA:JA+N-1) = P*L*U as computed by PZGETRF; IPIV(i) -> The global row local row i was swapped with. This array must be aligned with A( IA:IA+N-1, * ).

If FACT = 'N', then IPIV is an output argument and on exit contains the pivot indices from the factorization A(IA:IA+N-1,JA:JA+N-1) = P*L*U of the original matrix A(IA:IA+N-1,JA:JA+N-1).

🫶 okay that's the tea so far and we're only partway through the arguments — hold on bestie, there's more coming. don't go anywhere. 💖

If FACT = 'E', then IPIV is an output argument, and on exit it contains the pivot indices from the factorization A(IA:IA+N-1,JA:JA+N-1) = P*L*U of the equilibrated matrix A(IA:IA+N-1,JA:JA+N-1). 💫

  • EQUED — global input or global output, type CHARACTER. Specifies the form of equilibration that was done. Bestie, this is literally the "which flavor of scaling did we do" flag:
    • = 'N': no equilibration at all (always true if FACT = 'N').
    • = 'R': row equilibration, i.e. A(IA:IA+N-1,JA:JA+N-1) has been premultiplied by diag(R).
    • = 'C': column equilibration, i.e. A(IA:IA+N-1,JA:JA+N-1) has been postmultiplied by diag(C).
    • = 'B': both row and column equilibration, i.e. A(IA:IA+N-1,JA:JA+N-1) has been replaced by diag(R) * A(IA:IA+N-1,JA:JA+N-1) * diag(C).

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

  • R — local input or local output, DOUBLE PRECISION array, dimension LOCr(M_A). The row scale factors for A(IA:IA+N-1,JA:JA+N-1). If EQUED = 'R' or 'B', then A(IA:IA+N-1,JA:JA+N-1) is multiplied on the left by diag(R); if EQUED='N' or 'C', R is not accessed. R is an input variable if FACT = 'F'; otherwise, R is an output variable. If FACT = 'F' and EQUED = 'R' or 'B', each element of R must be positive. R is replicated in every process column, and is aligned with the distributed matrix A. ✨
  • C — local input or local output, DOUBLE PRECISION array, dimension LOCc(N_A). The column scale factors for A(IA:IA+N-1,JA:JA+N-1). If EQUED = 'C' or 'B', then A(IA:IA+N-1,JA:JA+N-1) is multiplied on the right by diag(C); if EQUED = 'N' or 'R', C is not accessed. C is an input variable if FACT = 'F'; otherwise, C is an output variable. If FACT = 'F' and EQUED = 'C' or C is replicated in every process row, and is aligned with the distributed matrix A. 💜
  • B — local input/local output, COMPLEX*16 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(IB:IB+N-1,JB:JB+NRHS-1). On exit, if EQUED = 'N', then B(IB:IB+N-1,JB:JB+NRHS-1) is not modified; if TRANS = 'N' and EQUED = 'R' or 'B', B is overwritten by diag(R)*B(IB:IB+N-1,JB:JB+NRHS-1); if TRANS = 'T' or 'C' and EQUED = 'C' or 'B', B(IB:IB+N-1,JB:JB+NRHS-1) is overwritten by diag(C)*B(IB:IB+N-1,JB:JB+NRHS-1). 🫶
  • 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, COMPLEX*16 pointer into the local memory to an array of local dimension (LLD_X, LOCc(JX+NRHS-1)). If INFO = 0, this is the N-by-NRHS solution matrix X(IX:IX+N-1,JX:JX+NRHS-1) to the original system of equations. Heads up though, bestie: note that A(IA:IA+N-1,JA:JA+N-1) and B(IB:IB+N-1,JB:JB+NRHS-1) are modified on exit if EQUED .ne. 'N', and the solution to the equilibrated system is inv(diag(C))*X(IX:IX+N-1,JX:JX+NRHS-1) if TRANS = 'N' and EQUED = 'C' or 'B', or inv(diag(R))*X(IX:IX+N-1,JX:JX+NRHS-1) if TRANS = 'T' or 'C' and EQUED = 'R' or 'B'. 💅
  • 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(IA:IA+N-1,JA:JA+N-1) after equilibration (if it was 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. 🚨
  • FERR — local output, DOUBLE PRECISION array, dimension LOCc(N_B). The estimated forward error bounds for each solution vector X(j) (that's the j-th column of the solution matrix X(IX:IX+N-1,JX:JX+NRHS-1)). 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 estimate is as reliable as the estimate for RCOND, and is almost always a slight overestimate of the true error. FERR is replicated in every process row, and is aligned with the matrices B and X. 📏
  • BERR — local output, DOUBLE PRECISION array, dimension LOCc(N_B). The componentwise relative backward error of each solution vector X(j) (i.e., the smallest relative change in any entry of A(IA:IA+N-1,JA:JA+N-1) or B(IB:IB+N-1,JB:JB+NRHS-1) that makes X(j) an exact solution). BERR is replicated in every process row, and is aligned with the matrices B and X. 🌸
  • 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 = MAX( PZGECON( LWORK ), PZGERFS( LWORK ) ) + LOCr( N_A )

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_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. The receipts:
    • = 0: successful exit ✨
    • < 0: if INFO = -i, the i-th argument had an illegal value 😭
    • > 0: if INFO = i, and i is
      • <= N: U(IA+I-1,IA+I-1) is exactly zero. The factorization has been completed, but the factor U is exactly singular, so 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. 💀
Bestie, real talk: INFO > 0 is not a vibe. If you see it, the matrix said "no" and the solution + error bounds simply don't exist. Don't come for me, come for your conditioning. 🙈

And that's a wrap, gorgeous — this page is from LAPACK version 1.5, dated 12 May 1997, and it's the man page for PZGESVX(l). The 90s really served us this distributed-matrix realness and honestly? Iconic. Go forth and solve, bestie. 👑💖

📦 where this page comes from