PSGESVX(l)

📦 version 1.5, in ubuntu jammy

💖 PSGESVX(l) — LAPACK routine (version 1.5) 💖

okay bestie, sit down, hydrate, because we're about to solve a real system of linear equations and we're gonna look iconic doing it. this is PSGESVX, a LAPACK routine, and she is a whole main character. 🫶

🌸 NAME 🌸

PSGESVX — bestie, she uses the LU factorization 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),

like. that's her whole thing. that's the vibe. ✨

🎀 SYNOPSIS 🎀

SUBROUTINE PSGESVX( 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,
                    IWORK, LIWORK, INFO )

    CHARACTER       EQUED, FACT, TRANS

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

    REAL            RCOND

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

    REAL            A( * ), AF( * ), B( * ), BERR( * ), C( * ), FERR( *
                    ), R( * ), WORK( * ), X( * )

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

✨ PURPOSE ✨

PSGESVX uses the LU factorization 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. she does it ALL. 😭

📝 Notes 📝

okay so before we go anywhere, you need to know how distributed arrays work, because this is ScaLAPACK and everything is ✨spread out✨.

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". iykyk 🦋

  • 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 🧚

💜 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. 💀 (yeah, she just. skips. them.)
  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. glow-up arc ✨
  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.

👑 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) (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) REAL 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) REAL 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 PSGETRF. 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 PSGETRF; 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 so IPIV is like: if FACT = 'E', 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 equilibrated matrix A(IA:IA+N-1,JA:JA+N-1). Periodt. 💅

  • EQUED (global input or global output) CHARACTER — Specifies the form of equilibration that was done.
    • = 'N': No equilibration (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 is an output variable. So if you set FACT = 'F', you're telling it what's up; otherwise it tells you. 👑
  • R (local input or local output) REAL array, dimension LOCr(M_A) — The row scale factors for A(IA:IA+N-1,JA:JA+N-1). If EQUED = 'R' or 'B', 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) REAL array, dimension LOCc(N_A) — The column scale factors for A(IA:IA+N-1,JA:JA+N-1). If EQUED = 'C' or 'B', 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) REAL 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', 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). The way that B gets overwritten depending on TRANS and EQUED is giving main character energy. 🌈
  • 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) REAL 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(IX:IX+N-1,JX:JX+NRHS-1) to the original system of equations. 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) REAL — The estimate of the reciprocal condition number of the matrix A(IA:IA+N-1,JA:JA+N-1) 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. 😭
  • FERR (local output) REAL array, dimension LOCc(N_B) — The estimated forward error bounds for each solution vector X(j) (the j-th column of the solution matrix X(IX:IX+N-1,JX:JX+NRHS-1). If XTRUEis the true solution,FERR(j)bounds the magnitude of the largest entry in(X(j) - XTRUE)divided by the magnitude of the largest entry inX(j). The estimate is as reliable as the estimate for RCOND, and is almost always a slight overestimate of the true error. FERRis replicated in every process row, and is aligned with the matricesBandX`. 🍓
  • BERR (local output) REAL 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) REAL 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( PSGECON( LWORK ), PSGERFS( 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. 💅
  • 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 = 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 —
    • = 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. 💀

And that's the whole vibe for PSGESVX(l), bestie! This masterpiece is from LAPACK version 1.5, dated 12 May 1997. You absolutely ate this read, periodt. 💅✨

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