PSGETRI(l)

📦 version 1.5, in ubuntu jammy

💖 NAME 💖

Bestie, gather round. This is the manual page for PSGETRI (yes, from the l library section, LAPACK routine, version 1.5, we stan a classic) — and what it does is compute the inverse of a distributed matrix using the LU factorization computed by PSGETRF. That's it. That's the whole vibe. 🦋

✨ SYNOPSIS ✨

SUBROUTINE PSGETRI( N, A, IA, JA, DESCA, IPIV, WORK, LWORK, IWORK,
                    LIWORK, INFO )

    INTEGER         IA, INFO, JA, LIWORK, LWORK, N

    INTEGER         DESCA( * ), IPIV( * ), IWORK( * )

    REAL            A( * ), WORK( * )

Look at her. The signature is giving main character energy, fr fr. 👑

🌸 PURPOSE 🌸

PSGETRI computes the inverse of a distributed matrix using the LU factorization computed by PSGETRF. The way it works is so clever: this method inverts U first, and then computes the inverse of sub( A ) = A(IA:IA+N-1,JA:JA+N-1) — denoted InvA — by solving the system InvA*L = inv(U) for InvA. Periodt. 💅

🧚 Notes 🧚

Okay so this is the part everyone skips and then everyone regrets skipping. Pay attention bestie, iykyk.

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".

  • 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. 😤✨

💜 ARGUMENTS 💜

  • N — (global input) INTEGER. The number of rows and columns to be operated on, i.e. the order of the distributed submatrix sub( A ). N >= 0.
  • A — (local input/local output) REAL pointer into the local memory to an array of dimension (LLD_A,LOCc(JA+N-1)). On entry, the local pieces of the L and U obtained by the factorization sub( A ) = P*L*U computed by PSGETRF. On exit, if INFO = 0, sub( A ) contains the inverse of the original distributed matrix sub( A ).
  • 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.
  • IPIV — (local input) INTEGER array, dimension LOCr(M_A)+MB_A. Keeps track of the pivoting information. IPIV(i) is the global row index the local row i was swapped with. This array is tied to the distributed matrix A.
  • 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 = LOCr(N+MOD(IA-1,MB_A))*NB_A. WORK is used to keep a copy of at most an entire column block of sub( 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.

Heads up bestie: LWORK = -1 is not "give me negative space", it's the secret handshake that says "just tell me how much room I need, no cap." 🫶
  • 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 used as workspace for physically transposing the pivots. LIWORK is local input and must be at least:
if NPROW == NPCOL then
   LIWORK = LOCc( N_A + MOD(JA-1, NB_A) ) + NB_A
else
   LIWORK = LOCc( N_A + MOD(JA-1, NB_A) ) +
            MAX( CEIL(CEIL(LOCr(M_A)/MB_A)/(LCM/NPROW)), NB_A )
end if

where LCM is the least common multiple of process rows and columns (NPROW and NPCOL).

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. We love a clean 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.
    • > 0: If INFO = K, U(IA+K-1,IA+K-1) is exactly zero; the matrix is singular and its inverse could not be computed.
Okay real talk: a positive INFO means your matrix is singular and the inverse straight up could not be computed. Not a vibe. Not recoverable. Check your input, bestie. 💀

And that's the whole page, gorgeous. Every flag, every formula, every MOD, CEIL and LCM — all accounted for. Nothing skipped, nothing invented. 💅

This has been PSGETRI(l), from LAPACK version 1.5, dated 12 May 1997. Yes she's an elder but she still serves. 🦋✨ Thanks for reading, ily, go invert something responsibly. 💖

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