PDGETRI(l)

📦 version 1.5, in ubuntu jammy

PDGETRI 💖 the LAPACK girlies know

okay bestie, gather round, because today we're talking about PDGETRI — a LAPACK routine, version 1.5, and she is lowkey the moment. This page is literally titled PDGETRI(l) in the man pages, so now you know her full government name. 👑

✨ NAME ✨

PDGETRI — compute the inverse of a distributed matrix using the LU factorization computed by PDGETRF. That's it, that's the whole vibe. 🦋

✨ SYNOPSIS ✨

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

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

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

    DOUBLE PRECISION A( * ), WORK( * )

✨ PURPOSE ✨

PDGETRI computes the inverse of a distributed matrix using the LU factorization computed by PDGETRF. No cap. The method is: it inverts U first, and then it gets the inverse of sub( A ) = A(IA:IA+N-1,JA:JA+N-1) — which we call InvA, cute nickname, love that for her — by solving the system InvA*L = inv(U) for InvA. periodt. 💅

Notes 📝

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. iykyk.

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". 🧚

Here's the whole notation situation, and yes I'm giving you every single line because nothing gets skipped in this house:

  • 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 ✨

okay this is the part where you actually pay attention, bestie, because every one of these matters 👀

  • 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) DOUBLE PRECISION 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 PDGETRF. On exit, if INFO = 0, sub( A ) contains the inverse of the original distributed matrix sub( 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.
  • 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) 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 = 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. sneaky little query mode, we love her 🫶

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

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. same energy, different array 💀

  • 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.
    • > 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.
🚨 bestie listen: if INFO = K > 0, then U(IA+K-1,IA+K-1) is exactly zero, the matrix is singular, and its inverse could not be computed. That's not a vibe check you pass, that's a vibe check you fail. 😭

and that's the whole page, fr fr. this one's from LAPACK version 1.5, dated 12 May 1997 — vintage icon behavior honestly, she's been serving since the 90s and she's still in rotation 👑🌸 thanks for coming to my ted talk, mwah 💖

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