PZGETRI(l)

📦 version 1.5, in ubuntu jammy

PZGETRI 💖✨

okay so bestie, welcome to the LAPACK routine page for PZGETRI — it lives in manual section l, it's a LAPACK routine, version 1.5, and yes this is the glow-up version of that man page. grab a snack, we're inverting matrices 🫶

🌸 NAME 🌸

PZGETRI — compute the inverse of a distributed matrix using the LU factorization computed by PZGETRF. like. it un-inverts. it flips the matrix back. iconic behavior fr fr 💅

✨ SYNOPSIS ✨

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

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

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

    COMPLEX*16      A( * ), WORK( * )

🦋 PURPOSE 🦋

PZGETRI computes the inverse of a distributed matrix using the LU factorization computed by PZGETRF. the method is: it inverts U, and then it computes the inverse of sub( A ) = A(IA:IA+N-1,JA:JA+N-1), which we're calling InvA, by solving the system InvA*L = inv(U) for InvA. slay, linear algebra, we love her 🎀

💜 Notes 💜

every global data object is described by an associated description vector. this vector stores all the info you need to establish the mapping between an object element and its corresponding process and memory location. it's giving very organized, very main character 📋✨

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 breakdown, bestie, every single line:

  • 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)).

now let K be the number of rows or columns of a distributed matrix, and assume 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.
  • 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 👑

  • 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) COMPLEX*16 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 PZGETRF. on exit, if INFO = 0, sub( A ) contains the inverse of the original distributed matrix sub( A ). glow-up complete ✨
  • 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) 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 = 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.

  • 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. the receipts, bestie:
    • = 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. 💀 no inverse for you, the math said no.
lowkey important reminder: this whole thing only makes sense after PZGETRF has done its thing. no LU factorization, no inverse, no cap. 🫶

okay that's the whole page, and here's your cute lil sign-off: this was LAPACK version 1.5, dated 12 May 1997, straight out of manual section l. love that for us. bye bestie, go invert something responsibly 💖🎀🦋✨

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