PCPOTRI(l)

👑 PCPOTRI 👑 (the (l) LAPACK routine, version 1.5, bestie edition)

okay so first of all, welcome to PCPOTRI — she's a LAPACK routine and she's that girl. Here's the whole vibe.

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✨ NAME ✨

PCPOTRI — compute the inverse of a complex Hermitian positive definite distributed matrix sub( A ) = A(IA:IA+N-1,JA:JA+N-1) using the Cholesky factorization sub( A ) = U**H*U or L*L**H computed by PCPOTRF. 💅

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🌸 SYNOPSIS 🌸

SUBROUTINE PCPOTRI( UPLO, N, A, IA, JA, DESCA, INFO )

    CHARACTER       UPLO

    INTEGER         IA, INFO, JA, N

    INTEGER         DESCA( * )

    COMPLEX         A( * )

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🎀 PURPOSE 🎀

PCPOTRI computes the inverse of a complex Hermitian positive definite distributed matrix sub( A ) = A(IA:IA+N-1,JA:JA+N-1) using the Cholesky factorization sub( A ) = U**H*U or L*L**H computed by PCPOTRF. no cap, that's literally the whole job. 🫶

💜 Notes 💜

okay so this part is the lore, and you need it, so lock in bestie 🧚

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".

the notation tea ☕:

• DTYPE_A (global) → DESCA( DTYPE_ ): the descriptor type. In this case, DTYPE_A = 1.
• CTXT_A (global) → 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) → DESCA( M_ ): the number of rows in the global array A.
• N_A (global) → DESCA( N_ ): the number of columns in the global array A.
• MB_A (global) → DESCA( MB_ ): the blocking factor used to distribute the rows of the array.
• NB_A (global) → DESCA( NB_ ): the blocking factor used to distribute the columns of the array.
• RSRC_A (global) → DESCA( RSRC_ ): the process row over which the first row of the array A is distributed.
• CSRC_A (global) → DESCA( CSRC_ ): the process column over which the first column of the array A is distributed.
• LLD_A (local) → 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

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🌈 ARGUMENTS 🌈

• UPLO — (global input) CHARACTER*1
  • = 'U': Upper triangle of sub( A ) is stored;
  • = 'L': Lower triangle of sub( A ) is stored.
• 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. zero is valid bestie, don't come for her 🍓
• A — (local input/local output) COMPLEX pointer into the local memory to an array of dimension (LLD_A, LOCc(JA+N-1)). On entry, the local pieces of the triangular factor U or L from the Cholesky factorization of the distributed matrix sub( A ) = U**H*U or L*L**H, as computed by PCPOTRF. On exit, the local pieces of the upper or lower triangle of the (Hermitian) inverse of sub( A ), overwriting the input factor U or L. she overwrites her input, that's just who she is 💀
• 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.
• 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 = i, the (i,i) element of the factor U or L is zero, and the inverse could not be computed.

⚠️ heads up bestie: if INFO = i > 0, that (i,i) element of the factor U or L is zero and the inverse genuinely could not be computed. that's not a vibe, that's a hard stop. 😭

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💖 no extra tips here, just the facts, because inventing behaviour would be not the move. everything above is the real deal, fr fr.

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that's the whole page bestie, go forth and invert responsibly 🦋💜✨

This was LAPACK routine version 1.5, dated 12 May 1997 — iconic, timeless, and honestly the blueprint. mwah 💅🎀🌸