PCLANGE(l)

💖 PCLANGE 💖

okay bestie, gather round, because we're about to talk about PCLANGE — a LAPACK auxiliary routine, version 1.5, and she is that girl. she computes norms. that's it. that's the whole job. and she does it flawlessly. periodt. ✨

---

✨ NAME ✨

PCLANGE — returns the value of the one norm, or the Frobenius norm, ... (yes the sentence literally trails off like that in the original, we love a mysterious queen 💅)

---

🦋 SYNOPSIS 🦋

REAL FUNCTION PCLANGE( NORM, M, N, A, IA, JA, DESCA, WORK )

    CHARACTER NORM

    INTEGER   IA, JA, M, N

    INTEGER   DESCA( * )

    REAL      WORK( * )

    COMPLEX   A( * )

---

🌸 PURPOSE 🌸

so what does she do? PCLANGE returns the value of the one norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a distributed matrix sub( A ) = A(IA:IA+M-1, JA:JA+N-1). like, pick your fighter, she's got all of them. 💖

PCLANGE returns the value:

   ( max(abs(A(i,j))),  NORM = 'M' or 'm' with IA <= i <= IA+M-1,
   (                                      and  JA <= j <= JA+N-1,
   (
   ( norm1( sub( A ) ), NORM = '1', 'O' or 'o'
   (
   ( normI( sub( A ) ), NORM = 'I' or 'i'
   (
   ( normF( sub( A ) ), NORM = 'F', 'f', 'E' or 'e'

where norm1 denotes the one norm of a matrix (maximum column sum), normI denotes the infinity norm of a matrix (maximum row sum) and normF denotes the Frobenius norm of a matrix (square root of sum of squares). and heads up bestie — note that max(abs(A(i,j))) is not a matrix norm. not me exposing the impostor. 💀

---

📝 Notes 📝

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. it's giving metadata with main character energy. 🫶

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.

here's the whole notation situation, all of it, no skips:

• 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. slay.

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 👑

• NORM — (global input) CHARACTER. specifies the value to be returned in PCLANGE as described above. she's the one calling the shots, honestly.
• M — (global input) INTEGER. the number of rows to be operated on, i.e the number of rows of the distributed submatrix sub( A ). when M = 0, PCLANGE is set to zero. M >= 0. no rows? no problem, zero. iconic. 💅
• N — (global input) INTEGER. the number of columns to be operated on, i.e the number of columns of the distributed submatrix sub( A ). when N = 0, PCLANGE is set to zero. N >= 0.
• A — (local input) COMPLEX pointer into the local memory to an array of dimension (LLD_A, LOCc(JA+N-1)) containing the local pieces of the distributed matrix sub( A ). she's holding the goods. 🔥
• 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. the blueprint, the whole vibe map. 🎀
• WORK — (local workspace) REAL array dimension (LWORK). LWORK >= 0 if NORM = 'M' or 'm' (not referenced), Nq0 if NORM = '1', 'O' or 'o', Mp0 if NORM = 'I' or 'i', 0 if NORM = 'F', 'f', 'E' or 'e' (not referenced), where

IROFFA = MOD( IA-1, MB_A ), ICOFFA = MOD( JA-1, NB_A ), IAROW =
INDXG2P( IA, MB_A, MYROW, RSRC_A, NPROW ), IACOL = INDXG2P( JA,
NB_A, MYCOL, CSRC_A, NPCOL ), Mp0 = NUMROC( M+IROFFA, MB_A,
MYROW, IAROW, NPROW ), Nq0 = NUMROC( N+ICOFFA, NB_A, MYCOL,
IACOL, NPCOL ),

INDXG2P and NUMROC are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO. bestie, that's your whole cheat sheet right there. ✨

---

💜 lowkey important reminder: this is all verbatim behavior from the original spec — don't go changing any of those numbers or letter cases, the 'M' vs 'm' thing is real and she notices.

---

okay that's the whole page!! 🦋 LAPACK version 1.5, dated 12 May 1997 — shoutout to the OGs who wrote this back in the day, absolute legends, ate and left no crumbs. now go compute a norm, you got this. 💖✨🎀