SRC/stpqrt.f(3)

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version 3.12.1 · 7 versions
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SRC/stpqrt.f 💅

okay bestie, welcome to SRC/stpqrt.f from the Library Functions Manual — this is the STPQRT page and it’s lowkey iconic for blocked QR vibes 🦋✨

🌸 NAME

SRC/stpqrt.f — that’s the whole name, bestie, no notes 🎀

✨ SYNOPSIS

Functions/Subroutines

subroutine stpqrt (m, n, l, nb, a, lda, b, ldb, t, ldt, work, info)

STPQRT 👑

🛠️ Function/Subroutine Documentation

subroutine stpqrt (integer m, integer n, integer l, integer nb, real, dimension( lda, * ) a, integer lda, real, dimension( ldb, * ) b, integer ldb, real, dimension( ldt, * ) t, integer ldt, real, dimension( * ) work, integer info)

STPQRT 💖

💡 Purpose

ok so STPQRT computes a blocked QR factorization of a real matrix C, and C is made of two parts: a triangular block A and a pentagonal block B, using the compact WY representation for Q. periodt ✨

🧸 Parameters

  • M — M is INTEGER. It’s the number of rows of the matrix B. Constraint: M >= 0. no cap 🫶
  • N — N is INTEGER. It’s the number of columns of the matrix B, and it’s also the order of the triangular matrix A. Constraint: N >= 0. slay 🌸
  • L — L is INTEGER. It’s the number of rows of the upper trapezoidal part of B. Constraint: MIN(M,N) >= L >= 0. See Further Details. 💅
  • NB — NB is INTEGER. It’s the block size used in the blocked QR. Constraint: N >= NB >= 1. big block energy 🔥
  • A — A is REAL array, dimension (LDA,N). On entry, it’s the upper triangular N-by-N matrix A. On exit, the elements on and above the diagonal of the array contain the upper triangular matrix R. iconic behavior 👑
  • LDA — LDA is INTEGER. It’s the leading dimension of the array A. Constraint: LDA >= max(1,N). 💜
  • B — B is REAL array, dimension (LDB,N). On entry, it’s the pentagonal M-by-N matrix B. The first M-L rows are rectangular, and the last L rows are upper trapezoidal. On exit, B contains the pentagonal matrix V. See Further Details. 🦋
  • LDB — LDB is INTEGER. It’s the leading dimension of the array B. Constraint: LDB >= max(1,M). ✨
  • T — T is REAL array, dimension (LDT,N). It holds the upper triangular block reflectors stored in compact form as a sequence of upper triangular blocks. See Further Details. 🎀
  • LDT — LDT is INTEGER. It’s the leading dimension of the array T. Constraint: LDT >= NB. 🫶
  • WORK — WORK is REAL array, dimension (NB*N). it’s giving workspace, and it’s doing its job 💅
  • INFO — INFO is INTEGER.
    • = 0: successful exit 🌈
    • < 0: if INFO = -i, the i-th argument had an illegal value 😭

👑 Author

  • Univ. of Tennessee
  • Univ. of California Berkeley
  • Univ. of Colorado Denver
  • NAG Ltd.

🔍 Further Details

The input matrix C is a (N+M)-by-N matrix:

C = [ A ]
    [ B ]

where A is an upper triangular N-by-N matrix, and B is an M-by-N pentagonal matrix consisting of a (M-L)-by-N rectangular matrix B1 on top of a L-by-N upper trapezoidal matrix B2:

B = [ B1 ]  <- (M-L)-by-N rectangular
    [ B2 ]  <-     L-by-N upper trapezoidal.

The upper trapezoidal matrix B2 consists of the first L rows of a N-by-N upper triangular matrix, where 0 <= L <= MIN(M,N). If L=0, B is rectangular M-by-N; if M=L=N, B is upper triangular. 💀✨

The matrix W stores the elementary reflectors H(i) in the i-th column below the diagonal (of A) in the (N+M)-by-N input matrix C:

C = [ A ]  <- upper triangular N-by-N
    [ B ]  <- M-by-N pentagonal

so that W can be represented as:

W = [ I ]  <- identity, N-by-N
    [ V ]  <- M-by-N, same form as B.

Thus, all of information needed for W is contained on exit in B, which we call V above. Note that V has the same form as B; that is,

V = [ V1 ] <- (M-L)-by-N rectangular
    [ V2 ] <-     L-by-N upper trapezoidal.

The columns of V represent the vectors which define the H(i)'s. 🧚

The number of blocks is B = ceiling(N/NB), where each block is of order NB except for the last block, which is of order IB = N - (B-1)*NB. For each of the B blocks, a upper triangular block reflector factor is computed: T1, T2, ..., TB. The NB-by-NB (and IB-by-IB for the last block) T's are stored in the NB-by-N matrix T as:

T = [T1 T2 ... TB].

Definition at line 187 of file stpqrt.f. 🎯

👑 Author

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