mirror of https://github.com/k3s-io/k3s
128 lines
3.4 KiB
Go
128 lines
3.4 KiB
Go
// Copyright ©2017 The Gonum Authors. All rights reserved.
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// Use of this source code is governed by a BSD-style
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// license that can be found in the LICENSE file.
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package gonum
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import (
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"math"
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"gonum.org/v1/gonum/blas"
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"gonum.org/v1/gonum/blas/blas64"
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)
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// Dlaqp2 computes a QR factorization with column pivoting of the block A[offset:m, 0:n]
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// of the m×n matrix A. The block A[0:offset, 0:n] is accordingly pivoted, but not factorized.
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//
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// On exit, the upper triangle of block A[offset:m, 0:n] is the triangular factor obtained.
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// The elements in block A[offset:m, 0:n] below the diagonal, together with tau, represent
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// the orthogonal matrix Q as a product of elementary reflectors.
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//
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// offset is number of rows of the matrix A that must be pivoted but not factorized.
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// offset must not be negative otherwise Dlaqp2 will panic.
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//
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// On exit, jpvt holds the permutation that was applied; the jth column of A*P was the
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// jpvt[j] column of A. jpvt must have length n, otherwise Dlaqp2 will panic.
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//
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// On exit tau holds the scalar factors of the elementary reflectors. It must have length
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// at least min(m-offset, n) otherwise Dlaqp2 will panic.
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//
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// vn1 and vn2 hold the partial and complete column norms respectively. They must have length n,
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// otherwise Dlaqp2 will panic.
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//
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// work must have length n, otherwise Dlaqp2 will panic.
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//
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// Dlaqp2 is an internal routine. It is exported for testing purposes.
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func (impl Implementation) Dlaqp2(m, n, offset int, a []float64, lda int, jpvt []int, tau, vn1, vn2, work []float64) {
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switch {
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case m < 0:
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panic(mLT0)
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case n < 0:
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panic(nLT0)
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case offset < 0:
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panic(offsetLT0)
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case offset > m:
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panic(offsetGTM)
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case lda < max(1, n):
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panic(badLdA)
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}
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// Quick return if possible.
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if m == 0 || n == 0 {
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return
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}
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mn := min(m-offset, n)
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switch {
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case len(a) < (m-1)*lda+n:
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panic(shortA)
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case len(jpvt) != n:
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panic(badLenJpvt)
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case len(tau) < mn:
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panic(shortTau)
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case len(vn1) < n:
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panic(shortVn1)
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case len(vn2) < n:
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panic(shortVn2)
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case len(work) < n:
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panic(shortWork)
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}
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tol3z := math.Sqrt(dlamchE)
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bi := blas64.Implementation()
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// Compute factorization.
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for i := 0; i < mn; i++ {
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offpi := offset + i
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// Determine ith pivot column and swap if necessary.
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p := i + bi.Idamax(n-i, vn1[i:], 1)
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if p != i {
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bi.Dswap(m, a[p:], lda, a[i:], lda)
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jpvt[p], jpvt[i] = jpvt[i], jpvt[p]
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vn1[p] = vn1[i]
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vn2[p] = vn2[i]
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}
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// Generate elementary reflector H_i.
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if offpi < m-1 {
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a[offpi*lda+i], tau[i] = impl.Dlarfg(m-offpi, a[offpi*lda+i], a[(offpi+1)*lda+i:], lda)
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} else {
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tau[i] = 0
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}
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if i < n-1 {
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// Apply H_iᵀ to A[offset+i:m, i:n] from the left.
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aii := a[offpi*lda+i]
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a[offpi*lda+i] = 1
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impl.Dlarf(blas.Left, m-offpi, n-i-1, a[offpi*lda+i:], lda, tau[i], a[offpi*lda+i+1:], lda, work)
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a[offpi*lda+i] = aii
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}
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// Update partial column norms.
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for j := i + 1; j < n; j++ {
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if vn1[j] == 0 {
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continue
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}
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// The following marked lines follow from the
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// analysis in Lapack Working Note 176.
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r := math.Abs(a[offpi*lda+j]) / vn1[j] // *
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temp := math.Max(0, 1-r*r) // *
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r = vn1[j] / vn2[j] // *
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temp2 := temp * r * r // *
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if temp2 < tol3z {
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var v float64
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if offpi < m-1 {
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v = bi.Dnrm2(m-offpi-1, a[(offpi+1)*lda+j:], lda)
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}
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vn1[j] = v
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vn2[j] = v
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} else {
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vn1[j] *= math.Sqrt(temp) // *
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}
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}
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}
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}
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