--- /dev/null
+// Copyright ©2017 The Gonum Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+package gonum
+
+import (
+ "math"
+
+ "gonum.org/v1/gonum/blas"
+ "gonum.org/v1/gonum/blas/blas64"
+)
+
+// Dlaqp2 computes a QR factorization with column pivoting of the block A[offset:m, 0:n]
+// of the m×n matrix A. The block A[0:offset, 0:n] is accordingly pivoted, but not factorized.
+//
+// On exit, the upper triangle of block A[offset:m, 0:n] is the triangular factor obtained.
+// The elements in block A[offset:m, 0:n] below the diagonal, together with tau, represent
+// the orthogonal matrix Q as a product of elementary reflectors.
+//
+// offset is number of rows of the matrix A that must be pivoted but not factorized.
+// offset must not be negative otherwise Dlaqp2 will panic.
+//
+// On exit, jpvt holds the permutation that was applied; the jth column of A*P was the
+// jpvt[j] column of A. jpvt must have length n, otherwise Dlaqp2 will panic.
+//
+// On exit tau holds the scalar factors of the elementary reflectors. It must have length
+// at least min(m-offset, n) otherwise Dlaqp2 will panic.
+//
+// vn1 and vn2 hold the partial and complete column norms respectively. They must have length n,
+// otherwise Dlaqp2 will panic.
+//
+// work must have length n, otherwise Dlaqp2 will panic.
+//
+// Dlaqp2 is an internal routine. It is exported for testing purposes.
+func (impl Implementation) Dlaqp2(m, n, offset int, a []float64, lda int, jpvt []int, tau, vn1, vn2, work []float64) {
+ checkMatrix(m, n, a, lda)
+ if len(jpvt) != n {
+ panic(badIpiv)
+ }
+ mn := min(m-offset, n)
+ if len(tau) < mn {
+ panic(badTau)
+ }
+ if len(vn1) < n {
+ panic(badVn1)
+ }
+ if len(vn2) < n {
+ panic(badVn2)
+ }
+ if len(work) < n {
+ panic(badWork)
+ }
+
+ tol3z := math.Sqrt(dlamchE)
+
+ bi := blas64.Implementation()
+
+ // Compute factorization.
+ for i := 0; i < mn; i++ {
+ offpi := offset + i
+
+ // Determine ith pivot column and swap if necessary.
+ p := i + bi.Idamax(n-i, vn1[i:], 1)
+ if p != i {
+ bi.Dswap(m, a[p:], lda, a[i:], lda)
+ jpvt[p], jpvt[i] = jpvt[i], jpvt[p]
+ vn1[p] = vn1[i]
+ vn2[p] = vn2[i]
+ }
+
+ // Generate elementary reflector H_i.
+ if offpi < m-1 {
+ a[offpi*lda+i], tau[i] = impl.Dlarfg(m-offpi, a[offpi*lda+i], a[(offpi+1)*lda+i:], lda)
+ } else {
+ tau[i] = 0
+ }
+
+ if i < n-1 {
+ // Apply H_i^T to A[offset+i:m, i:n] from the left.
+ aii := a[offpi*lda+i]
+ a[offpi*lda+i] = 1
+ impl.Dlarf(blas.Left, m-offpi, n-i-1, a[offpi*lda+i:], lda, tau[i], a[offpi*lda+i+1:], lda, work)
+ a[offpi*lda+i] = aii
+ }
+
+ // Update partial column norms.
+ for j := i + 1; j < n; j++ {
+ if vn1[j] == 0 {
+ continue
+ }
+
+ // The following marked lines follow from the
+ // analysis in Lapack Working Note 176.
+ r := math.Abs(a[offpi*lda+j]) / vn1[j] // *
+ temp := math.Max(0, 1-r*r) // *
+ r = vn1[j] / vn2[j] // *
+ temp2 := temp * r * r // *
+ if temp2 < tol3z {
+ var v float64
+ if offpi < m-1 {
+ v = bi.Dnrm2(m-offpi-1, a[(offpi+1)*lda+j:], lda)
+ }
+ vn1[j] = v
+ vn2[j] = v
+ } else {
+ vn1[j] *= math.Sqrt(temp) // *
+ }
+ }
+ }
+}
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