98 lines
3.2 KiB
Go
98 lines
3.2 KiB
Go
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// Copyright ©2016 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 "gonum.org/v1/gonum/blas"
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// Dgehd2 reduces a block of a general n×n matrix A to upper Hessenberg form H
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// by an orthogonal similarity transformation Q^T * A * Q = H.
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//
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// The matrix Q is represented as a product of (ihi-ilo) elementary
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// reflectors
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// Q = H_{ilo} H_{ilo+1} ... H_{ihi-1}.
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// Each H_i has the form
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// H_i = I - tau[i] * v * v^T
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// where v is a real vector with v[0:i+1] = 0, v[i+1] = 1 and v[ihi+1:n] = 0.
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// v[i+2:ihi+1] is stored on exit in A[i+2:ihi+1,i].
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//
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// On entry, a contains the n×n general matrix to be reduced. On return, the
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// upper triangle and the first subdiagonal of A are overwritten with the upper
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// Hessenberg matrix H, and the elements below the first subdiagonal, with the
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// slice tau, represent the orthogonal matrix Q as a product of elementary
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// reflectors.
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//
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// The contents of A are illustrated by the following example, with n = 7, ilo =
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// 1 and ihi = 5.
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// On entry,
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// [ a a a a a a a ]
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// [ a a a a a a ]
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// [ a a a a a a ]
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// [ a a a a a a ]
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// [ a a a a a a ]
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// [ a a a a a a ]
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// [ a ]
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// on return,
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// [ a a h h h h a ]
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// [ a h h h h a ]
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// [ h h h h h h ]
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// [ v1 h h h h h ]
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// [ v1 v2 h h h h ]
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// [ v1 v2 v3 h h h ]
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// [ a ]
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// where a denotes an element of the original matrix A, h denotes a
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// modified element of the upper Hessenberg matrix H, and vi denotes an
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// element of the vector defining H_i.
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//
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// ilo and ihi determine the block of A that will be reduced to upper Hessenberg
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// form. It must hold that 0 <= ilo <= ihi <= max(0, n-1), otherwise Dgehd2 will
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// panic.
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//
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// On return, tau will contain the scalar factors of the elementary reflectors.
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// It must have length equal to n-1, otherwise Dgehd2 will panic.
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//
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// work must have length at least n, otherwise Dgehd2 will panic.
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//
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// Dgehd2 is an internal routine. It is exported for testing purposes.
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func (impl Implementation) Dgehd2(n, ilo, ihi int, a []float64, lda int, tau, work []float64) {
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switch {
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case n < 0:
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panic(nLT0)
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case ilo < 0 || max(0, n-1) < ilo:
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panic(badIlo)
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case ihi < min(ilo, n-1) || n <= ihi:
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panic(badIhi)
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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 n == 0 {
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return
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}
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switch {
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case len(a) < (n-1)*lda+n:
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panic(shortA)
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case len(tau) != n-1:
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panic(badLenTau)
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case len(work) < n:
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panic(shortWork)
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}
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for i := ilo; i < ihi; i++ {
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// Compute elementary reflector H_i to annihilate A[i+2:ihi+1,i].
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var aii float64
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aii, tau[i] = impl.Dlarfg(ihi-i, a[(i+1)*lda+i], a[min(i+2, n-1)*lda+i:], lda)
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a[(i+1)*lda+i] = 1
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// Apply H_i to A[0:ihi+1,i+1:ihi+1] from the right.
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impl.Dlarf(blas.Right, ihi+1, ihi-i, a[(i+1)*lda+i:], lda, tau[i], a[i+1:], lda, work)
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// Apply H_i to A[i+1:ihi+1,i+1:n] from the left.
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impl.Dlarf(blas.Left, ihi-i, n-i-1, a[(i+1)*lda+i:], lda, tau[i], a[(i+1)*lda+i+1:], lda, work)
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a[(i+1)*lda+i] = aii
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}
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}
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