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qr.go
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/
qr.go
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// Copyright ©2013 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.
// Based on the QRDecomposition class from Jama 1.0.3.
package mat64
import (
"math"
)
type QRFactor struct {
QR *Dense
rDiag []float64
}
// QR computes a QR Decomposition for an m-by-n matrix a with m >= n by Householder
// reflections, the QR decomposition is an m-by-n orthogonal matrix q and an n-by-n
// upper triangular matrix r so that a = q.r. QR will panic with ErrShape if m < n.
//
// The QR decomposition always exists, even if the matrix does not have full rank,
// so QR will never fail unless m < n. The primary use of the QR decomposition is
// in the least squares solution of non-square systems of simultaneous linear equations.
// This will fail if QRIsFullRank() returns false. The matrix a is overwritten by the
// decomposition.
func QR(a *Dense) QRFactor {
// Initialize.
m, n := a.Dims()
if m < n {
panic(ErrShape)
}
qr := a
rDiag := make([]float64, n)
// Main loop.
for k := 0; k < n; k++ {
// Compute 2-norm of k-th column without under/overflow.
var norm float64
for i := k; i < m; i++ {
norm = math.Hypot(norm, qr.at(i, k))
}
if norm != 0 {
// Form k-th Householder vector.
if qr.at(k, k) < 0 {
norm = -norm
}
for i := k; i < m; i++ {
qr.set(i, k, qr.at(i, k)/norm)
}
qr.set(k, k, qr.at(k, k)+1)
// Apply transformation to remaining columns.
for j := k + 1; j < n; j++ {
var s float64
for i := k; i < m; i++ {
s += qr.at(i, k) * qr.at(i, j)
}
s /= -qr.at(k, k)
for i := k; i < m; i++ {
qr.set(i, j, qr.at(i, j)+s*qr.at(i, k))
}
}
}
rDiag[k] = -norm
}
return QRFactor{qr, rDiag}
}
// IsFullRank returns whether the R matrix and hence a has full rank.
func (f QRFactor) IsFullRank() bool {
for _, v := range f.rDiag {
if v == 0 {
return false
}
}
return true
}
// H returns the Householder vectors in a lower trapezoidal matrix
// whose columns define the reflections.
func (f QRFactor) H() *Dense {
qr := f.QR
m, n := qr.Dims()
h := NewDense(m, n, nil)
for i := 0; i < m; i++ {
for j := 0; j < n; j++ {
if i >= j {
h.set(i, j, qr.at(i, j))
}
}
}
return h
}
// R returns the upper triangular factor for the QR decomposition.
func (f QRFactor) R() *Dense {
qr, rDiag := f.QR, f.rDiag
_, n := qr.Dims()
r := NewDense(n, n, nil)
for i, v := range rDiag[:n] {
for j := 0; j < n; j++ {
if i < j {
r.set(i, j, qr.at(i, j))
} else if i == j {
r.set(i, j, v)
}
}
}
return r
}
// Q generates and returns the (economy-sized) orthogonal factor.
func (f QRFactor) Q() *Dense {
qr := f.QR
m, n := qr.Dims()
q := NewDense(m, n, nil)
for k := n - 1; k >= 0; k-- {
q.set(k, k, 1)
for j := k; j < n; j++ {
if qr.at(k, k) != 0 {
var s float64
for i := k; i < m; i++ {
s += qr.at(i, k) * q.at(i, j)
}
s /= -qr.at(k, k)
for i := k; i < m; i++ {
q.set(i, j, q.at(i, j)+s*qr.at(i, k))
}
}
}
}
return q
}
// Solve computes a least squares solution of a.x = b where b has as many rows as a.
// A matrix x is returned that minimizes the two norm of Q*R*X-B. Solve will panic
// if a is not full rank. The matrix b is overwritten during the call.
func (f QRFactor) Solve(b *Dense) (x *Dense) {
qr := f.QR
rDiag := f.rDiag
m, n := qr.Dims()
bm, bn := b.Dims()
if bm != m {
panic(ErrShape)
}
if !f.IsFullRank() {
panic(ErrSingular)
}
// Compute Y = transpose(Q)*B
for k := 0; k < n; k++ {
for j := 0; j < bn; j++ {
var s float64
for i := k; i < m; i++ {
s += qr.at(i, k) * b.at(i, j)
}
s /= -qr.at(k, k)
for i := k; i < m; i++ {
b.set(i, j, b.at(i, j)+s*qr.at(i, k))
}
}
}
// Solve R*X = Y;
for k := n - 1; k >= 0; k-- {
row := b.rowView(k)
for j := range row[:bn] {
row[j] /= rDiag[k]
}
for i := 0; i < k; i++ {
row := b.rowView(i)
for j := range row[:bn] {
row[j] -= b.at(k, j) * qr.at(i, k)
}
}
}
return b.View(0, 0, n, bn).(*Dense)
}