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@vladimir-ch @btracey @kortschak
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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.
package mat
import (
"gonum.org/v1/gonum/blas/blas64"
"gonum.org/v1/gonum/lapack"
"gonum.org/v1/gonum/lapack/lapack64"
)
// SVD is a type for creating and using the Singular Value Decomposition (SVD)
// of a matrix.
type SVD struct {
kind SVDKind
s []float64
u blas64.General
vt blas64.General
}
// SVDKind specifies the treatment of singular vectors during an SVD
// factorization.
type SVDKind int
const (
// SVDNone specifies that no singular vectors should be computed during
// the decomposition.
SVDNone SVDKind = 0
// SVDThinU specifies the thin decomposition for U should be computed.
SVDThinU SVDKind = 1 << (iota - 1)
// SVDFullU specifies the full decomposition for U should be computed.
SVDFullU
// SVDThinV specifies the thin decomposition for V should be computed.
SVDThinV
// SVDFullV specifies the full decomposition for V should be computed.
SVDFullV
// SVDThin is a convenience value for computing both thin vectors.
SVDThin SVDKind = SVDThinU | SVDThinV
// SVDThin is a convenience value for computing both full vectors.
SVDFull SVDKind = SVDFullU | SVDFullV
)
// succFact returns whether the receiver contains a successful factorization.
func (svd *SVD) succFact() bool {
return len(svd.s) != 0
}
// Factorize computes the singular value decomposition (SVD) of the input matrix A.
// The singular values of A are computed in all cases, while the singular
// vectors are optionally computed depending on the input kind.
//
// The full singular value decomposition (kind == SVDFull) is a factorization
// of an m×n matrix A of the form
// A = U * Σ * Vᵀ
// where Σ is an m×n diagonal matrix, U is an m×m orthogonal matrix, and V is an
// n×n orthogonal matrix. The diagonal elements of Σ are the singular values of A.
// The first min(m,n) columns of U and V are, respectively, the left and right
// singular vectors of A.
//
// Significant storage space can be saved by using the thin representation of
// the SVD (kind == SVDThin) instead of the full SVD, especially if
// m >> n or m << n. The thin SVD finds
// A = U~ * Σ * V~ᵀ
// where U~ is of size m×min(m,n), Σ is a diagonal matrix of size min(m,n)×min(m,n)
// and V~ is of size n×min(m,n).
//
// Factorize returns whether the decomposition succeeded. If the decomposition
// failed, routines that require a successful factorization will panic.
func (svd *SVD) Factorize(a Matrix, kind SVDKind) (ok bool) {
// kill previous factorization
svd.s = svd.s[:0]
svd.kind = kind
m, n := a.Dims()
var jobU, jobVT lapack.SVDJob
// TODO(btracey): This code should be modified to have the smaller
// matrix written in-place into aCopy when the lapack/native/dgesvd
// implementation is complete.
switch {
case kind&SVDFullU != 0:
jobU = lapack.SVDAll
svd.u = blas64.General{
Rows: m,
Cols: m,
Stride: m,
Data: use(svd.u.Data, m*m),
}
case kind&SVDThinU != 0:
jobU = lapack.SVDStore
svd.u = blas64.General{
Rows: m,
Cols: min(m, n),
Stride: min(m, n),
Data: use(svd.u.Data, m*min(m, n)),
}
default:
jobU = lapack.SVDNone
}
switch {
case kind&SVDFullV != 0:
svd.vt = blas64.General{
Rows: n,
Cols: n,
Stride: n,
Data: use(svd.vt.Data, n*n),
}
jobVT = lapack.SVDAll
case kind&SVDThinV != 0:
svd.vt = blas64.General{
Rows: min(m, n),
Cols: n,
Stride: n,
Data: use(svd.vt.Data, min(m, n)*n),
}
jobVT = lapack.SVDStore
default:
jobVT = lapack.SVDNone
}
// A is destroyed on call, so copy the matrix.
aCopy := DenseCopyOf(a)
svd.kind = kind
svd.s = use(svd.s, min(m, n))
work := []float64{0}
lapack64.Gesvd(jobU, jobVT, aCopy.mat, svd.u, svd.vt, svd.s, work, -1)
work = getFloats(int(work[0]), false)
ok = lapack64.Gesvd(jobU, jobVT, aCopy.mat, svd.u, svd.vt, svd.s, work, len(work))
putFloats(work)
if !ok {
svd.kind = 0
}
return ok
}
// Kind returns the SVDKind of the decomposition. If no decomposition has been
// computed, Kind returns -1.
func (svd *SVD) Kind() SVDKind {
if !svd.succFact() {
return -1
}
return svd.kind
}
// Cond returns the 2-norm condition number for the factorized matrix. Cond will
// panic if the receiver does not contain a successful factorization.
func (svd *SVD) Cond() float64 {
if !svd.succFact() {
panic(badFact)
}
return svd.s[0] / svd.s[len(svd.s)-1]
}
// Values returns the singular values of the factorized matrix in descending order.
//
// If the input slice is non-nil, the values will be stored in-place into
// the slice. In this case, the slice must have length min(m,n), and Values will
// panic with ErrSliceLengthMismatch otherwise. If the input slice is nil, a new
// slice of the appropriate length will be allocated and returned.
//
// Values will panic if the receiver does not contain a successful factorization.
func (svd *SVD) Values(s []float64) []float64 {
if !svd.succFact() {
panic(badFact)
}
if s == nil {
s = make([]float64, len(svd.s))
}
if len(s) != len(svd.s) {
panic(ErrSliceLengthMismatch)
}
copy(s, svd.s)
return s
}
// UTo extracts the matrix U from the singular value decomposition. The first
// min(m,n) columns are the left singular vectors and correspond to the singular
// values as returned from SVD.Values.
//
// If dst is empty, UTo will resize dst to be m×m if the full U was computed
// and size m×min(m,n) if the thin U was computed. When dst is non-empty, then
// UTo will panic if dst is not the appropriate size. UTo will also panic if
// the receiver does not contain a successful factorization, or if U was
// not computed during factorization.
func (svd *SVD) UTo(dst *Dense) {
if !svd.succFact() {
panic(badFact)
}
kind := svd.kind
if kind&SVDThinU == 0 && kind&SVDFullU == 0 {
panic("svd: u not computed during factorization")
}
r := svd.u.Rows
c := svd.u.Cols
if dst.IsEmpty() {
dst.ReuseAs(r, c)
} else {
r2, c2 := dst.Dims()
if r != r2 || c != c2 {
panic(ErrShape)
}
}
tmp := &Dense{
mat: svd.u,
capRows: r,
capCols: c,
}
dst.Copy(tmp)
}
// VTo extracts the matrix V from the singular value decomposition. The first
// min(m,n) columns are the right singular vectors and correspond to the singular
// values as returned from SVD.Values.
//
// If dst is empty, VTo will resize dst to be n×n if the full V was computed
// and size n×min(m,n) if the thin V was computed. When dst is non-empty, then
// VTo will panic if dst is not the appropriate size. VTo will also panic if
// the receiver does not contain a successful factorization, or if V was
// not computed during factorization.
func (svd *SVD) VTo(dst *Dense) {
if !svd.succFact() {
panic(badFact)
}
kind := svd.kind
if kind&SVDThinU == 0 && kind&SVDFullV == 0 {
panic("svd: v not computed during factorization")
}
r := svd.vt.Rows
c := svd.vt.Cols
if dst.IsEmpty() {
dst.ReuseAs(c, r)
} else {
r2, c2 := dst.Dims()
if c != r2 || r != c2 {
panic(ErrShape)
}
}
tmp := &Dense{
mat: svd.vt,
capRows: r,
capCols: c,
}
dst.Copy(tmp.T())
}
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