+++ /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"
- "gonum.org/v1/gonum/lapack"
-)
-
-// Dtgsja computes the generalized singular value decomposition (GSVD)
-// of two real upper triangular or trapezoidal matrices A and B.
-//
-// A and B have the following forms, which may be obtained by the
-// preprocessing subroutine Dggsvp from a general m×n matrix A and p×n
-// matrix B:
-//
-// n-k-l k l
-// A = k [ 0 A12 A13 ] if m-k-l >= 0;
-// l [ 0 0 A23 ]
-// m-k-l [ 0 0 0 ]
-//
-// n-k-l k l
-// A = k [ 0 A12 A13 ] if m-k-l < 0;
-// m-k [ 0 0 A23 ]
-//
-// n-k-l k l
-// B = l [ 0 0 B13 ]
-// p-l [ 0 0 0 ]
-//
-// where the k×k matrix A12 and l×l matrix B13 are non-singular
-// upper triangular. A23 is l×l upper triangular if m-k-l >= 0,
-// otherwise A23 is (m-k)×l upper trapezoidal.
-//
-// On exit,
-//
-// U^T*A*Q = D1*[ 0 R ], V^T*B*Q = D2*[ 0 R ],
-//
-// where U, V and Q are orthogonal matrices.
-// R is a non-singular upper triangular matrix, and D1 and D2 are
-// diagonal matrices, which are of the following structures:
-//
-// If m-k-l >= 0,
-//
-// k l
-// D1 = k [ I 0 ]
-// l [ 0 C ]
-// m-k-l [ 0 0 ]
-//
-// k l
-// D2 = l [ 0 S ]
-// p-l [ 0 0 ]
-//
-// n-k-l k l
-// [ 0 R ] = k [ 0 R11 R12 ] k
-// l [ 0 0 R22 ] l
-//
-// where
-//
-// C = diag( alpha_k, ... , alpha_{k+l} ),
-// S = diag( beta_k, ... , beta_{k+l} ),
-// C^2 + S^2 = I.
-//
-// R is stored in
-// A[0:k+l, n-k-l:n]
-// on exit.
-//
-// If m-k-l < 0,
-//
-// k m-k k+l-m
-// D1 = k [ I 0 0 ]
-// m-k [ 0 C 0 ]
-//
-// k m-k k+l-m
-// D2 = m-k [ 0 S 0 ]
-// k+l-m [ 0 0 I ]
-// p-l [ 0 0 0 ]
-//
-// n-k-l k m-k k+l-m
-// [ 0 R ] = k [ 0 R11 R12 R13 ]
-// m-k [ 0 0 R22 R23 ]
-// k+l-m [ 0 0 0 R33 ]
-//
-// where
-// C = diag( alpha_k, ... , alpha_m ),
-// S = diag( beta_k, ... , beta_m ),
-// C^2 + S^2 = I.
-//
-// R = [ R11 R12 R13 ] is stored in A[0:m, n-k-l:n]
-// [ 0 R22 R23 ]
-// and R33 is stored in
-// B[m-k:l, n+m-k-l:n] on exit.
-//
-// The computation of the orthogonal transformation matrices U, V or Q
-// is optional. These matrices may either be formed explicitly, or they
-// may be post-multiplied into input matrices U1, V1, or Q1.
-//
-// Dtgsja essentially uses a variant of Kogbetliantz algorithm to reduce
-// min(l,m-k)×l triangular or trapezoidal matrix A23 and l×l
-// matrix B13 to the form:
-//
-// U1^T*A13*Q1 = C1*R1; V1^T*B13*Q1 = S1*R1,
-//
-// where U1, V1 and Q1 are orthogonal matrices. C1 and S1 are diagonal
-// matrices satisfying
-//
-// C1^2 + S1^2 = I,
-//
-// and R1 is an l×l non-singular upper triangular matrix.
-//
-// jobU, jobV and jobQ are options for computing the orthogonal matrices. The behavior
-// is as follows
-// jobU == lapack.GSVDU Compute orthogonal matrix U
-// jobU == lapack.GSVDUnit Use unit-initialized matrix
-// jobU == lapack.GSVDNone Do not compute orthogonal matrix.
-// The behavior is the same for jobV and jobQ with the exception that instead of
-// lapack.GSVDU these accept lapack.GSVDV and lapack.GSVDQ respectively.
-// The matrices U, V and Q must be m×m, p×p and n×n respectively unless the
-// relevant job parameter is lapack.GSVDNone.
-//
-// k and l specify the sub-blocks in the input matrices A and B:
-// A23 = A[k:min(k+l,m), n-l:n) and B13 = B[0:l, n-l:n]
-// of A and B, whose GSVD is going to be computed by Dtgsja.
-//
-// tola and tolb are the convergence criteria for the Jacobi-Kogbetliantz
-// iteration procedure. Generally, they are the same as used in the preprocessing
-// step, for example,
-// tola = max(m, n)*norm(A)*eps,
-// tolb = max(p, n)*norm(B)*eps,
-// where eps is the machine epsilon.
-//
-// work must have length at least 2*n, otherwise Dtgsja will panic.
-//
-// alpha and beta must have length n or Dtgsja will panic. On exit, alpha and
-// beta contain the generalized singular value pairs of A and B
-// alpha[0:k] = 1,
-// beta[0:k] = 0,
-// if m-k-l >= 0,
-// alpha[k:k+l] = diag(C),
-// beta[k:k+l] = diag(S),
-// if m-k-l < 0,
-// alpha[k:m]= C, alpha[m:k+l]= 0
-// beta[k:m] = S, beta[m:k+l] = 1.
-// if k+l < n,
-// alpha[k+l:n] = 0 and
-// beta[k+l:n] = 0.
-//
-// On exit, A[n-k:n, 0:min(k+l,m)] contains the triangular matrix R or part of R
-// and if necessary, B[m-k:l, n+m-k-l:n] contains a part of R.
-//
-// Dtgsja returns whether the routine converged and the number of iteration cycles
-// that were run.
-//
-// Dtgsja is an internal routine. It is exported for testing purposes.
-func (impl Implementation) Dtgsja(jobU, jobV, jobQ lapack.GSVDJob, m, p, n, k, l int, a []float64, lda int, b []float64, ldb int, tola, tolb float64, alpha, beta, u []float64, ldu int, v []float64, ldv int, q []float64, ldq int, work []float64) (cycles int, ok bool) {
- const maxit = 40
-
- checkMatrix(m, n, a, lda)
- checkMatrix(p, n, b, ldb)
-
- if len(alpha) != n {
- panic(badAlpha)
- }
- if len(beta) != n {
- panic(badBeta)
- }
-
- initu := jobU == lapack.GSVDUnit
- wantu := initu || jobU == lapack.GSVDU
- if !initu && !wantu && jobU != lapack.GSVDNone {
- panic(badGSVDJob + "U")
- }
- if jobU != lapack.GSVDNone {
- checkMatrix(m, m, u, ldu)
- }
-
- initv := jobV == lapack.GSVDUnit
- wantv := initv || jobV == lapack.GSVDV
- if !initv && !wantv && jobV != lapack.GSVDNone {
- panic(badGSVDJob + "V")
- }
- if jobV != lapack.GSVDNone {
- checkMatrix(p, p, v, ldv)
- }
-
- initq := jobQ == lapack.GSVDUnit
- wantq := initq || jobQ == lapack.GSVDQ
- if !initq && !wantq && jobQ != lapack.GSVDNone {
- panic(badGSVDJob + "Q")
- }
- if jobQ != lapack.GSVDNone {
- checkMatrix(n, n, q, ldq)
- }
-
- if len(work) < 2*n {
- panic(badWork)
- }
-
- // Initialize U, V and Q, if necessary
- if initu {
- impl.Dlaset(blas.All, m, m, 0, 1, u, ldu)
- }
- if initv {
- impl.Dlaset(blas.All, p, p, 0, 1, v, ldv)
- }
- if initq {
- impl.Dlaset(blas.All, n, n, 0, 1, q, ldq)
- }
-
- bi := blas64.Implementation()
- minTol := math.Min(tola, tolb)
-
- // Loop until convergence.
- upper := false
- for cycles = 1; cycles <= maxit; cycles++ {
- upper = !upper
-
- for i := 0; i < l-1; i++ {
- for j := i + 1; j < l; j++ {
- var a1, a2, a3 float64
- if k+i < m {
- a1 = a[(k+i)*lda+n-l+i]
- }
- if k+j < m {
- a3 = a[(k+j)*lda+n-l+j]
- }
-
- b1 := b[i*ldb+n-l+i]
- b3 := b[j*ldb+n-l+j]
-
- var b2 float64
- if upper {
- if k+i < m {
- a2 = a[(k+i)*lda+n-l+j]
- }
- b2 = b[i*ldb+n-l+j]
- } else {
- if k+j < m {
- a2 = a[(k+j)*lda+n-l+i]
- }
- b2 = b[j*ldb+n-l+i]
- }
-
- csu, snu, csv, snv, csq, snq := impl.Dlags2(upper, a1, a2, a3, b1, b2, b3)
-
- // Update (k+i)-th and (k+j)-th rows of matrix A: U^T*A.
- if k+j < m {
- bi.Drot(l, a[(k+j)*lda+n-l:], 1, a[(k+i)*lda+n-l:], 1, csu, snu)
- }
-
- // Update i-th and j-th rows of matrix B: V^T*B.
- bi.Drot(l, b[j*ldb+n-l:], 1, b[i*ldb+n-l:], 1, csv, snv)
-
- // Update (n-l+i)-th and (n-l+j)-th columns of matrices
- // A and B: A*Q and B*Q.
- bi.Drot(min(k+l, m), a[n-l+j:], lda, a[n-l+i:], lda, csq, snq)
- bi.Drot(l, b[n-l+j:], ldb, b[n-l+i:], ldb, csq, snq)
-
- if upper {
- if k+i < m {
- a[(k+i)*lda+n-l+j] = 0
- }
- b[i*ldb+n-l+j] = 0
- } else {
- if k+j < m {
- a[(k+j)*lda+n-l+i] = 0
- }
- b[j*ldb+n-l+i] = 0
- }
-
- // Update orthogonal matrices U, V, Q, if desired.
- if wantu && k+j < m {
- bi.Drot(m, u[k+j:], ldu, u[k+i:], ldu, csu, snu)
- }
- if wantv {
- bi.Drot(p, v[j:], ldv, v[i:], ldv, csv, snv)
- }
- if wantq {
- bi.Drot(n, q[n-l+j:], ldq, q[n-l+i:], ldq, csq, snq)
- }
- }
- }
-
- if !upper {
- // The matrices A13 and B13 were lower triangular at the start
- // of the cycle, and are now upper triangular.
- //
- // Convergence test: test the parallelism of the corresponding
- // rows of A and B.
- var error float64
- for i := 0; i < min(l, m-k); i++ {
- bi.Dcopy(l-i, a[(k+i)*lda+n-l+i:], 1, work, 1)
- bi.Dcopy(l-i, b[i*ldb+n-l+i:], 1, work[l:], 1)
- ssmin := impl.Dlapll(l-i, work, 1, work[l:], 1)
- error = math.Max(error, ssmin)
- }
- if math.Abs(error) <= minTol {
- // The algorithm has converged.
- // Compute the generalized singular value pairs (alpha, beta)
- // and set the triangular matrix R to array A.
- for i := 0; i < k; i++ {
- alpha[i] = 1
- beta[i] = 0
- }
-
- for i := 0; i < min(l, m-k); i++ {
- a1 := a[(k+i)*lda+n-l+i]
- b1 := b[i*ldb+n-l+i]
-
- if a1 != 0 {
- gamma := b1 / a1
-
- // Change sign if necessary.
- if gamma < 0 {
- bi.Dscal(l-i, -1, b[i*ldb+n-l+i:], 1)
- if wantv {
- bi.Dscal(p, -1, v[i:], ldv)
- }
- }
- beta[k+i], alpha[k+i], _ = impl.Dlartg(math.Abs(gamma), 1)
-
- if alpha[k+i] >= beta[k+i] {
- bi.Dscal(l-i, 1/alpha[k+i], a[(k+i)*lda+n-l+i:], 1)
- } else {
- bi.Dscal(l-i, 1/beta[k+i], b[i*ldb+n-l+i:], 1)
- bi.Dcopy(l-i, b[i*ldb+n-l+i:], 1, a[(k+i)*lda+n-l+i:], 1)
- }
- } else {
- alpha[k+i] = 0
- beta[k+i] = 1
- bi.Dcopy(l-i, b[i*ldb+n-l+i:], 1, a[(k+i)*lda+n-l+i:], 1)
- }
- }
-
- for i := m; i < k+l; i++ {
- alpha[i] = 0
- beta[i] = 1
- }
- if k+l < n {
- for i := k + l; i < n; i++ {
- alpha[i] = 0
- beta[i] = 0
- }
- }
-
- return cycles, true
- }
- }
- }
-
- // The algorithm has not converged after maxit cycles.
- return cycles, false
-}
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