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// 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,
//
// Uᵀ*A*Q = D1*[ 0 R ], Vᵀ*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:
//
// U1ᵀ*A13*Q1 = C1*R1; V1ᵀ*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
initu := jobU == lapack.GSVDUnit
wantu := initu || jobU == lapack.GSVDU
initv := jobV == lapack.GSVDUnit
wantv := initv || jobV == lapack.GSVDV
initq := jobQ == lapack.GSVDUnit
wantq := initq || jobQ == lapack.GSVDQ
switch {
case !initu && !wantu && jobU != lapack.GSVDNone:
panic(badGSVDJob + "U")
case !initv && !wantv && jobV != lapack.GSVDNone:
panic(badGSVDJob + "V")
case !initq && !wantq && jobQ != lapack.GSVDNone:
panic(badGSVDJob + "Q")
case m < 0:
panic(mLT0)
case p < 0:
panic(pLT0)
case n < 0:
panic(nLT0)
case lda < max(1, n):
panic(badLdA)
case len(a) < (m-1)*lda+n:
panic(shortA)
case ldb < max(1, n):
panic(badLdB)
case len(b) < (p-1)*ldb+n:
panic(shortB)
case len(alpha) != n:
panic(badLenAlpha)
case len(beta) != n:
panic(badLenBeta)
case ldu < 1, wantu && ldu < m:
panic(badLdU)
case wantu && len(u) < (m-1)*ldu+m:
panic(shortU)
case ldv < 1, wantv && ldv < p:
panic(badLdV)
case wantv && len(v) < (p-1)*ldv+p:
panic(shortV)
case ldq < 1, wantq && ldq < n:
panic(badLdQ)
case wantq && len(q) < (n-1)*ldq+n:
panic(shortQ)
case len(work) < 2*n:
panic(shortWork)
}
// 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: Uᵀ*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: Vᵀ*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
}