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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 }