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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/lapack" ) // Dggsvp3 computes orthogonal matrices U, V and Q such that // // n-k-l k l // Uᵀ*A*Q = k [ 0 A12 A13 ] if m-k-l >= 0; // l [ 0 0 A23 ] // m-k-l [ 0 0 0 ] // // n-k-l k l // Uᵀ*A*Q = k [ 0 A12 A13 ] if m-k-l < 0; // m-k [ 0 0 A23 ] // // n-k-l k l // Vᵀ*B*Q = 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. // // Dggsvp3 returns k and l, the dimensions of the sub-blocks. k+l // is the effective numerical rank of the (m+p)×n matrix [ Aᵀ Bᵀ ]ᵀ. // // 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.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. // // 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. // // iwork must have length n, work must have length at least max(1, lwork), and // lwork must be -1 or greater than zero, otherwise Dggsvp3 will panic. // // Dggsvp3 is an internal routine. It is exported for testing purposes. func (impl Implementation) Dggsvp3(jobU, jobV, jobQ lapack.GSVDJob, m, p, n int, a []float64, lda int, b []float64, ldb int, tola, tolb float64, u []float64, ldu int, v []float64, ldv int, q []float64, ldq int, iwork []int, tau, work []float64, lwork int) (k, l int) { wantu := jobU == lapack.GSVDU wantv := jobV == lapack.GSVDV wantq := jobQ == lapack.GSVDQ switch { case !wantu && jobU != lapack.GSVDNone: panic(badGSVDJob + "U") case !wantv && jobV != lapack.GSVDNone: panic(badGSVDJob + "V") case !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 ldb < max(1, n): panic(badLdB) case ldu < 1, wantu && ldu < m: panic(badLdU) case ldv < 1, wantv && ldv < p: panic(badLdV) case ldq < 1, wantq && ldq < n: panic(badLdQ) case len(iwork) != n: panic(shortWork) case lwork < 1 && lwork != -1: panic(badLWork) case len(work) < max(1, lwork): panic(shortWork) } var lwkopt int impl.Dgeqp3(p, n, b, ldb, iwork, tau, work, -1) lwkopt = int(work[0]) if wantv { lwkopt = max(lwkopt, p) } lwkopt = max(lwkopt, min(n, p)) lwkopt = max(lwkopt, m) if wantq { lwkopt = max(lwkopt, n) } impl.Dgeqp3(m, n, a, lda, iwork, tau, work, -1) lwkopt = max(lwkopt, int(work[0])) lwkopt = max(1, lwkopt) if lwork == -1 { work[0] = float64(lwkopt) return 0, 0 } switch { case len(a) < (m-1)*lda+n: panic(shortA) case len(b) < (p-1)*ldb+n: panic(shortB) case wantu && len(u) < (m-1)*ldu+m: panic(shortU) case wantv && len(v) < (p-1)*ldv+p: panic(shortV) case wantq && len(q) < (n-1)*ldq+n: panic(shortQ) case len(tau) < n: // tau check must come after lwkopt query since // the Dggsvd3 call for lwkopt query may have // lwork == -1, and tau is provided by work. panic(shortTau) } const forward = true // QR with column pivoting of B: B*P = V*[ S11 S12 ]. // [ 0 0 ] for i := range iwork[:n] { iwork[i] = 0 } impl.Dgeqp3(p, n, b, ldb, iwork, tau, work, lwork) // Update A := A*P. impl.Dlapmt(forward, m, n, a, lda, iwork) // Determine the effective rank of matrix B. for i := 0; i < min(p, n); i++ { if math.Abs(b[i*ldb+i]) > tolb { l++ } } if wantv { // Copy the details of V, and form V. impl.Dlaset(blas.All, p, p, 0, 0, v, ldv) if p > 1 { impl.Dlacpy(blas.Lower, p-1, min(p, n), b[ldb:], ldb, v[ldv:], ldv) } impl.Dorg2r(p, p, min(p, n), v, ldv, tau, work) } // Clean up B. for i := 1; i < l; i++ { r := b[i*ldb : i*ldb+i] for j := range r { r[j] = 0 } } if p > l { impl.Dlaset(blas.All, p-l, n, 0, 0, b[l*ldb:], ldb) } if wantq { // Set Q = I and update Q := Q*P. impl.Dlaset(blas.All, n, n, 0, 1, q, ldq) impl.Dlapmt(forward, n, n, q, ldq, iwork) } if p >= l && n != l { // RQ factorization of [ S11 S12 ]: [ S11 S12 ] = [ 0 S12 ]*Z. impl.Dgerq2(l, n, b, ldb, tau, work) // Update A := A*Zᵀ. impl.Dormr2(blas.Right, blas.Trans, m, n, l, b, ldb, tau, a, lda, work) if wantq { // Update Q := Q*Zᵀ. impl.Dormr2(blas.Right, blas.Trans, n, n, l, b, ldb, tau, q, ldq, work) } // Clean up B. impl.Dlaset(blas.All, l, n-l, 0, 0, b, ldb) for i := 1; i < l; i++ { r := b[i*ldb+n-l : i*ldb+i+n-l] for j := range r { r[j] = 0 } } } // Let N-L L // A = [ A11 A12 ] M, // // then the following does the complete QR decomposition of A11: // // A11 = U*[ 0 T12 ]*P1ᵀ. // [ 0 0 ] for i := range iwork[:n-l] { iwork[i] = 0 } impl.Dgeqp3(m, n-l, a, lda, iwork[:n-l], tau, work, lwork) // Determine the effective rank of A11. for i := 0; i < min(m, n-l); i++ { if math.Abs(a[i*lda+i]) > tola { k++ } } // Update A12 := Uᵀ*A12, where A12 = A[0:m, n-l:n]. impl.Dorm2r(blas.Left, blas.Trans, m, l, min(m, n-l), a, lda, tau, a[n-l:], lda, work) if wantu { // Copy the details of U, and form U. impl.Dlaset(blas.All, m, m, 0, 0, u, ldu) if m > 1 { impl.Dlacpy(blas.Lower, m-1, min(m, n-l), a[lda:], lda, u[ldu:], ldu) } impl.Dorg2r(m, m, min(m, n-l), u, ldu, tau, work) } if wantq { // Update Q[0:n, 0:n-l] := Q[0:n, 0:n-l]*P1. impl.Dlapmt(forward, n, n-l, q, ldq, iwork[:n-l]) } // Clean up A: set the strictly lower triangular part of // A[0:k, 0:k] = 0, and A[k:m, 0:n-l] = 0. for i := 1; i < k; i++ { r := a[i*lda : i*lda+i] for j := range r { r[j] = 0 } } if m > k { impl.Dlaset(blas.All, m-k, n-l, 0, 0, a[k*lda:], lda) } if n-l > k { // RQ factorization of [ T11 T12 ] = [ 0 T12 ]*Z1. impl.Dgerq2(k, n-l, a, lda, tau, work) if wantq { // Update Q[0:n, 0:n-l] := Q[0:n, 0:n-l]*Z1ᵀ. impl.Dorm2r(blas.Right, blas.Trans, n, n-l, k, a, lda, tau, q, ldq, work) } // Clean up A. impl.Dlaset(blas.All, k, n-l-k, 0, 0, a, lda) for i := 1; i < k; i++ { r := a[i*lda+n-k-l : i*lda+i+n-k-l] for j := range r { a[j] = 0 } } } if m > k { // QR factorization of A[k:m, n-l:n]. impl.Dgeqr2(m-k, l, a[k*lda+n-l:], lda, tau, work) if wantu { // Update U[:, k:m) := U[:, k:m]*U1. impl.Dorm2r(blas.Right, blas.NoTrans, m, m-k, min(m-k, l), a[k*lda+n-l:], lda, tau, u[k:], ldu, work) } // Clean up A. for i := k + 1; i < m; i++ { r := a[i*lda+n-l : i*lda+min(n-l+i-k, n)] for j := range r { r[j] = 0 } } } work[0] = float64(lwkopt) return k, l }