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+// matrix/jama-svd.h
+
+// Copyright 2009-2011 Microsoft Corporation
+
+// See ../../COPYING for clarification regarding multiple authors
+//
+// Licensed under the Apache License, Version 2.0 (the "License");
+// you may not use this file except in compliance with the License.
+// You may obtain a copy of the License at
+//
+// http://www.apache.org/licenses/LICENSE-2.0
+//
+// THIS CODE IS PROVIDED *AS IS* BASIS, WITHOUT WARRANTIES OR CONDITIONS OF ANY
+// KIND, EITHER EXPRESS OR IMPLIED, INCLUDING WITHOUT LIMITATION ANY IMPLIED
+// WARRANTIES OR CONDITIONS OF TITLE, FITNESS FOR A PARTICULAR PURPOSE,
+// MERCHANTABLITY OR NON-INFRINGEMENT.
+// See the Apache 2 License for the specific language governing permissions and
+// limitations under the License.
+
+// This file consists of a port and modification of materials from
+// JAMA: A Java Matrix Package
+// under the following notice: This software is a cooperative product of
+// The MathWorks and the National Institute of Standards and Technology (NIST)
+// which has been released to the public. This notice and the original code are
+// available at http://math.nist.gov/javanumerics/jama/domain.notice
+
+
+#ifndef KALDI_MATRIX_JAMA_SVD_H_
+#define KALDI_MATRIX_JAMA_SVD_H_ 1
+
+
+#include "matrix/kaldi-matrix.h"
+#include "matrix/sp-matrix.h"
+#include "matrix/cblas-wrappers.h"
+
+namespace kaldi {
+
+#if defined(HAVE_ATLAS) || defined(USE_KALDI_SVD)
+// using ATLAS as our math library, which doesn't have SVD -> need
+// to implement it.
+
+// This routine is a modified form of jama_svd.h which is part of the TNT distribution.
+// (originally comes from JAMA).
+
+/** Singular Value Decomposition.
+ * <P>
+ * For an m-by-n matrix A with m >= n, the singular value decomposition is
+ * an m-by-n orthogonal matrix U, an n-by-n diagonal matrix S, and
+ * an n-by-n orthogonal matrix V so that A = U*S*V'.
+ * <P>
+ * The singular values, sigma[k] = S(k, k), are ordered so that
+ * sigma[0] >= sigma[1] >= ... >= sigma[n-1].
+ * <P>
+ * The singular value decompostion always exists, so the constructor will
+ * never fail. The matrix condition number and the effective numerical
+ * rank can be computed from this decomposition.
+
+ * <p>
+ * (Adapted from JAMA, a Java Matrix Library, developed by jointly
+ * by the Mathworks and NIST; see http://math.nist.gov/javanumerics/jama).
+ */
+
+
+template<typename Real>
+bool MatrixBase<Real>::JamaSvd(VectorBase<Real> *s_in,
+ MatrixBase<Real> *U_in,
+ MatrixBase<Real> *V_in) { // Destructive!
+ KALDI_ASSERT(s_in != NULL && U_in != this && V_in != this);
+ int wantu = (U_in != NULL), wantv = (V_in != NULL);
+ Matrix<Real> Utmp, Vtmp;
+ MatrixBase<Real> &U = (U_in ? *U_in : Utmp), &V = (V_in ? *V_in : Vtmp);
+ VectorBase<Real> &s = *s_in;
+
+ int m = num_rows_, n = num_cols_;
+ KALDI_ASSERT(m>=n && m != 0 && n != 0);
+ if (wantu) KALDI_ASSERT((int)U.num_rows_ == m && (int)U.num_cols_ == n);
+ if (wantv) KALDI_ASSERT((int)V.num_rows_ == n && (int)V.num_cols_ == n);
+ KALDI_ASSERT((int)s.Dim() == n); // n<=m so n is min.
+
+ int nu = n;
+ U.SetZero(); // make sure all zero.
+ Vector<Real> e(n);
+ Vector<Real> work(m);
+ MatrixBase<Real> &A(*this);
+ Real *adata = A.Data(), *workdata = work.Data(), *edata = e.Data(),
+ *udata = U.Data(), *vdata = V.Data();
+ int astride = static_cast<int>(A.Stride()),
+ ustride = static_cast<int>(U.Stride()),
+ vstride = static_cast<int>(V.Stride());
+ int i = 0, j = 0, k = 0;
+
+ // Reduce A to bidiagonal form, storing the diagonal elements
+ // in s and the super-diagonal elements in e.
+
+ int nct = std::min(m-1, n);
+ int nrt = std::max(0, std::min(n-2, m));
+ for (k = 0; k < std::max(nct, nrt); k++) {
+ if (k < nct) {
+
+ // Compute the transformation for the k-th column and
+ // place the k-th diagonal in s(k).
+ // Compute 2-norm of k-th column without under/overflow.
+ s(k) = 0;
+ for (i = k; i < m; i++) {
+ s(k) = hypot(s(k), A(i, k));
+ }
+ if (s(k) != 0.0) {
+ if (A(k, k) < 0.0) {
+ s(k) = -s(k);
+ }
+ for (i = k; i < m; i++) {
+ A(i, k) /= s(k);
+ }
+ A(k, k) += 1.0;
+ }
+ s(k) = -s(k);
+ }
+ for (j = k+1; j < n; j++) {
+ if ((k < nct) && (s(k) != 0.0)) {
+
+ // Apply the transformation.
+
+ Real t = cblas_Xdot(m - k, adata + astride*k + k, astride,
+ adata + astride*k + j, astride);
+ /*for (i = k; i < m; i++) {
+ t += adata[i*astride + k]*adata[i*astride + j]; // A(i, k)*A(i, j); // 3
+ }*/
+ t = -t/A(k, k);
+ cblas_Xaxpy(m - k, t, adata + k*astride + k, astride,
+ adata + k*astride + j, astride);
+ /*for (i = k; i < m; i++) {
+ adata[i*astride + j] += t*adata[i*astride + k]; // A(i, j) += t*A(i, k); // 5
+ }*/
+ }
+
+ // Place the k-th row of A into e for the
+ // subsequent calculation of the row transformation.
+
+ e(j) = A(k, j);
+ }
+ if (wantu & (k < nct)) {
+
+ // Place the transformation in U for subsequent back
+ // multiplication.
+
+ for (i = k; i < m; i++) {
+ U(i, k) = A(i, k);
+ }
+ }
+ if (k < nrt) {
+
+ // Compute the k-th row transformation and place the
+ // k-th super-diagonal in e(k).
+ // Compute 2-norm without under/overflow.
+ e(k) = 0;
+ for (i = k+1; i < n; i++) {
+ e(k) = hypot(e(k), e(i));
+ }
+ if (e(k) != 0.0) {
+ if (e(k+1) < 0.0) {
+ e(k) = -e(k);
+ }
+ for (i = k+1; i < n; i++) {
+ e(i) /= e(k);
+ }
+ e(k+1) += 1.0;
+ }
+ e(k) = -e(k);
+ if ((k+1 < m) & (e(k) != 0.0)) {
+
+ // Apply the transformation.
+
+ for (i = k+1; i < m; i++) {
+ work(i) = 0.0;
+ }
+ for (j = k+1; j < n; j++) {
+ for (i = k+1; i < m; i++) {
+ workdata[i] += edata[j] * adata[i*astride + j]; // work(i) += e(j)*A(i, j); // 5
+ }
+ }
+ for (j = k+1; j < n; j++) {
+ Real t(-e(j)/e(k+1));
+ cblas_Xaxpy(m - (k+1), t, workdata + (k+1), 1,
+ adata + (k+1)*astride + j, astride);
+ /*
+ for (i = k+1; i < m; i++) {
+ adata[i*astride + j] += t*workdata[i]; // A(i, j) += t*work(i); // 5
+ }*/
+ }
+ }
+ if (wantv) {
+
+ // Place the transformation in V for subsequent
+ // back multiplication.
+
+ for (i = k+1; i < n; i++) {
+ V(i, k) = e(i);
+ }
+ }
+ }
+ }
+
+ // Set up the final bidiagonal matrix or order p.
+
+ int p = std::min(n, m+1);
+ if (nct < n) {
+ s(nct) = A(nct, nct);
+ }
+ if (m < p) {
+ s(p-1) = 0.0;
+ }
+ if (nrt+1 < p) {
+ e(nrt) = A(nrt, p-1);
+ }
+ e(p-1) = 0.0;
+
+ // If required, generate U.
+
+ if (wantu) {
+ for (j = nct; j < nu; j++) {
+ for (i = 0; i < m; i++) {
+ U(i, j) = 0.0;
+ }
+ U(j, j) = 1.0;
+ }
+ for (k = nct-1; k >= 0; k--) {
+ if (s(k) != 0.0) {
+ for (j = k+1; j < nu; j++) {
+ Real t = cblas_Xdot(m - k, udata + k*ustride + k, ustride, udata + k*ustride + j, ustride);
+ //for (i = k; i < m; i++) {
+ // t += udata[i*ustride + k]*udata[i*ustride + j]; // t += U(i, k)*U(i, j); // 8
+ // }
+ t = -t/U(k, k);
+ cblas_Xaxpy(m - k, t, udata + ustride*k + k, ustride,
+ udata + k*ustride + j, ustride);
+ /*for (i = k; i < m; i++) {
+ udata[i*ustride + j] += t*udata[i*ustride + k]; // U(i, j) += t*U(i, k); // 4
+ }*/
+ }
+ for (i = k; i < m; i++ ) {
+ U(i, k) = -U(i, k);
+ }
+ U(k, k) = 1.0 + U(k, k);
+ for (i = 0; i < k-1; i++) {
+ U(i, k) = 0.0;
+ }
+ } else {
+ for (i = 0; i < m; i++) {
+ U(i, k) = 0.0;
+ }
+ U(k, k) = 1.0;
+ }
+ }
+ }
+
+ // If required, generate V.
+
+ if (wantv) {
+ for (k = n-1; k >= 0; k--) {
+ if ((k < nrt) & (e(k) != 0.0)) {
+ for (j = k+1; j < nu; j++) {
+ Real t = cblas_Xdot(n - (k+1), vdata + (k+1)*vstride + k, vstride,
+ vdata + (k+1)*vstride + j, vstride);
+ /*Real t (0.0);
+ for (i = k+1; i < n; i++) {
+ t += vdata[i*vstride + k]*vdata[i*vstride + j]; // t += V(i, k)*V(i, j); // 7
+ }*/
+ t = -t/V(k+1, k);
+ cblas_Xaxpy(n - (k+1), t, vdata + (k+1)*vstride + k, vstride,
+ vdata + (k+1)*vstride + j, vstride);
+ /*for (i = k+1; i < n; i++) {
+ vdata[i*vstride + j] += t*vdata[i*vstride + k]; // V(i, j) += t*V(i, k); // 7
+ }*/
+ }
+ }
+ for (i = 0; i < n; i++) {
+ V(i, k) = 0.0;
+ }
+ V(k, k) = 1.0;
+ }
+ }
+
+ // Main iteration loop for the singular values.
+
+ int pp = p-1;
+ int iter = 0;
+ // note: -52.0 is from Jama code; the -23 is the extension
+ // to float, because mantissa length in (double, float)
+ // is (52, 23) bits respectively.
+ Real eps(pow(2.0, sizeof(Real) == 4 ? -23.0 : -52.0));
+ // Note: the -966 was taken from Jama code, but the -120 is a guess
+ // of how to extend this to float... the exponent in double goes
+ // from -1022 .. 1023, and in float from -126..127. I'm not sure
+ // what the significance of 966 is, so -120 just represents a number
+ // that's a bit less negative than -126. If we get convergence
+ // failure in float only, this may mean that we have to make the
+ // -120 value less negative.
+ Real tiny(pow(2.0, sizeof(Real) == 4 ? -120.0: -966.0 ));
+
+ while (p > 0) {
+ int k = 0;
+ int kase = 0;
+
+ if (iter == 500 || iter == 750) {
+ KALDI_WARN << "Svd taking a long time: making convergence criterion less exact.";
+ eps = pow(static_cast<Real>(0.8), eps);
+ tiny = pow(static_cast<Real>(0.8), tiny);
+ }
+ if (iter > 1000) {
+ KALDI_WARN << "Svd not converging on matrix of size " << m << " by " <<n;
+ return false;
+ }
+
+ // This section of the program inspects for
+ // negligible elements in the s and e arrays. On
+ // completion the variables kase and k are set as follows.
+
+ // kase = 1 if s(p) and e(k-1) are negligible and k < p
+ // kase = 2 if s(k) is negligible and k < p
+ // kase = 3 if e(k-1) is negligible, k < p, and
+ // s(k), ..., s(p) are not negligible (qr step).
+ // kase = 4 if e(p-1) is negligible (convergence).
+
+ for (k = p-2; k >= -1; k--) {
+ if (k == -1) {
+ break;
+ }
+ if (std::abs(e(k)) <=
+ tiny + eps*(std::abs(s(k)) + std::abs(s(k+1)))) {
+ e(k) = 0.0;
+ break;
+ }
+ }
+ if (k == p-2) {
+ kase = 4;
+ } else {
+ int ks;
+ for (ks = p-1; ks >= k; ks--) {
+ if (ks == k) {
+ break;
+ }
+ Real t( (ks != p ? std::abs(e(ks)) : 0.) +
+ (ks != k+1 ? std::abs(e(ks-1)) : 0.));
+ if (std::abs(s(ks)) <= tiny + eps*t) {
+ s(ks) = 0.0;
+ break;
+ }
+ }
+ if (ks == k) {
+ kase = 3;
+ } else if (ks == p-1) {
+ kase = 1;
+ } else {
+ kase = 2;
+ k = ks;
+ }
+ }
+ k++;
+
+ // Perform the task indicated by kase.
+
+ switch (kase) {
+
+ // Deflate negligible s(p).
+
+ case 1: {
+ Real f(e(p-2));
+ e(p-2) = 0.0;
+ for (j = p-2; j >= k; j--) {
+ Real t( hypot(s(j), f));
+ Real cs(s(j)/t);
+ Real sn(f/t);
+ s(j) = t;
+ if (j != k) {
+ f = -sn*e(j-1);
+ e(j-1) = cs*e(j-1);
+ }
+ if (wantv) {
+ for (i = 0; i < n; i++) {
+ t = cs*V(i, j) + sn*V(i, p-1);
+ V(i, p-1) = -sn*V(i, j) + cs*V(i, p-1);
+ V(i, j) = t;
+ }
+ }
+ }
+ }
+ break;
+
+ // Split at negligible s(k).
+
+ case 2: {
+ Real f(e(k-1));
+ e(k-1) = 0.0;
+ for (j = k; j < p; j++) {
+ Real t(hypot(s(j), f));
+ Real cs( s(j)/t);
+ Real sn(f/t);
+ s(j) = t;
+ f = -sn*e(j);
+ e(j) = cs*e(j);
+ if (wantu) {
+ for (i = 0; i < m; i++) {
+ t = cs*U(i, j) + sn*U(i, k-1);
+ U(i, k-1) = -sn*U(i, j) + cs*U(i, k-1);
+ U(i, j) = t;
+ }
+ }
+ }
+ }
+ break;
+
+ // Perform one qr step.
+
+ case 3: {
+
+ // Calculate the shift.
+
+ Real scale = std::max(std::max(std::max(std::max(
+ std::abs(s(p-1)), std::abs(s(p-2))), std::abs(e(p-2))),
+ std::abs(s(k))), std::abs(e(k)));
+ Real sp = s(p-1)/scale;
+ Real spm1 = s(p-2)/scale;
+ Real epm1 = e(p-2)/scale;
+ Real sk = s(k)/scale;
+ Real ek = e(k)/scale;
+ Real b = ((spm1 + sp)*(spm1 - sp) + epm1*epm1)/2.0;
+ Real c = (sp*epm1)*(sp*epm1);
+ Real shift = 0.0;
+ if ((b != 0.0) || (c != 0.0)) {
+ shift = std::sqrt(b*b + c);
+ if (b < 0.0) {
+ shift = -shift;
+ }
+ shift = c/(b + shift);
+ }
+ Real f = (sk + sp)*(sk - sp) + shift;
+ Real g = sk*ek;
+
+ // Chase zeros.
+
+ for (j = k; j < p-1; j++) {
+ Real t = hypot(f, g);
+ Real cs = f/t;
+ Real sn = g/t;
+ if (j != k) {
+ e(j-1) = t;
+ }
+ f = cs*s(j) + sn*e(j);
+ e(j) = cs*e(j) - sn*s(j);
+ g = sn*s(j+1);
+ s(j+1) = cs*s(j+1);
+ if (wantv) {
+ cblas_Xrot(n, vdata + j, vstride, vdata + j+1, vstride, cs, sn);
+ /*for (i = 0; i < n; i++) {
+ t = cs*vdata[i*vstride + j] + sn*vdata[i*vstride + j+1]; // t = cs*V(i, j) + sn*V(i, j+1); // 13
+ vdata[i*vstride + j+1] = -sn*vdata[i*vstride + j] + cs*vdata[i*vstride + j+1]; // V(i, j+1) = -sn*V(i, j) + cs*V(i, j+1); // 5
+ vdata[i*vstride + j] = t; // V(i, j) = t; // 4
+ }*/
+ }
+ t = hypot(f, g);
+ cs = f/t;
+ sn = g/t;
+ s(j) = t;
+ f = cs*e(j) + sn*s(j+1);
+ s(j+1) = -sn*e(j) + cs*s(j+1);
+ g = sn*e(j+1);
+ e(j+1) = cs*e(j+1);
+ if (wantu && (j < m-1)) {
+ cblas_Xrot(m, udata + j, ustride, udata + j+1, ustride, cs, sn);
+ /*for (i = 0; i < m; i++) {
+ t = cs*udata[i*ustride + j] + sn*udata[i*ustride + j+1]; // t = cs*U(i, j) + sn*U(i, j+1); // 7
+ udata[i*ustride + j+1] = -sn*udata[i*ustride + j] +cs*udata[i*ustride + j+1]; // U(i, j+1) = -sn*U(i, j) + cs*U(i, j+1); // 8
+ udata[i*ustride + j] = t; // U(i, j) = t; // 1
+ }*/
+ }
+ }
+ e(p-2) = f;
+ iter = iter + 1;
+ }
+ break;
+
+ // Convergence.
+
+ case 4: {
+
+ // Make the singular values positive.
+
+ if (s(k) <= 0.0) {
+ s(k) = (s(k) < 0.0 ? -s(k) : 0.0);
+ if (wantv) {
+ for (i = 0; i <= pp; i++) {
+ V(i, k) = -V(i, k);
+ }
+ }
+ }
+
+ // Order the singular values.
+
+ while (k < pp) {
+ if (s(k) >= s(k+1)) {
+ break;
+ }
+ Real t = s(k);
+ s(k) = s(k+1);
+ s(k+1) = t;
+ if (wantv && (k < n-1)) {
+ for (i = 0; i < n; i++) {
+ t = V(i, k+1); V(i, k+1) = V(i, k); V(i, k) = t;
+ }
+ }
+ if (wantu && (k < m-1)) {
+ for (i = 0; i < m; i++) {
+ t = U(i, k+1); U(i, k+1) = U(i, k); U(i, k) = t;
+ }
+ }
+ k++;
+ }
+ iter = 0;
+ p--;
+ }
+ break;
+ }
+ }
+ return true;
+}
+
+#endif // defined(HAVE_ATLAS) || defined(USE_KALDI_SVD)
+
+} // namespace kaldi
+
+#endif // KALDI_MATRIX_JAMA_SVD_H_
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