001    /*
002     * Licensed to the Apache Software Foundation (ASF) under one or more
003     * contributor license agreements.  See the NOTICE file distributed with
004     * this work for additional information regarding copyright ownership.
005     * The ASF licenses this file to You under the Apache License, Version 2.0
006     * (the "License"); you may not use this file except in compliance with
007     * the License.  You may obtain a copy of the License at
008     *
009     *      http://www.apache.org/licenses/LICENSE-2.0
010     *
011     * Unless required by applicable law or agreed to in writing, software
012     * distributed under the License is distributed on an "AS IS" BASIS,
013     * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
014     * See the License for the specific language governing permissions and
015     * limitations under the License.
016     */
017    package org.apache.commons.math3.linear;
018    
019    import org.apache.commons.math3.exception.NumberIsTooLargeException;
020    import org.apache.commons.math3.exception.util.LocalizedFormats;
021    import org.apache.commons.math3.util.FastMath;
022    import org.apache.commons.math3.util.Precision;
023    
024    /**
025     * Calculates the compact Singular Value Decomposition of a matrix.
026     * <p>
027     * The Singular Value Decomposition of matrix A is a set of three matrices: U,
028     * &Sigma; and V such that A = U &times; &Sigma; &times; V<sup>T</sup>. Let A be
029     * a m &times; n matrix, then U is a m &times; p orthogonal matrix, &Sigma; is a
030     * p &times; p diagonal matrix with positive or null elements, V is a p &times;
031     * n orthogonal matrix (hence V<sup>T</sup> is also orthogonal) where
032     * p=min(m,n).
033     * </p>
034     * <p>This class is similar to the class with similar name from the
035     * <a href="http://math.nist.gov/javanumerics/jama/">JAMA</a> library, with the
036     * following changes:</p>
037     * <ul>
038     *   <li>the {@code norm2} method which has been renamed as {@link #getNorm()
039     *   getNorm},</li>
040     *   <li>the {@code cond} method which has been renamed as {@link
041     *   #getConditionNumber() getConditionNumber},</li>
042     *   <li>the {@code rank} method which has been renamed as {@link #getRank()
043     *   getRank},</li>
044     *   <li>a {@link #getUT() getUT} method has been added,</li>
045     *   <li>a {@link #getVT() getVT} method has been added,</li>
046     *   <li>a {@link #getSolver() getSolver} method has been added,</li>
047     *   <li>a {@link #getCovariance(double) getCovariance} method has been added.</li>
048     * </ul>
049     * @see <a href="http://mathworld.wolfram.com/SingularValueDecomposition.html">MathWorld</a>
050     * @see <a href="http://en.wikipedia.org/wiki/Singular_value_decomposition">Wikipedia</a>
051     * @version $Id: SingularValueDecomposition.java 1416643 2012-12-03 19:37:14Z tn $
052     * @since 2.0 (changed to concrete class in 3.0)
053     */
054    public class SingularValueDecomposition {
055        /** Relative threshold for small singular values. */
056        private static final double EPS = 0x1.0p-52;
057        /** Absolute threshold for small singular values. */
058        private static final double TINY = 0x1.0p-966;
059        /** Computed singular values. */
060        private final double[] singularValues;
061        /** max(row dimension, column dimension). */
062        private final int m;
063        /** min(row dimension, column dimension). */
064        private final int n;
065        /** Indicator for transposed matrix. */
066        private final boolean transposed;
067        /** Cached value of U matrix. */
068        private final RealMatrix cachedU;
069        /** Cached value of transposed U matrix. */
070        private RealMatrix cachedUt;
071        /** Cached value of S (diagonal) matrix. */
072        private RealMatrix cachedS;
073        /** Cached value of V matrix. */
074        private final RealMatrix cachedV;
075        /** Cached value of transposed V matrix. */
076        private RealMatrix cachedVt;
077        /**
078         * Tolerance value for small singular values, calculated once we have
079         * populated "singularValues".
080         **/
081        private final double tol;
082    
083        /**
084         * Calculates the compact Singular Value Decomposition of the given matrix.
085         *
086         * @param matrix Matrix to decompose.
087         */
088        public SingularValueDecomposition(final RealMatrix matrix) {
089            final double[][] A;
090    
091             // "m" is always the largest dimension.
092            if (matrix.getRowDimension() < matrix.getColumnDimension()) {
093                transposed = true;
094                A = matrix.transpose().getData();
095                m = matrix.getColumnDimension();
096                n = matrix.getRowDimension();
097            } else {
098                transposed = false;
099                A = matrix.getData();
100                m = matrix.getRowDimension();
101                n = matrix.getColumnDimension();
102            }
103    
104            singularValues = new double[n];
105            final double[][] U = new double[m][n];
106            final double[][] V = new double[n][n];
107            final double[] e = new double[n];
108            final double[] work = new double[m];
109            // Reduce A to bidiagonal form, storing the diagonal elements
110            // in s and the super-diagonal elements in e.
111            final int nct = FastMath.min(m - 1, n);
112            final int nrt = FastMath.max(0, n - 2);
113            for (int k = 0; k < FastMath.max(nct, nrt); k++) {
114                if (k < nct) {
115                    // Compute the transformation for the k-th column and
116                    // place the k-th diagonal in s[k].
117                    // Compute 2-norm of k-th column without under/overflow.
118                    singularValues[k] = 0;
119                    for (int i = k; i < m; i++) {
120                        singularValues[k] = FastMath.hypot(singularValues[k], A[i][k]);
121                    }
122                    if (singularValues[k] != 0) {
123                        if (A[k][k] < 0) {
124                            singularValues[k] = -singularValues[k];
125                        }
126                        for (int i = k; i < m; i++) {
127                            A[i][k] /= singularValues[k];
128                        }
129                        A[k][k] += 1;
130                    }
131                    singularValues[k] = -singularValues[k];
132                }
133                for (int j = k + 1; j < n; j++) {
134                    if (k < nct &&
135                        singularValues[k] != 0) {
136                        // Apply the transformation.
137                        double t = 0;
138                        for (int i = k; i < m; i++) {
139                            t += A[i][k] * A[i][j];
140                        }
141                        t = -t / A[k][k];
142                        for (int i = k; i < m; i++) {
143                            A[i][j] += t * A[i][k];
144                        }
145                    }
146                    // Place the k-th row of A into e for the
147                    // subsequent calculation of the row transformation.
148                    e[j] = A[k][j];
149                }
150                if (k < nct) {
151                    // Place the transformation in U for subsequent back
152                    // multiplication.
153                    for (int i = k; i < m; i++) {
154                        U[i][k] = A[i][k];
155                    }
156                }
157                if (k < nrt) {
158                    // Compute the k-th row transformation and place the
159                    // k-th super-diagonal in e[k].
160                    // Compute 2-norm without under/overflow.
161                    e[k] = 0;
162                    for (int i = k + 1; i < n; i++) {
163                        e[k] = FastMath.hypot(e[k], e[i]);
164                    }
165                    if (e[k] != 0) {
166                        if (e[k + 1] < 0) {
167                            e[k] = -e[k];
168                        }
169                        for (int i = k + 1; i < n; i++) {
170                            e[i] /= e[k];
171                        }
172                        e[k + 1] += 1;
173                    }
174                    e[k] = -e[k];
175                    if (k + 1 < m &&
176                        e[k] != 0) {
177                        // Apply the transformation.
178                        for (int i = k + 1; i < m; i++) {
179                            work[i] = 0;
180                        }
181                        for (int j = k + 1; j < n; j++) {
182                            for (int i = k + 1; i < m; i++) {
183                                work[i] += e[j] * A[i][j];
184                            }
185                        }
186                        for (int j = k + 1; j < n; j++) {
187                            final double t = -e[j] / e[k + 1];
188                            for (int i = k + 1; i < m; i++) {
189                                A[i][j] += t * work[i];
190                            }
191                        }
192                    }
193    
194                    // Place the transformation in V for subsequent
195                    // back multiplication.
196                    for (int i = k + 1; i < n; i++) {
197                        V[i][k] = e[i];
198                    }
199                }
200            }
201            // Set up the final bidiagonal matrix or order p.
202            int p = n;
203            if (nct < n) {
204                singularValues[nct] = A[nct][nct];
205            }
206            if (m < p) {
207                singularValues[p - 1] = 0;
208            }
209            if (nrt + 1 < p) {
210                e[nrt] = A[nrt][p - 1];
211            }
212            e[p - 1] = 0;
213    
214            // Generate U.
215            for (int j = nct; j < n; j++) {
216                for (int i = 0; i < m; i++) {
217                    U[i][j] = 0;
218                }
219                U[j][j] = 1;
220            }
221            for (int k = nct - 1; k >= 0; k--) {
222                if (singularValues[k] != 0) {
223                    for (int j = k + 1; j < n; j++) {
224                        double t = 0;
225                        for (int i = k; i < m; i++) {
226                            t += U[i][k] * U[i][j];
227                        }
228                        t = -t / U[k][k];
229                        for (int i = k; i < m; i++) {
230                            U[i][j] += t * U[i][k];
231                        }
232                    }
233                    for (int i = k; i < m; i++) {
234                        U[i][k] = -U[i][k];
235                    }
236                    U[k][k] = 1 + U[k][k];
237                    for (int i = 0; i < k - 1; i++) {
238                        U[i][k] = 0;
239                    }
240                } else {
241                    for (int i = 0; i < m; i++) {
242                        U[i][k] = 0;
243                    }
244                    U[k][k] = 1;
245                }
246            }
247    
248            // Generate V.
249            for (int k = n - 1; k >= 0; k--) {
250                if (k < nrt &&
251                    e[k] != 0) {
252                    for (int j = k + 1; j < n; j++) {
253                        double t = 0;
254                        for (int i = k + 1; i < n; i++) {
255                            t += V[i][k] * V[i][j];
256                        }
257                        t = -t / V[k + 1][k];
258                        for (int i = k + 1; i < n; i++) {
259                            V[i][j] += t * V[i][k];
260                        }
261                    }
262                }
263                for (int i = 0; i < n; i++) {
264                    V[i][k] = 0;
265                }
266                V[k][k] = 1;
267            }
268    
269            // Main iteration loop for the singular values.
270            final int pp = p - 1;
271            int iter = 0;
272            while (p > 0) {
273                int k;
274                int kase;
275                // Here is where a test for too many iterations would go.
276                // This section of the program inspects for
277                // negligible elements in the s and e arrays.  On
278                // completion the variables kase and k are set as follows.
279                // kase = 1     if s(p) and e[k-1] are negligible and k<p
280                // kase = 2     if s(k) is negligible and k<p
281                // kase = 3     if e[k-1] is negligible, k<p, and
282                //              s(k), ..., s(p) are not negligible (qr step).
283                // kase = 4     if e(p-1) is negligible (convergence).
284                for (k = p - 2; k >= 0; k--) {
285                    final double threshold
286                        = TINY + EPS * (FastMath.abs(singularValues[k]) +
287                                        FastMath.abs(singularValues[k + 1]));
288                    if (FastMath.abs(e[k]) <= threshold) {
289                        e[k] = 0;
290                        break;
291                    }
292                }
293    
294                if (k == p - 2) {
295                    kase = 4;
296                } else {
297                    int ks;
298                    for (ks = p - 1; ks >= k; ks--) {
299                        if (ks == k) {
300                            break;
301                        }
302                        final double t = (ks != p ? FastMath.abs(e[ks]) : 0) +
303                            (ks != k + 1 ? FastMath.abs(e[ks - 1]) : 0);
304                        if (FastMath.abs(singularValues[ks]) <= TINY + EPS * t) {
305                            singularValues[ks] = 0;
306                            break;
307                        }
308                    }
309                    if (ks == k) {
310                        kase = 3;
311                    } else if (ks == p - 1) {
312                        kase = 1;
313                    } else {
314                        kase = 2;
315                        k = ks;
316                    }
317                }
318                k++;
319                // Perform the task indicated by kase.
320                switch (kase) {
321                    // Deflate negligible s(p).
322                    case 1: {
323                        double f = e[p - 2];
324                        e[p - 2] = 0;
325                        for (int j = p - 2; j >= k; j--) {
326                            double t = FastMath.hypot(singularValues[j], f);
327                            final double cs = singularValues[j] / t;
328                            final double sn = f / t;
329                            singularValues[j] = t;
330                            if (j != k) {
331                                f = -sn * e[j - 1];
332                                e[j - 1] = cs * e[j - 1];
333                            }
334    
335                            for (int i = 0; i < n; i++) {
336                                t = cs * V[i][j] + sn * V[i][p - 1];
337                                V[i][p - 1] = -sn * V[i][j] + cs * V[i][p - 1];
338                                V[i][j] = t;
339                            }
340                        }
341                    }
342                    break;
343                    // Split at negligible s(k).
344                    case 2: {
345                        double f = e[k - 1];
346                        e[k - 1] = 0;
347                        for (int j = k; j < p; j++) {
348                            double t = FastMath.hypot(singularValues[j], f);
349                            final double cs = singularValues[j] / t;
350                            final double sn = f / t;
351                            singularValues[j] = t;
352                            f = -sn * e[j];
353                            e[j] = cs * e[j];
354    
355                            for (int i = 0; i < m; i++) {
356                                t = cs * U[i][j] + sn * U[i][k - 1];
357                                U[i][k - 1] = -sn * U[i][j] + cs * U[i][k - 1];
358                                U[i][j] = t;
359                            }
360                        }
361                    }
362                    break;
363                    // Perform one qr step.
364                    case 3: {
365                        // Calculate the shift.
366                        final double maxPm1Pm2 = FastMath.max(FastMath.abs(singularValues[p - 1]),
367                                                              FastMath.abs(singularValues[p - 2]));
368                        final double scale = FastMath.max(FastMath.max(FastMath.max(maxPm1Pm2,
369                                                                                    FastMath.abs(e[p - 2])),
370                                                                       FastMath.abs(singularValues[k])),
371                                                          FastMath.abs(e[k]));
372                        final double sp = singularValues[p - 1] / scale;
373                        final double spm1 = singularValues[p - 2] / scale;
374                        final double epm1 = e[p - 2] / scale;
375                        final double sk = singularValues[k] / scale;
376                        final double ek = e[k] / scale;
377                        final double b = ((spm1 + sp) * (spm1 - sp) + epm1 * epm1) / 2.0;
378                        final double c = (sp * epm1) * (sp * epm1);
379                        double shift = 0;
380                        if (b != 0 ||
381                            c != 0) {
382                            shift = FastMath.sqrt(b * b + c);
383                            if (b < 0) {
384                                shift = -shift;
385                            }
386                            shift = c / (b + shift);
387                        }
388                        double f = (sk + sp) * (sk - sp) + shift;
389                        double g = sk * ek;
390                        // Chase zeros.
391                        for (int j = k; j < p - 1; j++) {
392                            double t = FastMath.hypot(f, g);
393                            double cs = f / t;
394                            double sn = g / t;
395                            if (j != k) {
396                                e[j - 1] = t;
397                            }
398                            f = cs * singularValues[j] + sn * e[j];
399                            e[j] = cs * e[j] - sn * singularValues[j];
400                            g = sn * singularValues[j + 1];
401                            singularValues[j + 1] = cs * singularValues[j + 1];
402    
403                            for (int i = 0; i < n; i++) {
404                                t = cs * V[i][j] + sn * V[i][j + 1];
405                                V[i][j + 1] = -sn * V[i][j] + cs * V[i][j + 1];
406                                V[i][j] = t;
407                            }
408                            t = FastMath.hypot(f, g);
409                            cs = f / t;
410                            sn = g / t;
411                            singularValues[j] = t;
412                            f = cs * e[j] + sn * singularValues[j + 1];
413                            singularValues[j + 1] = -sn * e[j] + cs * singularValues[j + 1];
414                            g = sn * e[j + 1];
415                            e[j + 1] = cs * e[j + 1];
416                            if (j < m - 1) {
417                                for (int i = 0; i < m; i++) {
418                                    t = cs * U[i][j] + sn * U[i][j + 1];
419                                    U[i][j + 1] = -sn * U[i][j] + cs * U[i][j + 1];
420                                    U[i][j] = t;
421                                }
422                            }
423                        }
424                        e[p - 2] = f;
425                        iter = iter + 1;
426                    }
427                    break;
428                    // Convergence.
429                    default: {
430                        // Make the singular values positive.
431                        if (singularValues[k] <= 0) {
432                            singularValues[k] = singularValues[k] < 0 ? -singularValues[k] : 0;
433    
434                            for (int i = 0; i <= pp; i++) {
435                                V[i][k] = -V[i][k];
436                            }
437                        }
438                        // Order the singular values.
439                        while (k < pp) {
440                            if (singularValues[k] >= singularValues[k + 1]) {
441                                break;
442                            }
443                            double t = singularValues[k];
444                            singularValues[k] = singularValues[k + 1];
445                            singularValues[k + 1] = t;
446                            if (k < n - 1) {
447                                for (int i = 0; i < n; i++) {
448                                    t = V[i][k + 1];
449                                    V[i][k + 1] = V[i][k];
450                                    V[i][k] = t;
451                                }
452                            }
453                            if (k < m - 1) {
454                                for (int i = 0; i < m; i++) {
455                                    t = U[i][k + 1];
456                                    U[i][k + 1] = U[i][k];
457                                    U[i][k] = t;
458                                }
459                            }
460                            k++;
461                        }
462                        iter = 0;
463                        p--;
464                    }
465                    break;
466                }
467            }
468    
469            // Set the small value tolerance used to calculate rank and pseudo-inverse
470            tol = FastMath.max(m * singularValues[0] * EPS,
471                               FastMath.sqrt(Precision.SAFE_MIN));
472    
473            if (!transposed) {
474                cachedU = MatrixUtils.createRealMatrix(U);
475                cachedV = MatrixUtils.createRealMatrix(V);
476            } else {
477                cachedU = MatrixUtils.createRealMatrix(V);
478                cachedV = MatrixUtils.createRealMatrix(U);
479            }
480        }
481    
482        /**
483         * Returns the matrix U of the decomposition.
484         * <p>U is an orthogonal matrix, i.e. its transpose is also its inverse.</p>
485         * @return the U matrix
486         * @see #getUT()
487         */
488        public RealMatrix getU() {
489            // return the cached matrix
490            return cachedU;
491    
492        }
493    
494        /**
495         * Returns the transpose of the matrix U of the decomposition.
496         * <p>U is an orthogonal matrix, i.e. its transpose is also its inverse.</p>
497         * @return the U matrix (or null if decomposed matrix is singular)
498         * @see #getU()
499         */
500        public RealMatrix getUT() {
501            if (cachedUt == null) {
502                cachedUt = getU().transpose();
503            }
504            // return the cached matrix
505            return cachedUt;
506        }
507    
508        /**
509         * Returns the diagonal matrix &Sigma; of the decomposition.
510         * <p>&Sigma; is a diagonal matrix. The singular values are provided in
511         * non-increasing order, for compatibility with Jama.</p>
512         * @return the &Sigma; matrix
513         */
514        public RealMatrix getS() {
515            if (cachedS == null) {
516                // cache the matrix for subsequent calls
517                cachedS = MatrixUtils.createRealDiagonalMatrix(singularValues);
518            }
519            return cachedS;
520        }
521    
522        /**
523         * Returns the diagonal elements of the matrix &Sigma; of the decomposition.
524         * <p>The singular values are provided in non-increasing order, for
525         * compatibility with Jama.</p>
526         * @return the diagonal elements of the &Sigma; matrix
527         */
528        public double[] getSingularValues() {
529            return singularValues.clone();
530        }
531    
532        /**
533         * Returns the matrix V of the decomposition.
534         * <p>V is an orthogonal matrix, i.e. its transpose is also its inverse.</p>
535         * @return the V matrix (or null if decomposed matrix is singular)
536         * @see #getVT()
537         */
538        public RealMatrix getV() {
539            // return the cached matrix
540            return cachedV;
541        }
542    
543        /**
544         * Returns the transpose of the matrix V of the decomposition.
545         * <p>V is an orthogonal matrix, i.e. its transpose is also its inverse.</p>
546         * @return the V matrix (or null if decomposed matrix is singular)
547         * @see #getV()
548         */
549        public RealMatrix getVT() {
550            if (cachedVt == null) {
551                cachedVt = getV().transpose();
552            }
553            // return the cached matrix
554            return cachedVt;
555        }
556    
557        /**
558         * Returns the n &times; n covariance matrix.
559         * <p>The covariance matrix is V &times; J &times; V<sup>T</sup>
560         * where J is the diagonal matrix of the inverse of the squares of
561         * the singular values.</p>
562         * @param minSingularValue value below which singular values are ignored
563         * (a 0 or negative value implies all singular value will be used)
564         * @return covariance matrix
565         * @exception IllegalArgumentException if minSingularValue is larger than
566         * the largest singular value, meaning all singular values are ignored
567         */
568        public RealMatrix getCovariance(final double minSingularValue) {
569            // get the number of singular values to consider
570            final int p = singularValues.length;
571            int dimension = 0;
572            while (dimension < p &&
573                   singularValues[dimension] >= minSingularValue) {
574                ++dimension;
575            }
576    
577            if (dimension == 0) {
578                throw new NumberIsTooLargeException(LocalizedFormats.TOO_LARGE_CUTOFF_SINGULAR_VALUE,
579                                                    minSingularValue, singularValues[0], true);
580            }
581    
582            final double[][] data = new double[dimension][p];
583            getVT().walkInOptimizedOrder(new DefaultRealMatrixPreservingVisitor() {
584                /** {@inheritDoc} */
585                @Override
586                public void visit(final int row, final int column,
587                        final double value) {
588                    data[row][column] = value / singularValues[row];
589                }
590            }, 0, dimension - 1, 0, p - 1);
591    
592            RealMatrix jv = new Array2DRowRealMatrix(data, false);
593            return jv.transpose().multiply(jv);
594        }
595    
596        /**
597         * Returns the L<sub>2</sub> norm of the matrix.
598         * <p>The L<sub>2</sub> norm is max(|A &times; u|<sub>2</sub> /
599         * |u|<sub>2</sub>), where |.|<sub>2</sub> denotes the vectorial 2-norm
600         * (i.e. the traditional euclidian norm).</p>
601         * @return norm
602         */
603        public double getNorm() {
604            return singularValues[0];
605        }
606    
607        /**
608         * Return the condition number of the matrix.
609         * @return condition number of the matrix
610         */
611        public double getConditionNumber() {
612            return singularValues[0] / singularValues[n - 1];
613        }
614    
615        /**
616         * Computes the inverse of the condition number.
617         * In cases of rank deficiency, the {@link #getConditionNumber() condition
618         * number} will become undefined.
619         *
620         * @return the inverse of the condition number.
621         */
622        public double getInverseConditionNumber() {
623            return singularValues[n - 1] / singularValues[0];
624        }
625    
626        /**
627         * Return the effective numerical matrix rank.
628         * <p>The effective numerical rank is the number of non-negligible
629         * singular values. The threshold used to identify non-negligible
630         * terms is max(m,n) &times; ulp(s<sub>1</sub>) where ulp(s<sub>1</sub>)
631         * is the least significant bit of the largest singular value.</p>
632         * @return effective numerical matrix rank
633         */
634        public int getRank() {
635            int r = 0;
636            for (int i = 0; i < singularValues.length; i++) {
637                if (singularValues[i] > tol) {
638                    r++;
639                }
640            }
641            return r;
642        }
643    
644        /**
645         * Get a solver for finding the A &times; X = B solution in least square sense.
646         * @return a solver
647         */
648        public DecompositionSolver getSolver() {
649            return new Solver(singularValues, getUT(), getV(), getRank() == m, tol);
650        }
651    
652        /** Specialized solver. */
653        private static class Solver implements DecompositionSolver {
654            /** Pseudo-inverse of the initial matrix. */
655            private final RealMatrix pseudoInverse;
656            /** Singularity indicator. */
657            private boolean nonSingular;
658    
659            /**
660             * Build a solver from decomposed matrix.
661             *
662             * @param singularValues Singular values.
663             * @param uT U<sup>T</sup> matrix of the decomposition.
664             * @param v V matrix of the decomposition.
665             * @param nonSingular Singularity indicator.
666             * @param tol tolerance for singular values
667             */
668            private Solver(final double[] singularValues, final RealMatrix uT,
669                           final RealMatrix v, final boolean nonSingular, final double tol) {
670                final double[][] suT = uT.getData();
671                for (int i = 0; i < singularValues.length; ++i) {
672                    final double a;
673                    if (singularValues[i] > tol) {
674                        a = 1 / singularValues[i];
675                    } else {
676                        a = 0;
677                    }
678                    final double[] suTi = suT[i];
679                    for (int j = 0; j < suTi.length; ++j) {
680                        suTi[j] *= a;
681                    }
682                }
683                pseudoInverse = v.multiply(new Array2DRowRealMatrix(suT, false));
684                this.nonSingular = nonSingular;
685            }
686    
687            /**
688             * Solve the linear equation A &times; X = B in least square sense.
689             * <p>
690             * The m&times;n matrix A may not be square, the solution X is such that
691             * ||A &times; X - B|| is minimal.
692             * </p>
693             * @param b Right-hand side of the equation A &times; X = B
694             * @return a vector X that minimizes the two norm of A &times; X - B
695             * @throws org.apache.commons.math3.exception.DimensionMismatchException
696             * if the matrices dimensions do not match.
697             */
698            public RealVector solve(final RealVector b) {
699                return pseudoInverse.operate(b);
700            }
701    
702            /**
703             * Solve the linear equation A &times; X = B in least square sense.
704             * <p>
705             * The m&times;n matrix A may not be square, the solution X is such that
706             * ||A &times; X - B|| is minimal.
707             * </p>
708             *
709             * @param b Right-hand side of the equation A &times; X = B
710             * @return a matrix X that minimizes the two norm of A &times; X - B
711             * @throws org.apache.commons.math3.exception.DimensionMismatchException
712             * if the matrices dimensions do not match.
713             */
714            public RealMatrix solve(final RealMatrix b) {
715                return pseudoInverse.multiply(b);
716            }
717    
718            /**
719             * Check if the decomposed matrix is non-singular.
720             *
721             * @return {@code true} if the decomposed matrix is non-singular.
722             */
723            public boolean isNonSingular() {
724                return nonSingular;
725            }
726    
727            /**
728             * Get the pseudo-inverse of the decomposed matrix.
729             *
730             * @return the inverse matrix.
731             */
732            public RealMatrix getInverse() {
733                return pseudoInverse;
734            }
735        }
736    }