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 * Σ and V such that A = U × Σ × V<sup>T</sup>. Let A be
029 * a m × n matrix, then U is a m × p orthogonal matrix, Σ is a
030 * p × p diagonal matrix with positive or null elements, V is a p ×
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 Σ of the decomposition.
510 * <p>Σ is a diagonal matrix. The singular values are provided in
511 * non-increasing order, for compatibility with Jama.</p>
512 * @return the Σ 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 Σ 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 Σ 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 × n covariance matrix.
559 * <p>The covariance matrix is V × J × 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 × 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) × 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 × 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 × X = B in least square sense.
689 * <p>
690 * The m×n matrix A may not be square, the solution X is such that
691 * ||A × X - B|| is minimal.
692 * </p>
693 * @param b Right-hand side of the equation A × X = B
694 * @return a vector X that minimizes the two norm of A × 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 × X = B in least square sense.
704 * <p>
705 * The m×n matrix A may not be square, the solution X is such that
706 * ||A × X - B|| is minimal.
707 * </p>
708 *
709 * @param b Right-hand side of the equation A × X = B
710 * @return a matrix X that minimizes the two norm of A × 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 }