1 /* Copyright 2002-2025 CS GROUP
2 * Licensed to CS GROUP (CS) under one or more
3 * contributor license agreements. See the NOTICE file distributed with
4 * this work for additional information regarding copyright ownership.
5 * CS licenses this file to You under the Apache License, Version 2.0
6 * (the "License"); you may not use this file except in compliance with
7 * the License. You may obtain a copy of the License at
8 *
9 * http://www.apache.org/licenses/LICENSE-2.0
10 *
11 * Unless required by applicable law or agreed to in writing, software
12 * distributed under the License is distributed on an "AS IS" BASIS,
13 * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
14 * See the License for the specific language governing permissions and
15 * limitations under the License.
16 */
17 package org.orekit.propagation.semianalytical.dsst.utilities.hansen;
18
19 import org.hipparchus.analysis.polynomials.PolynomialFunction;
20
21 /**
22 * Utilities class.
23 *
24 * @author Lucian Barbulescu
25 */
26 public class HansenUtilities {
27
28 /** 1 represented as a polynomial. */
29 public static final PolynomialFunction ONE = new PolynomialFunction(new double[] {
30 1
31 });
32
33 /** 0 represented as a polynomial. */
34 public static final PolynomialFunction ZERO = new PolynomialFunction(new double[] {
35 0
36 });
37
38 /** Private constructor as class is a utility.
39 */
40 private HansenUtilities() {
41 }
42
43 /**
44 * Build the identity matrix of order 2.
45 *
46 * <pre>
47 * / 1 0 \
48 * I₂ = | |
49 * \ 0 1 /
50 * </pre>
51 *
52 * @return the identity matrix of order 2
53 */
54 public static PolynomialFunctionMatrix buildIdentityMatrix2() {
55 final PolynomialFunctionMatrix matrix = new PolynomialFunctionMatrix(2);
56 matrix.setMatrix(new PolynomialFunction[][] {
57 {
58 ONE, ZERO
59 },
60 {
61 ZERO, ONE
62 }
63 });
64 return matrix;
65 }
66
67 /**
68 * Build the empty matrix of order 2.
69 *
70 * <pre>
71 * / 0 0 \
72 * E₂ = | |
73 * \ 0 0 /
74 * </pre>
75 *
76 * @return the identity matrix of order 2
77 */
78 public static PolynomialFunctionMatrix buildZeroMatrix2() {
79 final PolynomialFunctionMatrix matrix = new PolynomialFunctionMatrix(2);
80 matrix.setMatrix(new PolynomialFunction[][] {
81 {
82 ZERO, ZERO
83 },
84 {
85 ZERO, ZERO
86 }
87 });
88 return matrix;
89 }
90
91
92 /**
93 * Build the identity matrix of order 4.
94 *
95 * <pre>
96 * / 1 0 0 0 \
97 * | |
98 * | 0 1 0 0 |
99 * I₄ = | |
100 * | 0 0 1 0 |
101 * | |
102 * \ 0 0 0 1 /
103 * </pre>
104 *
105 * @return the identity matrix of order 4
106 */
107 public static PolynomialFunctionMatrix buildIdentityMatrix4() {
108 final PolynomialFunctionMatrix matrix = new PolynomialFunctionMatrix(4);
109 matrix.setMatrix(new PolynomialFunction[][] {
110 {
111 ONE, ZERO, ZERO, ZERO
112 },
113 {
114 ZERO, ONE, ZERO, ZERO
115 },
116 {
117 ZERO, ZERO, ONE, ZERO
118 },
119 {
120 ZERO, ZERO, ZERO, ONE
121 }
122 });
123 return matrix;
124 }
125
126 /**
127 * Build the empty matrix of order 4.
128 *
129 * <pre>
130 * / 0 0 0 0 \
131 * | |
132 * | 0 0 0 0 |
133 * E₄ = | |
134 * | 0 0 0 0 |
135 * | |
136 * \ 0 0 0 0 /
137 * </pre>
138 *
139 * @return the identity matrix of order 4
140 */
141 public static PolynomialFunctionMatrix buildZeroMatrix4() {
142 final PolynomialFunctionMatrix matrix = new PolynomialFunctionMatrix(4);
143 matrix.setMatrix(new PolynomialFunction[][] {
144 {
145 ZERO, ZERO, ZERO, ZERO
146 },
147 {
148 ZERO, ZERO, ZERO, ZERO
149 },
150 {
151 ZERO, ZERO, ZERO, ZERO
152 },
153 {
154 ZERO, ZERO, ZERO, ZERO
155 }
156 } );
157 return matrix;
158 }
159
160 /**
161 * Compute polynomial coefficient a.
162 *
163 * <p>
164 * It is used to generate the coefficient for K₀<sup>-n, s</sup> when computing K₀<sup>-n-1, s</sup>
165 * and the coefficient for dK₀<sup>-n, s</sup> / de² when computing dK₀<sup>-n-1, s</sup> / de²
166 * </p>
167 *
168 * <p>
169 * See Danielson 2.7.3-(6) and Collins 4-242 and 4-245
170 * </p>
171 *
172 * @param s the s coefficient
173 * @param mnm1 -n-1 value
174 * @return the polynomial
175 */
176 private static PolynomialFunction aZonal(final int s, final int mnm1) {
177 // from recurrence Collins 4-242
178 final double d1 = (mnm1 + 2) * (2 * mnm1 + 5);
179 final double d2 = (mnm1 + 2 - s) * (mnm1 + 2 + s);
180 return new PolynomialFunction(new double[] {
181 0.0, 0.0, d1 / d2
182 });
183 }
184
185 /**
186 * Compute polynomial coefficient b.
187 *
188 * <p>
189 * It is used to generate the coefficient for K₀<sup>-n+1, s</sup> when computing K₀<sup>-n-1, s</sup>
190 * and the coefficient for dK₀<sup>-n+1, s</sup> / de² when computing dK₀<sup>-n-1, s</sup> / de²
191 * </p>
192 *
193 * <p>
194 * See Danielson 2.7.3-(6) and Collins 4-242 and 4-245
195 * </p>
196 *
197 * @param s the s coefficient
198 * @param mnm1 -n-1 value
199 * @return the polynomial
200 */
201 private static PolynomialFunction bZonal(final int s, final int mnm1) {
202 // from recurence Collins 4-242
203 final double d1 = (mnm1 + 2) * (mnm1 + 3);
204 final double d2 = (mnm1 + 2 - s) * (mnm1 + 2 + s);
205 return new PolynomialFunction(new double[] {
206 0.0, 0.0, -d1 / d2
207 });
208 }
209
210 /**
211 * Generate the polynomials needed in the linear transformation.
212 *
213 * @param n0 the index of the initial condition, Petre's paper
214 * @param nMin rhe minimum value for the order
215 * @param offset offset used to identify the polynomial that corresponds
216 * to a negative value of n in the internal array that
217 * starts at 0
218 * @param slice number of coefficients that will be computed with a set
219 * of roots
220 * @param s the s coefficient
221 * @param mpvec array to store the first vector of polynomials
222 * associated to Hansen coefficients and derivatives.
223 * @param mpvecDeriv array to store the second vector of polynomials
224 * associated only to derivatives.
225 * <p>
226 * See Petre's paper
227 * </p>
228 */
229 public static void generateZonalPolynomials(final int n0, final int nMin,
230 final int offset, final int slice,
231 final int s,
232 final PolynomialFunction[][] mpvec,
233 final PolynomialFunction[][] mpvecDeriv) {
234
235 int sliceCounter = 0;
236 int index;
237
238 // Initialisation of matrix for linear transformmations
239 // The final configuration of these matrix are obtained by composition
240 // of linear transformations
241 PolynomialFunctionMatrix A = HansenUtilities.buildIdentityMatrix2();
242 PolynomialFunctionMatrix D = HansenUtilities.buildZeroMatrix2();
243 PolynomialFunctionMatrix E = HansenUtilities.buildIdentityMatrix2();
244
245 // generation of polynomials associated to Hansen coefficients and to
246 // their derivatives
247 final PolynomialFunctionMatrix a = HansenUtilities.buildZeroMatrix2();
248 a.setElem(0, 1, HansenUtilities.ONE);
249
250 //The B matrix is constant.
251 final PolynomialFunctionMatrix B = HansenUtilities.buildZeroMatrix2();
252 // from Collins 4-245 and Petre's paper
253 B.setElem(1, 1, new PolynomialFunction(new double[] {
254 2.0
255 }));
256
257 for (int i = n0 - 1; i > nMin - 1; i--) {
258 index = i + offset;
259 // Matrix of the current linear transformation
260 // Petre's paper
261 a.setMatrixLine(1, new PolynomialFunction[] {
262 bZonal(s, i), aZonal(s, i)
263 });
264 // composition of linear transformations
265 // see Petre's paper
266 A = A.multiply(a);
267 // store the polynomials for Hansen coefficients
268 mpvec[index] = A.getMatrixLine(1);
269
270 D = D.multiply(a);
271 E = E.multiply(a);
272 D = D.add(E.multiply(B));
273
274 // store the polynomials for Hansen coefficients from the expressions
275 // of derivatives
276 mpvecDeriv[index] = D.getMatrixLine(1);
277
278 if (++sliceCounter % slice == 0) {
279 // Re-Initialisation of matrix for linear transformmations
280 // The final configuration of these matrix are obtained by composition
281 // of linear transformations
282 A = HansenUtilities.buildIdentityMatrix2();
283 D = HansenUtilities.buildZeroMatrix2();
284 E = HansenUtilities.buildIdentityMatrix2();
285 }
286
287 }
288 }
289
290 /**
291 * Compute polynomial coefficient a.
292 *
293 * <p>
294 * It is used to generate the coefficient for K<sub>j</sub><sup>-n, s</sup> when computing K<sub>j</sub><sup>-n-1, s</sup>
295 * and the coefficient for dK<sub>j</sub><sup>-n, s</sup> / de² when computing dK<sub>j</sub><sup>-n-1, s</sup> / de²
296 * </p>
297 *
298 * <p>
299 * See Danielson 2.7.3-(9) and Collins 4-236 and 4-240
300 * </p>
301 *
302 * @param s the s coefficient
303 * @param mnm1 -n-1
304 * @return the polynomial
305 */
306 private static PolynomialFunction aTesseral(final int s, final int mnm1) {
307 // Collins 4-236, Danielson 2.7.3-(9)
308 final double r1 = (mnm1 + 2.) * (2. * mnm1 + 5.);
309 final double r2 = (2. + mnm1 + s) * (2. + mnm1 - s);
310 return new PolynomialFunction(new double[] {
311 0.0, 0.0, r1 / r2
312 });
313 }
314
315 /**
316 * Compute polynomial coefficient b.
317 *
318 * <p>
319 * It is used to generate the coefficient for K<sub>j</sub><sup>-n+1, s</sup> when computing K<sub>j</sub><sup>-n-1, s</sup>
320 * and the coefficient for dK<sub>j</sub><sup>-n+1, s</sup> / de² when computing dK<sub>j</sub><sup>-n-1, s</sup> / de²
321 * </p>
322 *
323 * <p>
324 * See Danielson 2.7.3-(9) and Collins 4-236 and 4-240
325 * </p>
326 *
327 * @param j the j coefficient
328 * @param s the s coefficient
329 * @param mnm1 -n-1
330 * @return the polynomial
331 */
332 private static PolynomialFunction bTesseral(final int j, final int s, final int mnm1) {
333 // Collins 4-236, Danielson 2.7.3-(9)
334 final double r2 = (2. + mnm1 + s) * (2. + mnm1 - s);
335 final double d1 = (mnm1 + 3.) * 2. * j * s / (r2 * (mnm1 + 4.));
336 final double d2 = (mnm1 + 3.) * (mnm1 + 2.) / r2;
337 return new PolynomialFunction(new double[] {
338 0.0, -d1, -d2
339 });
340 }
341
342 /**
343 * Compute polynomial coefficient c.
344 *
345 * <p>
346 * It is used to generate the coefficient for K<sub>j</sub><sup>-n+3, s</sup> when computing K<sub>j</sub><sup>-n-1, s</sup>
347 * and the coefficient for dK<sub>j</sub><sup>-n+3, s</sup> / de² when computing dK<sub>j</sub><sup>-n-1, s</sup> / de²
348 * </p>
349 *
350 * <p>
351 * See Danielson 2.7.3-(9) and Collins 4-236 and 4-240
352 * </p>
353 *
354 * @param j the j coefficient
355 * @param s the s coefficient
356 * @param mnm1 -n-1
357 * @return the polynomial
358 */
359 private static PolynomialFunction cTesseral(final int j, final int s, final int mnm1) {
360 // Collins 4-236, Danielson 2.7.3-(9)
361 final double r1 = j * j * (mnm1 + 2.);
362 final double r2 = (mnm1 + 4.) * (2. + mnm1 + s) * (2. + mnm1 - s);
363
364 return new PolynomialFunction(new double[] {
365 0.0, 0.0, r1 / r2
366 });
367 }
368
369 /**
370 * Compute polynomial coefficient d.
371 * <p>
372 * It is used to generate the coefficient for K<sub>j</sub><sup>-n-1,
373 * s</sup> / dχ when computing dK<sub>j</sub><sup>-n-1, s</sup> / de²
374 * </p>
375 * <p>
376 * See Danielson 2.7.3-(9) and Collins 4-236 and 4-240
377 * </p>
378 *
379 * @return the polynomial
380 */
381 private static PolynomialFunction dTesseral() {
382 // Collins 4-236, Danielson 2.7.3-(9)
383 return new PolynomialFunction(new double[] {
384 0.0, 0.0, 1.0
385 });
386 }
387
388 /**
389 * Compute polynomial coefficient f.
390 *
391 * <p>
392 * It is used to generate the coefficient for K<sub>j</sub><sup>-n+1, s</sup> / dχ when computing dK<sub>j</sub><sup>-n-1, s</sup> / de²
393 * </p>
394 *
395 * <p>
396 * See Danielson 2.7.3-(9) and Collins 4-236 and 4-240
397 * </p>
398 *
399 * @param j the j coefficient
400 * @param s the s coefficient
401 * @param n index
402 * @return the polynomial
403 */
404 private static PolynomialFunction fTesseral(final int j, final int s, final int n) {
405 // Collins 4-236, Danielson 2.7.3-(9)
406 final double r1 = (n + 3.0) * j * s;
407 final double r2 = (n + 4.0) * (2.0 + n + s) * (2.0 + n - s);
408 return new PolynomialFunction(new double[] {
409 0.0, 0.0, 0.0, r1 / r2
410 });
411 }
412
413 /**
414 * Generate the polynomials needed in the linear transformation.
415 *
416 * @param n0 the index of the initial condition, Petre's paper
417 * @param nMin rhe minimum value for the order
418 * @param offset offset used to identify the polynomial that corresponds
419 * to a negative value of n in the internal array that
420 * starts at 0
421 * @param slice number of coefficients that will be computed with a set
422 * of roots
423 * @param j the j coefficient
424 * @param s the s coefficient
425 * @param mpvec array to store the first vector of polynomials
426 * associated to Hansen coefficients and derivatives.
427 * @param mpvecDeriv array to store the second vector of polynomials
428 * associated only to derivatives.
429 */
430 public static void generateTesseralPolynomials(final int n0, final int nMin,
431 final int offset, final int slice,
432 final int j, final int s,
433 final PolynomialFunction[][] mpvec,
434 final PolynomialFunction[][] mpvecDeriv) {
435
436
437 // Initialization of the matrices for linear transformations
438 // The final configuration of these matrices are obtained by composition
439 // of linear transformations
440
441 // The matrix of polynomials associated to Hansen coefficients, Petre's
442 // paper
443 PolynomialFunctionMatrix A = HansenUtilities.buildIdentityMatrix4();
444
445 // The matrix of polynomials associated to derivatives, Petre's paper
446 final PolynomialFunctionMatrix B = HansenUtilities.buildZeroMatrix4();
447 PolynomialFunctionMatrix D = HansenUtilities.buildZeroMatrix4();
448 final PolynomialFunctionMatrix a = HansenUtilities.buildZeroMatrix4();
449
450 // The matrix of the current linear transformation
451 a.setMatrixLine(0, new PolynomialFunction[] {
452 HansenUtilities.ZERO, HansenUtilities.ONE, HansenUtilities.ZERO, HansenUtilities.ZERO
453 });
454 a.setMatrixLine(1, new PolynomialFunction[] {
455 HansenUtilities.ZERO, HansenUtilities.ZERO, HansenUtilities.ONE, HansenUtilities.ZERO
456 });
457 a.setMatrixLine(2, new PolynomialFunction[] {
458 HansenUtilities.ZERO, HansenUtilities.ZERO, HansenUtilities.ZERO, HansenUtilities.ONE
459 });
460 // The generation process
461 int index;
462 int sliceCounter = 0;
463 for (int i = n0 - 1; i > nMin - 1; i--) {
464 index = i + offset;
465 // The matrix of the current linear transformation is updated
466 // Petre's paper
467 a.setMatrixLine(3, new PolynomialFunction[] {
468 cTesseral(j, s, i), HansenUtilities.ZERO, bTesseral(j, s, i), aTesseral(s, i)
469 });
470
471 // composition of the linear transformations to calculate
472 // the polynomials associated to Hansen coefficients
473 // Petre's paper
474 A = A.multiply(a);
475 // store the polynomials for Hansen coefficients
476 mpvec[index] = A.getMatrixLine(3);
477 // composition of the linear transformations to calculate
478 // the polynomials associated to derivatives
479 // Petre's paper
480 D = D.multiply(a);
481
482 //Update the B matrix
483 B.setMatrixLine(3, new PolynomialFunction[] {
484 HansenUtilities.ZERO, fTesseral(j, s, i),
485 HansenUtilities.ZERO, dTesseral()
486 });
487 D = D.add(A.multiply(B));
488
489 // store the polynomials for Hansen coefficients from the
490 // expressions of derivatives
491 mpvecDeriv[index] = D.getMatrixLine(3);
492
493 if (++sliceCounter % slice == 0) {
494 // Re-Initialisation of matrix for linear transformmations
495 // The final configuration of these matrix are obtained by composition
496 // of linear transformations
497 A = HansenUtilities.buildIdentityMatrix4();
498 D = HansenUtilities.buildZeroMatrix4();
499 }
500 }
501 }
502
503 /**
504 * Compute polynomial coefficient a.
505 *
506 * <p>
507 * It is used to generate the coefficient for K₀<sup>n-1, s</sup> when computing K₀<sup>n, s</sup>
508 * and the coefficient for dK₀<sup>n-1, s</sup> / dΧ when computing dK₀<sup>n, s</sup> / dΧ
509 * </p>
510 *
511 * <p>
512 * See Danielson 2.7.3-(7c) and Collins 4-254 and 4-257
513 * </p>
514 *
515 * @param n n value
516 * @return the polynomial
517 */
518 private static PolynomialFunction aThirdBody(final int n) {
519 // from recurrence Danielson 2.7.3-(7c), Collins 4-254
520 final double r1 = 2 * n + 1;
521 final double r2 = n + 1;
522
523 return new PolynomialFunction(new double[] {
524 r1 / r2
525 });
526 }
527
528 /**
529 * Compute polynomial coefficient b.
530 *
531 * <p>
532 * It is used to generate the coefficient for K₀<sup>n-2, s</sup> when computing K₀<sup>n, s</sup>
533 * and the coefficient for dK₀<sup>n-2, s</sup> / dΧ when computing dK₀<sup>n, s</sup> / dΧ
534 * </p>
535 *
536 * <p>
537 * See Danielson 2.7.3-(7c) and Collins 4-254 and 4-257
538 * </p>
539 *
540 * @param s the s coefficient
541 * @param n n value
542 * @return the polynomial
543 */
544 private static PolynomialFunction bThirdBody(final int s, final int n) {
545 // from recurrence Danielson 2.7.3-(7c), Collins 4-254
546 final double r1 = (n + s) * (n - s);
547 final double r2 = n * (n + 1);
548
549 return new PolynomialFunction(new double[] {
550 0.0, 0.0, -r1 / r2
551 });
552 }
553
554 /**
555 * Compute polynomial coefficient d.
556 *
557 * <p>
558 * It is used to generate the coefficient for K₀<sup>n-2, s</sup> when computing dK₀<sup>n, s</sup> / dΧ
559 * </p>
560 *
561 * <p>
562 * See Danielson 2.7.3-(7c) and Collins 4-254 and 4-257
563 * </p>
564 *
565 * @param s the s coefficient
566 * @param n n value
567 * @return the polynomial
568 */
569 private static PolynomialFunction dThirdBody(final int s, final int n) {
570 // from Danielson 3.2-(3b)
571 final double r1 = 2 * (n + s) * (n - s);
572 final double r2 = n * (n + 1);
573
574 return new PolynomialFunction(new double[] {
575 0.0, 0.0, 0.0, r1 / r2
576 });
577 }
578
579 /**
580 * Generate the polynomials needed in the linear transformation.
581 *
582 * @param n0 the index of the initial condition, Petre's paper
583 * @param nMax the maximum order of n indexes
584 * @param slice number of coefficients that will be computed with a set of roots
585 * @param s the s coefficient
586 * @param mpvec array to store the first vector of polynomials
587 * associated to Hansen coefficients and derivatives.
588 * @param mpvecDeriv array to store the second vector of polynomials
589 * associated only to derivatives.
590 * <p>
591 * See Petre's paper
592 * </p>
593 */
594 public static void generateThirdBodyPolynomials(final int n0, final int nMax,
595 final int slice,
596 final int s,
597 final PolynomialFunction[][] mpvec,
598 final PolynomialFunction[][] mpvecDeriv) {
599
600 int sliceCounter = 0;
601
602 // Initialization of the matrices for linear transformations
603 // The final configuration of these matrices are obtained by composition
604 // of linear transformations
605
606 // the matrix A for the polynomials associated
607 // to Hansen coefficients, Petre's pater
608 PolynomialFunctionMatrix A = HansenUtilities.buildIdentityMatrix2();
609
610 // the matrix D for the polynomials associated
611 // to derivatives, Petre's paper
612 final PolynomialFunctionMatrix B = HansenUtilities.buildZeroMatrix2();
613 PolynomialFunctionMatrix D = HansenUtilities.buildZeroMatrix2();
614 PolynomialFunctionMatrix E = HansenUtilities.buildIdentityMatrix2();
615
616 // The matrix that contains the coefficients at each step
617 final PolynomialFunctionMatrix a = HansenUtilities.buildZeroMatrix2();
618 a.setElem(0, 1, HansenUtilities.ONE);
619
620 // The generation process
621 for (int i = n0 + 2; i <= nMax; i++) {
622 // Collins 4-254 or Danielson 2.7.3-(7)
623 // Petre's paper
624 // The matrix of the current linear transformation is actualized
625 a.setMatrixLine(1, new PolynomialFunction[] {
626 bThirdBody(s, i), aThirdBody(i)
627 });
628
629 // composition of the linear transformations to calculate
630 // the polynomials associated to Hansen coefficients
631 A = A.multiply(a);
632 // store the polynomials associated to Hansen coefficients
633 mpvec[i] = A.getMatrixLine(1);
634 // composition of the linear transformations to calculate
635 // the polynomials associated to derivatives
636 // Danielson 3.2-(3b) and Petre's paper
637 D = D.multiply(a);
638 if (sliceCounter % slice != 0) {
639 a.setMatrixLine(1, new PolynomialFunction[] {
640 bThirdBody(s, i - 1), aThirdBody(i - 1)
641 });
642 E = E.multiply(a);
643 }
644
645 B.setElem(1, 0, dThirdBody(s, i));
646 // F = E.prod(B);
647 D = D.add(E.multiply(B));
648 // store the polynomials associated to the derivatives
649 mpvecDeriv[i] = D.getMatrixLine(1);
650
651 if (++sliceCounter % slice == 0) {
652 // Re-Initialization of the matrices for linear transformations
653 // The final configuration of these matrices are obtained by composition
654 // of linear transformations
655 A = HansenUtilities.buildIdentityMatrix2();
656 D = HansenUtilities.buildZeroMatrix2();
657 E = HansenUtilities.buildIdentityMatrix2();
658 }
659 }
660 }
661
662 }