maths_fns.weights

1 ############################################################################### 2 # # 3 # Copyright (C) 2004-2005 Edward d'Auvergne # 4 # # 5 # This file is part of the program relax. # 6 # # 7 # relax is free software; you can redistribute it and/or modify # 8 # it under the terms of the GNU General Public License as published by # 9 # the Free Software Foundation; either version 2 of the License, or # 10 # (at your option) any later version. # 11 # # 12 # relax is distributed in the hope that it will be useful, # 13 # but WITHOUT ANY WARRANTY; without even the implied warranty of # 14 # MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the # 15 # GNU General Public License for more details. # 16 # # 17 # You should have received a copy of the GNU General Public License # 18 # along with relax; if not, write to the Free Software # 19 # Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA # 20 # # 21 ############################################################################### 22 23 # Python module imports. 24 from math import sqrt 25 from numpy import outer 26 27 28 ########## 29 # Sphere # 30 ########## 31 32 33 # Sphere weight. 34 ################ 35

36 -def calc_sphere_ci(data, diff_data):

37 """Weight for spherical diffusion. 38 39 The equation is:: 40 41 c0 = 1. 42 """ 43 44 data.ci[0] = 1.0

45 46 47 48 49 ############ 50 # Spheroid # 51 ############ 52 53 54 # Spheroid weights. 55 ################### 56

57 -def calc_spheroid_ci(data, diff_data):

58 """Weights for spheroidal diffusion. 59 60 The equations are:: 61 62 c-1 = 1/4 (3dz**2 - 1)**2, 63 64 c0 = 3dz**2 (1 - dz**2), 65 66 c1 = 3/4 (dz**2 - 1)**2, 67 68 where dz is the direction cosine of the unit bond vector along the z-axis of the diffusion 69 tensor which is calculated as the dot product of the unit bond vector with a unit vector along 70 Dpar. 71 """ 72 73 # Components. 74 data.three_dz2_one = 3.0 * data.dz**2 - 1.0 75 data.one_two_dz2 = 1.0 - 2.0 * data.dz**2 76 data.one_dz2 = 1.0 - data.dz**2 77 data.dz2_one = -data.one_dz2 78 79 # Weights. 80 data.ci[0] = 0.25 * data.three_dz2_one**2 81 data.ci[1] = 3.0 * data.dz**2 * data.one_dz2 82 data.ci[2] = 0.75 * data.dz2_one**2

83 84 85 86 # Spheroid weight gradient. 87 ########################### 88

89 -def calc_spheroid_dci(data, diff_data):

90 """Weight gradient for spheroidal diffusion. 91 92 The equations are:: 93 94 dc-1 ddz 95 ---- = 3dz (3dz**2 - 1) --- , 96 dOi dOi 97 98 dc0 ddz 99 --- = 6dz (1 - 2dz**2) --- , 100 dOi dOi 101 102 dc1 ddz 103 --- = 3dz (dz**2 - 1) --- , 104 dOi dOi 105 106 where the orientation parameter set O is {theta, phi}. 107 """ 108 109 # Components. 110 data.dci[2:, 0] = 3.0 * data.dz * data.three_dz2_one * data.ddz_dO 111 data.dci[2:, 1] = 6.0 * data.dz * data.one_two_dz2 * data.ddz_dO 112 data.dci[2:, 2] = 3.0 * data.dz * data.dz2_one * data.ddz_dO

113 114 115 116 # Spheroid weight Hessian. 117 ########################## 118

119 -def calc_spheroid_d2ci(data, diff_data):

120 """Weight Hessian for spheroidal diffusion. 121 122 The equations are:: 123 124 d2c-1 / ddz ddz d2dz \ 125 ------- = 3 |(9dz**2 - 1) --- . --- + dz (3dz**2 - 1) ------- | , 126 dOi.dOj \ dOi dOj dOi.dOj / 127 128 d2c0 / ddz ddz d2dz \ 129 ------- = 6 |(1 - 6dz**2) --- . --- + dz (1 - 2dz**2) ------- | , 130 dOi.dOj \ dOi dOj dOi.dOj / 131 132 d2c1 / ddz ddz d2dz \ 133 ------- = 3 |(3dz**2 - 1) --- . --- + dz (dz**2 - 1) ------- | , 134 dOi.dOj \ dOi dOj dOi.dOj / 135 136 where the orientation parameter set O is {theta, phi}. 137 """ 138 139 # Outer product. 140 op = outer(data.ddz_dO, data.ddz_dO) 141 142 # Hessian. 143 data.d2ci[2:, 2:, 0] = 3.0 * ((9.0 * data.dz**2 - 1.0) * op + data.dz * data.three_dz2_one * data.d2dz_dO2) 144 data.d2ci[2:, 2:, 1] = 6.0 * ((1.0 - 6.0*data.dz**2) * op + data.dz * data.one_two_dz2 * data.d2dz_dO2) 145 data.d2ci[2:, 2:, 2] = 3.0 * (data.three_dz2_one * op + data.dz * data.dz2_one * data.d2dz_dO2)

146 147 148 149 150 ############# 151 # Ellipsoid # 152 ############# 153 154 155 # Ellipsoid weights. 156 #################### 157

158 -def calc_ellipsoid_ci(data, diff_data):

159 """Weight equations for ellipsoidal diffusion. 160 161 The equations are:: 162 163 c-2 = 1/4 (d - e), 164 165 c-1 = 3dy**2.dz**2, 166 167 c0 = 3dx**2.dz**2, 168 169 c1 = 3dx**2.dy**2, 170 171 c2 = 1/4 (d + e), 172 173 where:: 174 175 d = 3(dx**4 + dy**4 + dz**4) - 1, 176 177 e = 1/R [(1 + 3Dr)(dx**4 + 2dy**2.dz**2) + (1 - 3Dr)(dy**4 + 2dx**2.dz**2) 178 - 2(dz**4 + 2dx**2.dy**2)], 179 180 and where the factor R is defined as:: 181 ___________ 182 R = V 1 + 3Dr**2. 183 184 dx, dy, and dz are the direction cosines of the XH bond vector along the x, y, and z-axes of the 185 diffusion tensor, calculated as the dot product of the unit bond vector and the unit vectors 186 along Dx, Dy, and Dz respectively. 187 """ 188 189 # Calculate R. 190 data.R = sqrt(1 + 3.0*diff_data.params[2]**2) 191 data.inv_R = 1.0 / data.R 192 193 # Factors. 194 data.one_3Dr = 1.0 + 3.0 * diff_data.params[2] 195 data.one_m3Dr = 1.0 - 3.0 * diff_data.params[2] 196 data.dx_sqrd = data.dx**2 197 data.dy_sqrd = data.dy**2 198 data.dz_sqrd = data.dz**2 199 data.dx_cubed = data.dx**3 200 data.dy_cubed = data.dy**3 201 data.dz_cubed = data.dz**3 202 data.dx_quar = data.dx**4 203 data.dy_quar = data.dy**4 204 data.dz_quar = data.dz**4 205 206 # Components. 207 data.ex = data.dx_quar + 2.0 * data.dy_sqrd * data.dz_sqrd 208 data.ey = data.dy_quar + 2.0 * data.dx_sqrd * data.dz_sqrd 209 data.ez = data.dz_quar + 2.0 * data.dx_sqrd * data.dy_sqrd 210 211 # Calculate d. 212 d = 3.0 * (data.dx_quar + data.dy_quar + data.dz_quar) - 1.0 213 214 # Calculate e. 215 e = data.inv_R * (data.one_3Dr * data.ex + data.one_m3Dr * data.ey - 2.0 * data.ez) 216 217 # Weight c-2. 218 data.ci[0] = 0.25 * (d - e) 219 220 # Weight c-1. 221 data.ci[1] = 3.0 * data.dy**2 * data.dz**2 222 223 # Weight c0. 224 data.ci[2] = 3.0 * data.dx**2 * data.dz**2 225 226 # Weight c1. 227 data.ci[3] = 3.0 * data.dx**2 * data.dy**2 228 229 # Weight c2. 230 data.ci[4] = 0.25 * (d + e)

231 232 233 234 # Ellipsoid weight gradient. 235 ############################ 236

237 -def calc_ellipsoid_dci(data, diff_data):

238 """Weight gradient for ellipsoidal diffusion. 239 240 Oi partial derivatives 241 ====================== 242 243 The equations are:: 244 245 dc-2 / ddx ddy ddz \ de 246 ---- = 3 | dx**3 --- + dy**3 --- + dz**3 --- | - --- , 247 dOi \ dOi dOi dOi / dOi 248 249 250 dc-1 / ddz ddy \ 251 ---- = 6dy.dz | dy --- + dz --- | , 252 dOi \ dOi dOi / 253 254 255 dc0 / ddz ddx \ 256 --- = 6dx.dz | dx --- + dz --- | , 257 dOi \ dOi dOi / 258 259 260 dc1 / ddy ddx \ 261 --- = 6dx.dy | dx --- + dy --- | , 262 dOi \ dOi dOi / 263 264 265 dc2 / ddx ddy ddz \ de 266 --- = 3 | dx**3 --- + dy**3 --- + dz**3 --- | + --- , 267 dOi \ dOi dOi dOi / dOi 268 269 270 where:: 271 272 de 1 / / ddx / ddz ddy \ \ 273 --- = - | (1 + 3Dr) |dx**3 --- + dy.dz | dy --- + dz --- | | 274 dOi R \ \ dOi \ dOi dOi / / 275 276 / ddy / ddz ddx \ \ 277 + (1 - 3Dr) | dy**3 --- + dx.dz | dx --- + dz --- | | 278 \ dOi \ dOi dOi / / 279 280 / ddz / ddy ddx \ \ \ 281 - 2 | dz**3 --- + dx.dy | dx --- + dy --- | | | , 282 \ dOi \ dOi dOi / / / 283 284 285 and where the orietation parameter set O is:: 286 287 O = {alpha, beta, gamma}. 288 289 290 tm partial derivatives 291 ====================== 292 293 The equations are:: 294 295 dc-2 296 ---- = 0, 297 dtm 298 299 dc-1 300 ---- = 0, 301 dtm 302 303 dc0 304 --- = 0, 305 dtm 306 307 dc1 308 --- = 0, 309 dtm 310 311 dc2 312 --- = 0. 313 dtm 314 315 316 Da partial derivatives 317 ====================== 318 319 The equations are:: 320 321 dc-2 322 ---- = 0, 323 dDa 324 325 dc-1 326 ---- = 0, 327 dDa 328 329 dc0 330 --- = 0, 331 dDa 332 333 dc1 334 --- = 0, 335 dDa 336 337 dc2 338 --- = 0. 339 dDa 340 341 342 343 Dr partial derivatives 344 ====================== 345 346 The equations are:: 347 348 dc-2 3 de 349 ---- = - - ---, 350 dDr 4 dDr 351 352 dc-1 353 ---- = 0, 354 dDr 355 356 dc0 357 --- = 0, 358 dDr 359 360 dc1 361 --- = 0, 362 dDr 363 364 dc2 3 de 365 --- = - ---, 366 dDr 4 dDr 367 368 where:: 369 370 de 1 / \ 371 --- = ---- | (1 - Dr) (dx**4 + 2dy**2.dz**2) - (1 + Dr) (dy**4 + 2dx**2.dz**2) + 2Dr (dz**4 + 2dx**2.dy**2) | . 372 dDr R**3 \ / 373 374 375 """ 376 377 # Factors. 378 data.inv_R_cubed = data.inv_R ** 3 379 data.one_Dr = 1.0 + diff_data.params[2] 380 data.one_mDr = 1.0 - diff_data.params[2] 381 382 383 # Oi partial derivative. 384 ########################## 385 386 # Components. 387 data.ci_xy = data.dx * data.dy * (data.dx * data.ddy_dO + data.dy * data.ddx_dO) 388 data.ci_xz = data.dx * data.dz * (data.dx * data.ddz_dO + data.dz * data.ddx_dO) 389 data.ci_yz = data.dy * data.dz * (data.dy * data.ddz_dO + data.dz * data.ddy_dO) 390 391 data.ci_x = data.dx_cubed * data.ddx_dO 392 data.ci_y = data.dy_cubed * data.ddy_dO 393 data.ci_z = data.dz_cubed * data.ddz_dO 394 395 data.ci_X = data.ci_x + data.ci_yz 396 data.ci_Y = data.ci_y + data.ci_xz 397 data.ci_Z = data.ci_z + data.ci_xy 398 399 # Calculate dd_dOi. 400 dd_dOi = 3.0 * (data.ci_x + data.ci_y + data.ci_z) 401 402 # Calculate de_dOi. 403 de_dOi = data.inv_R * (data.one_3Dr * data.ci_X + data.one_m3Dr * data.ci_Y - 2.0 * data.ci_Z) 404 405 # Weight c-2. 406 data.dci[3:, 0] = dd_dOi - de_dOi 407 408 # Weight c-1. 409 data.dci[3:, 1] = 6.0 * data.ci_yz 410 411 # Weight c0. 412 data.dci[3:, 2] = 6.0 * data.ci_xz 413 414 # Weight c1. 415 data.dci[3:, 3] = 6.0 * data.ci_xy 416 417 # Weight c2. 418 data.dci[3:, 4] = dd_dOi + de_dOi 419 420 421 # Dr partial derivative. 422 ######################## 423 424 # Calculate de_dDr. 425 de_dDr = data.inv_R_cubed * (data.one_mDr * data.ex - data.one_Dr * data.ey + 2.0 * diff_data.params[2] * data.ez) 426 427 # Weight c-2. 428 data.dci[2, 0] = -0.75 * de_dDr 429 430 # Weight c2. 431 data.dci[2, 4] = 0.75 * de_dDr

432 433 434 435 # Ellipsoid weight Hessian. 436 ########################### 437

438 -def calc_ellipsoid_d2ci(data, diff_data):

439 """Weight Hessian for ellipsoidal diffusion. 440 441 Oi-Oj partial derivatives 442 ========================= 443 444 The equations are:: 445 446 d2c-2 / / d2dx ddx ddx \ / d2dy ddy ddy \ / d2dz ddz ddz \ \ d2e 447 ------- = 3 | dx**2 | dx ------- + 3 --- . --- | + dy**2 | dy ------- + 3 --- . --- | + dz**2 | dz ------- + 3 --- . --- | | - ------- , 448 dOi.dOj \ \ dOi.dOj dOi dOj / \ dOi.dOj dOi dOj / \ dOi.dOj dOi dOj / / dOi.dOj 449 450 451 d2c-1 / d2dz ddz ddz \ / ddy ddz ddz ddy \ / d2dy ddy ddy \ 452 ------- = 6 dy**2 | dz ------- + --- . --- | + 12 dy.dz | --- . --- + --- . --- | + 6 dz**2 | dy ------- + --- . --- | , 453 dOi.dOj \ dOi.dOj dOi dOj / \ dOi dOj dOi dOj / \ dOi.dOj dOi dOj / 454 455 456 d2c0 / d2dz ddz ddz \ / ddx ddz ddz ddx \ / d2dx ddx ddx \ 457 ------- = 6 dx**2 | dz ------- + --- . --- | + 12 dx.dz | --- . --- + --- . --- | + 6 dz**2 | dx ------- + --- . --- | , 458 dOi.dOj \ dOi.dOj dOi dOj / \ dOi dOj dOi dOj / \ dOi.dOj dOi dOj / 459 460 461 d2c1 / d2dy ddy ddy \ / ddx ddy ddy ddx \ / d2dx ddx ddx \ 462 ------- = 6 dx**2 | dy ------- + --- . --- | + 12 dx.dy | --- . --- + --- . --- | + 6 dy**2 | dx ------- + --- . --- | , 463 dOi.dOj \ dOi.dOj dOi dOj / \ dOi dOj dOi dOj / \ dOi.dOj dOi dOj / 464 465 466 d2c2 / / d2dx ddx ddx \ / d2dy ddy ddy \ / d2dz ddz ddz \ \ d2e 467 ------- = 3 | dx**2 | dx ------- + 3 --- . --- | + dy**2 | dy ------- + 3 --- . --- | + dz**2 | dz ------- + 3 --- . --- | | + ------- , 468 dOi.dOj \ \ dOi.dOj dOi dOj / \ dOi.dOj dOi dOj / \ dOi.dOj dOi dOj / / dOi.dOj 469 470 where:: 471 472 d2e 1 / / / d2dx ddx ddx \ / d2dz ddz ddz \ 473 ------- = - | (1 + 3Dr) | dx**2 | dx ------- + 3 --- . --- | + dy**2 | dz ------- + --- . --- | 474 dOi.dOj R \ \ \ dOi.dOj dOi dOj / \ dOi.dOj dOi dOj / 475 476 / d2dy ddy ddy \ / ddy ddz ddz ddy \ \ 477 + dz**2 | dy ------- + --- . --- | + 2dy.dz | --- . --- + --- . --- | | 478 \ dOi.dOj dOi dOj / \ dOi dOj dOi dOj / / 479 480 / / d2dy ddy ddy \ / d2dz ddz ddz \ 481 + (1 - 3Dr) | dy**2 | dy ------- + 3 --- . --- | + dx**2 | dz ------- + --- . --- | 482 \ \ dOi.dOj dOi dOj / \ dOi.dOj dOi dOj / 483 484 / d2dx ddx ddx \ / ddx ddz ddz ddx \ \ 485 + dz**2 | dx ------- + --- . --- | + 2dx.dz | --- . --- + --- . --- | | 486 \ dOi.dOj dOi dOj / \ dOi dOj dOi dOj / / 487 488 / / d2dz ddz ddz \ / d2dy ddy ddy \ 489 - 2 | dz**2 | dz ------- + 3 --- . --- | + dx**2 | dy ------- + --- . --- | 490 \ \ dOi.dOj dOi dOj / \ dOi.dOj dOi dOj / 491 492 / d2dx ddx ddx \ / ddx ddy ddy ddx \ \ \ 493 + dy**2 | dx ------- + --- . --- | + 2dx.dy | --- . --- + --- . --- | | | 494 \ dOi.dOj dOi dOj / \ dOi dOj dOi dOj / / / 495 496 497 Oi-tm partial derivatives 498 ========================= 499 500 The equations are:: 501 502 d2c-2 503 ------- = 0, 504 dOi.dtm 505 506 507 d2c-1 508 ------- = 0, 509 dOi.dtm 510 511 512 d2c0 513 ------- = 0, 514 dOi.dtm 515 516 517 d2c1 518 ------- = 0, 519 dOi.dtm 520 521 522 d2c2 523 ------- = 0. 524 dOi.dtm 525 526 527 Oi-Da partial derivatives 528 ========================= 529 530 The equations are:: 531 532 d2c-2 533 ------- = 0, 534 dOi.dDa 535 536 537 d2c-1 538 ------- = 0, 539 dOi.dDa 540 541 542 d2c0 543 ------- = 0, 544 dOi.dDa 545 546 547 d2c1 548 ------- = 0, 549 dOi.dDa 550 551 552 d2c2 553 ------- = 0. 554 dOi.dDa 555 556 557 Oi-Dr partial derivatives 558 ========================= 559 560 The equations are:: 561 562 d2c-2 d2e 563 ------- = - 3 -------, 564 dOi.dDr dOi.dDr 565 566 567 d2c-1 568 ------- = 0, 569 dOi.dDr 570 571 572 d2c0 573 ------- = 0, 574 dOi.dDr 575 576 577 d2c1 578 ------- = 0, 579 dOi.dDr 580 581 582 d2c2 d2e 583 ------- = 3 -------, 584 dOi.dDr dOi.dDr 585 586 where:: 587 588 d2e 1 / / ddx / ddz ddy \ \ 589 ------- = ---- | (1 - Dr) | dx**3 --- + dy.dz | dy --- + dz --- | | 590 dOi.dDr R**3 \ \ dOi \ dOi dOi / / 591 592 / ddy / ddz ddx \ \ 593 - (1 + Dr) | dy**3 --- + dx.dz | dx --- + dz --- | | 594 \ dOi \ dOi dOi / / 595 596 / ddz / ddy ddx \ \ \ 597 + 2Dr | dz**3 --- + dx.dy | dx --- + dy --- | | | 598 \ dOi \ dOi dOi / / / 599 600 601 tm-tm partial derivatives 602 ========================= 603 604 The equations are:: 605 606 d2c-2 607 ----- = 0, 608 dtm2 609 610 611 d2c-1 612 ----- = 0, 613 dtm2 614 615 616 d2c0 617 ---- = 0, 618 dtm2 619 620 621 d2c1 622 ---- = 0, 623 dtm2 624 625 626 d2c2 627 ---- = 0. 628 dtm2 629 630 631 tm-Da partial derivatives 632 ========================= 633 634 The equations are:: 635 636 d2c-2 637 ------- = 0, 638 dtm.dDa 639 640 641 d2c-1 642 ------- = 0, 643 dtm.dDa 644 645 646 d2c0 647 ------- = 0, 648 dtm.dDa 649 650 651 d2c1 652 ------- = 0, 653 dtm.dDa 654 655 656 d2c2 657 ------- = 0. 658 dtm.dDa 659 660 661 tm-Dr partial derivatives 662 ========================= 663 664 The equations are:: 665 666 d2c-2 667 ------- = 0, 668 dtm.dDr 669 670 671 d2c-1 672 ------- = 0, 673 dtm.dDr 674 675 676 d2c0 677 ------- = 0, 678 dtm.dDr 679 680 681 d2c1 682 ------- = 0, 683 dtm.dDr 684 685 686 d2c2 687 ------- = 0. 688 dtm.dDr 689 690 691 Da-Da partial derivatives 692 ========================= 693 694 The equations are:: 695 696 d2c-2 697 ------ = 0, 698 dDa**2 699 700 701 d2c-1 702 ------ = 0, 703 dDa**2 704 705 706 d2c0 707 ------ = 0, 708 dDa**2 709 710 711 d2c1 712 ------ = 0, 713 dDa**2 714 715 716 d2c2 717 ------ = 0. 718 dDa**2 719 720 721 Da-Dr partial derivatives 722 ========================= 723 724 The equations are:: 725 726 d2c-2 727 ------- = 0, 728 dDa.dDr 729 730 731 d2c-1 732 ------- = 0, 733 dDa.dDr 734 735 736 d2c0 737 ------- = 0, 738 dDa.dDr 739 740 741 d2c1 742 ------- = 0, 743 dDa.dDr 744 745 746 d2c2 747 ------- = 0. 748 dDa.dDr 749 750 751 Dr-Dr partial derivatives 752 ========================= 753 754 The equations are:: 755 756 d2c-2 3 d2e 757 ------ = - - ------, 758 dDr**2 4 dDr**2 759 760 761 d2c-1 762 ------ = 0, 763 dDr**2 764 765 766 d2c0 767 ------ = 0, 768 dDr**2 769 770 771 d2c1 772 ------ = 0, 773 dDr**2 774 775 776 d2c2 3 d2e 777 ------ = - ------, 778 dDr**2 4 dDr**2 779 780 where:: 781 782 d2e 1 / \ 783 ------ = ---- | (6Dr**2 - 9Dr - 1)(dx**4 + 2dy**2.dz**2) + (6Dr**2 + 9Dr - 1)(dy**4 + 2dx**2.dz**2) - 2(6Dr**2 - 1)(ddz*4 + 2dx**2.dy**2) | 784 dDr**2 R**5 \ / 785 """ 786 787 # Oi-Oj partial derivative. 788 ############################### 789 790 # Outer products. 791 op_xx = outer(data.ddx_dO, data.ddx_dO) 792 op_yy = outer(data.ddy_dO, data.ddy_dO) 793 op_zz = outer(data.ddz_dO, data.ddz_dO) 794 795 op_xy = outer(data.ddx_dO, data.ddy_dO) 796 op_yx = outer(data.ddy_dO, data.ddx_dO) 797 798 op_xz = outer(data.ddx_dO, data.ddz_dO) 799 op_zx = outer(data.ddz_dO, data.ddx_dO) 800 801 op_yz = outer(data.ddy_dO, data.ddz_dO) 802 op_zy = outer(data.ddz_dO, data.ddy_dO) 803 804 # Components. 805 x_comp = data.dx * data.d2dx_dO2 + op_xx 806 y_comp = data.dy * data.d2dy_dO2 + op_yy 807 z_comp = data.dz * data.d2dz_dO2 + op_zz 808 809 x3_comp = data.dx_sqrd * (data.dx * data.d2dx_dO2 + 3.0 * op_xx) 810 y3_comp = data.dy_sqrd * (data.dy * data.d2dy_dO2 + 3.0 * op_yy) 811 z3_comp = data.dz_sqrd * (data.dz * data.d2dz_dO2 + 3.0 * op_zz) 812 813 xy_comp = data.dx_sqrd * y_comp + 2.0 * data.dx * data.dy * (op_xy + op_yx) + data.dy_sqrd * x_comp 814 xz_comp = data.dx_sqrd * z_comp + 2.0 * data.dx * data.dz * (op_xz + op_zx) + data.dz_sqrd * x_comp 815 yz_comp = data.dy_sqrd * z_comp + 2.0 * data.dy * data.dz * (op_yz + op_zy) + data.dz_sqrd * y_comp 816 817 # Calculate d2d_dOidOj. 818 d2d_dOidOj = 3.0 * (x3_comp + y3_comp + z3_comp) 819 820 # Calculate d2e_dOidOj. 821 d2e_dOidOj = data.inv_R * (data.one_3Dr * (x3_comp + yz_comp) + data.one_m3Dr * (y3_comp + xz_comp) - 2.0 * (z3_comp + xy_comp)) 822 823 # Weight c-2. 824 data.d2ci[3:, 3:, 0] = d2d_dOidOj - d2e_dOidOj 825 826 # Weight c-2. 827 data.d2ci[3:, 3:, 1] = 6.0 * yz_comp 828 829 # Weight c-1. 830 data.d2ci[3:, 3:, 2] = 6.0 * xz_comp 831 832 # Weight c1. 833 data.d2ci[3:, 3:, 3] = 6.0 * xy_comp 834 835 # Weight c2. 836 data.d2ci[3:, 3:, 4] = d2d_dOidOj + d2e_dOidOj 837 838 839 # Oi-Dr partial derivative. 840 ############################# 841 842 # Calculate d2e_dOidDr. 843 d2e_dOidDr = data.inv_R_cubed * (data.one_mDr * data.ci_X - data.one_Dr * data.ci_Y + 2.0 * diff_data.params[2] * data.ci_Z) 844 845 # Weight c0. 846 data.d2ci[3:, 2, 0] = data.d2ci[2, 3:, 0] = -3.0 * d2e_dOidDr 847 848 # Weight c2. 849 data.d2ci[3:, 2, 4] = data.d2ci[2, 3:, 4] = 3.0 * d2e_dOidDr 850 851 852 # Dr-Dr partial derivative. 853 ########################### 854 855 # Components 856 d2e1_dDr2 = 6.0 * diff_data.params[2]**2 - 9.0 * diff_data.params[2] - 1.0 857 d2e2_dDr2 = 6.0 * diff_data.params[2]**2 + 9.0 * diff_data.params[2] - 1.0 858 d2e3_dDr2 = 6.0 * diff_data.params[2]**2 - 1.0 859 860 # Calculate d2e_dDr2. 861 d2e_dDr2 = data.inv_R**5 * (d2e1_dDr2 * data.ex + d2e2_dDr2 * data.ey - 2.0 * d2e3_dDr2 * data.ez) 862 863 # Weight c0. 864 data.d2ci[2, 2, 0] = -0.75 * d2e_dDr2 865 866 # Weight c2. 867 data.d2ci[2, 2, 4] = 0.75 * d2e_dDr2

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Source Code for Module maths_fns.weights