maths_fns.jw_mf

1 ############################################################################### 2 # # 3 # Copyright (C) 2003-2005 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 24 from Numeric import Float64, zeros 25 26 27 28 ############################ 29 # Spectral density values. # 30 ############################ 31 32 33 # Original {S2, te}. 34 #################### 35

36 -def calc_S2_te_jw_comps(data, params):

37 """Spectral density component function. 38 39 Calculate the components of the spectral density value for the original model-free formula with 40 the parameters {S2, te}. 41 42 The model-free formula is 43 44 _n_ 45 2 \ / S2 (1 - S2)(te + ti)te \ 46 J(w) = - > ci . ti | ------------ + ------------------------- |. 47 5 /__ \ 1 + (w.ti)^2 (te + ti)^2 + (w.te.ti)^2 / 48 i=m 49 50 Replicated calculations are 51 52 w_ti_sqrd = (w.ti)^2 (pre-calculated during initialisation), 53 54 te_ti = te + ti, 55 te_ti_te = (te + ti).te. 56 57 58 Calculations which are replicated in the gradient equations are 59 60 fact_ti = 1 / (1 + (w.ti)^2) (pre-calculated during initialisation), 61 62 one_s2 = 1 - S2, 63 64 te_ti_sqrd = (te + ti)^2, 65 w_te_ti_sqrd = (w.te.ti)^2, 66 67 (te + ti)te 68 fact_te = -------------------------. 69 (te + ti)^2 + (w.te.ti)^2 70 71 """ 72 73 # Order parameter factors. 74 data.one_s2 = 1.0 - params[data.s2_i] 75 76 # Internal correlation time, te, factors. 77 data.te_ti = params[data.te_i] + data.ti 78 data.te_ti_te = data.te_ti * params[data.te_i] 79 data.te_ti_sqrd = data.te_ti**2 80 81 data.w_te_ti_sqrd = data.w_ti_sqrd * params[data.te_i]**2 82 data.inv_te_denom = 1.0 / (data.te_ti_sqrd + data.w_te_ti_sqrd) 83 data.fact_te = data.te_ti_te * data.inv_te_denom

84 85 86 87 # Extended {S2f, S2, ts}. 88 ######################### 89

90 -def calc_S2f_S2_ts_jw_comps(data, params):

91 """Spectral density component function. 92 93 Calculate the components of the spectral density value for the extended model-free formula with 94 the parameters {S2f, S2, ts}. 95 96 The model-free formula is 97 98 _n_ 99 2 \ / S2 (S2f - S2)(ts + ti)ts \ 100 J(w) = - > ci . ti | ------------ + ------------------------- |. 101 5 /__ \ 1 + (w.ti)^2 (ts + ti)^2 + (w.ts.ti)^2 / 102 i=m 103 104 Replicated calculations are 105 106 w_ti_sqrd = (w.ti)^2 (pre-calculated during initialisation), 107 108 ts_ti = ts + ti, 109 ts_ti_ts = (ts + ti).ts. 110 111 112 Calculations which are replicated in the gradient equations are 113 114 fact_ti = 1 / (1 + (w.ti)^2) (pre-calculated during initialisation), 115 116 s2f_s2 = S2f - S2, 117 118 ts_ti_sqrd = (ts + ti)^2, 119 w_ts_ti_sqrd = (w.ts.ti)^2, 120 inv_ts_denom = 1 / ((ts + ti)^2 + (w.ts.ti)^2), 121 122 (ts + ti)ts 123 fact_ts = -------------------------. 124 (ts + ti)^2 + (w.ts.ti)^2 125 126 """ 127 128 # Order parameter factors. 129 data.s2f_s2 = params[data.s2f_i] - params[data.s2_i] 130 131 # Internal correlation time, ts, factors. 132 data.ts_ti = params[data.ts_i] + data.ti 133 data.ts_ti_ts = data.ts_ti * params[data.ts_i] 134 data.ts_ti_sqrd = data.ts_ti**2 135 136 data.w_ts_ti_sqrd = data.w_ti_sqrd * params[data.ts_i]**2 137 data.inv_ts_denom = 1.0 / (data.ts_ti_sqrd + data.w_ts_ti_sqrd) 138 data.fact_ts = data.ts_ti_ts * data.inv_ts_denom

139 140 141 142 # Extended 2 {S2f, S2s, ts}. 143 ############################ 144

145 -def calc_S2f_S2s_ts_jw_comps(data, params):

146 """Spectral density component function. 147 148 Calculate the components of the spectral density value for the extended model-free formula with 149 the parameters {S2f, S2s, ts}. 150 151 The model-free formula is 152 153 _n_ 154 2 \ / S2s (1 - S2s)(ts + ti)ts \ 155 J(w) = - S2f > ci . ti | ------------ + ------------------------- |. 156 5 /__ \ 1 + (w.ti)^2 (ts + ti)^2 + (w.ts.ti)^2 / 157 i=m 158 159 Replicated calculations are 160 161 w_ti_sqrd = (w.ti)^2 (pre-calculated during initialisation), 162 163 ts_ti = ts + ti, 164 ts_ti_ts = (ts + ti).ts. 165 166 167 Calculations which are replicated in the gradient equations are 168 169 fact_ti = 1 / (1 + (w.ti)^2) (pre-calculated during initialisation), 170 171 one_s2s = 1 - S2s, 172 173 ts_ti_sqrd = (ts + ti)^2, 174 w_ts_ti_sqrd = (w.ts.ti)^2, 175 inv_ts_denom = 1 / ((ts + ti)^2 + (w.ts.ti)^2), 176 177 (ts + ti)ts 178 fact_ts = -------------------------. 179 (ts + ti)^2 + (w.ts.ti)^2 180 181 """ 182 183 # Order parameter factors. 184 data.one_s2s = 1.0 - params[data.s2s_i] 185 data.s2f_s2 = data.one_s2s * params[data.s2f_i] 186 187 # Internal correlation time, ts, factors. 188 data.ts_ti = params[data.ts_i] + data.ti 189 data.ts_ti_ts = data.ts_ti * params[data.ts_i] 190 data.ts_ti_sqrd = data.ts_ti**2 191 192 data.w_ts_ti_sqrd = data.w_ti_sqrd * params[data.ts_i]**2 193 data.inv_ts_denom = 1.0 / (data.ts_ti_sqrd + data.w_ts_ti_sqrd) 194 data.fact_ts = data.ts_ti_ts * data.inv_ts_denom

195 196 197 198 # Extended {S2f, tf, S2, ts}. 199 ############################# 200

201 -def calc_S2f_tf_S2_ts_jw_comps(data, params):

202 """Spectral density component function. 203 204 Calculate the components of the spectral density value for the extended model-free formula with 205 the parameters {S2f, tf, S2, ts}. 206 207 The model-free formula is 208 209 _n_ 210 2 \ / S2 (1 - S2f)(tf + ti)tf 211 J(w) = - > ci . ti | ------------ + ------------------------- 212 5 /__ \ 1 + (w.ti)^2 (tf + ti)^2 + (w.tf.ti)^2 213 i=m 214 215 (S2f - S2)(ts + ti)ts \ 216 + ------------------------- |. 217 (ts + ti)^2 + (w.ts.ti)^2 / 218 219 Replicated calculations are 220 221 w_ti_sqrd = (w.ti)^2 (pre-calculated during initialisation), 222 223 tf_ti = tf + ti, 224 ts_ti = ts + ti, 225 tf_ti_tf = (tf + ti).tf, 226 ts_ti_ts = (ts + ti).ts. 227 228 229 Calculations which are replicated in the gradient equations are 230 231 fact_ti = 1 / (1 + (w.ti)^2) (pre-calculated during initialisation), 232 233 one_s2f = 1 - S2f, 234 s2f_s2 = S2f - S2, 235 236 tf_ti_sqrd = (tf + ti)^2, 237 ts_ti_sqrd = (ts + ti)^2, 238 w_tf_ti_sqrd = (w.tf.ti)^2, 239 w_ts_ti_sqrd = (w.ts.ti)^2, 240 inv_tf_denom = 1 / ((tf + ti)^2 + (w.tf.ti)^2), 241 inv_ts_denom = 1 / ((ts + ti)^2 + (w.ts.ti)^2), 242 243 (tf + ti)tf 244 fact_tf = -------------------------, 245 (tf + ti)^2 + (w.tf.ti)^2 246 247 248 (ts + ti)ts 249 fact_ts = -------------------------. 250 (ts + ti)^2 + (w.ts.ti)^2 251 252 """ 253 254 # Order parameter factors. 255 data.one_s2f = 1.0 - params[data.s2f_i] 256 data.s2f_s2 = params[data.s2f_i] - params[data.s2_i] 257 258 # Internal correlation time, tf, factors. 259 data.tf_ti = params[data.tf_i] + data.ti 260 data.tf_ti_tf = data.tf_ti * params[data.tf_i] 261 data.tf_ti_sqrd = data.tf_ti**2 262 263 data.w_tf_ti_sqrd = data.w_ti_sqrd * params[data.tf_i]**2 264 data.inv_tf_denom = 1.0 / (data.tf_ti_sqrd + data.w_tf_ti_sqrd) 265 data.fact_tf = data.tf_ti_tf * data.inv_tf_denom 266 267 # Internal correlation time, ts, factors. 268 data.ts_ti = params[data.ts_i] + data.ti 269 data.ts_ti_ts = data.ts_ti * params[data.ts_i] 270 data.ts_ti_sqrd = data.ts_ti**2 271 272 data.w_ts_ti_sqrd = data.w_ti_sqrd * params[data.ts_i]**2 273 data.inv_ts_denom = 1.0 / (data.ts_ti_sqrd + data.w_ts_ti_sqrd) 274 data.fact_ts = data.ts_ti_ts * data.inv_ts_denom

275 276 277 278 # Extended 2 {S2f, tf, S2s, ts}. 279 ################################ 280

281 -def calc_S2f_tf_S2s_ts_jw_comps(data, params):

282 """Spectral density component function. 283 284 Calculate the components of the spectral density value for the extended model-free formula with 285 the parameters {S2f, tf, S2s, ts}. 286 287 The model-free formula is 288 289 _n_ 290 2 \ / S2f . S2s (1 - S2f)(tf + ti)tf 291 J(w) = - > ci . ti | ------------ + ------------------------- 292 5 /__ \ 1 + (w.ti)^2 (tf + ti)^2 + (w.tf.ti)^2 293 i=m 294 295 S2f(1 - S2s)(ts + ti)ts \ 296 + ------------------------- |. 297 (ts + ti)^2 + (w.ts.ti)^2 / 298 299 Replicated calculations are 300 301 w_ti_sqrd = (w.ti)^2 (pre-calculated during initialisation), 302 303 tf_ti = tf + ti, 304 ts_ti = ts + ti, 305 tf_ti_tf = (tf + ti).tf, 306 ts_ti_ts = (ts + ti).ts. 307 308 309 Calculations which are replicated in the gradient equations are 310 311 fact_ti = 1 / (1 + (w.ti)^2) (pre-calculated during initialisation), 312 313 one_s2s = 1 - S2s, 314 one_s2f = 1 - S2f, 315 s2f_s2 = S2f(1 - S2s) = S2f - S2, 316 317 tf_ti_sqrd = (tf + ti)^2, 318 ts_ti_sqrd = (ts + ti)^2, 319 w_tf_ti_sqrd = (w.tf.ti)^2, 320 w_ts_ti_sqrd = (w.ts.ti)^2, 321 inv_tf_denom = 1 / ((tf + ti)^2 + (w.tf.ti)^2), 322 inv_ts_denom = 1 / ((ts + ti)^2 + (w.ts.ti)^2), 323 324 (tf + ti)tf 325 fact_tf = -------------------------, 326 (tf + ti)^2 + (w.tf.ti)^2 327 328 329 (ts + ti)ts 330 fact_ts = -------------------------. 331 (ts + ti)^2 + (w.ts.ti)^2 332 333 """ 334 335 # Order parameter factors. 336 data.one_s2s = 1.0 - params[data.s2s_i] 337 data.one_s2f = 1.0 - params[data.s2f_i] 338 data.s2f_s2 = params[data.s2f_i] * data.one_s2s 339 340 # Internal correlation time, tf, factors. 341 data.tf_ti = params[data.tf_i] + data.ti 342 data.tf_ti_tf = data.tf_ti * params[data.tf_i] 343 data.tf_ti_sqrd = data.tf_ti**2 344 345 data.w_tf_ti_sqrd = data.w_ti_sqrd * params[data.tf_i]**2 346 data.inv_tf_denom = 1.0 / (data.tf_ti_sqrd + data.w_tf_ti_sqrd) 347 data.fact_tf = data.tf_ti_tf * data.inv_tf_denom 348 349 # Internal correlation time, ts, factors. 350 data.ts_ti = params[data.ts_i] + data.ti 351 data.ts_ti_ts = data.ts_ti * params[data.ts_i] 352 data.ts_ti_sqrd = data.ts_ti**2 353 354 data.w_ts_ti_sqrd = data.w_ti_sqrd * params[data.ts_i]**2 355 data.inv_ts_denom = 1.0 / (data.ts_ti_sqrd + data.w_ts_ti_sqrd) 356 data.fact_ts = data.ts_ti_ts * data.inv_ts_denom

357 358 359 360 361 ############################### 362 # Spectral density gradients. # 363 ############################### 364 365 366 # Original {} and {S2} with diffusion parameters. 367 ################################################# 368

369 -def calc_diff_djw_comps(data, params):

370 """Spectral density gradient component function. 371 372 Calculate the components of the spectral density gradient for the original model-free formula 373 with no parameters {} or the parameter {S2} together with diffusion tensor parameters. 374 375 Replicated calculations are 376 377 1 - (w.ti)^2 378 fact_ti_djw_dti = ----------------. 379 (1 + (w.ti)^2)^2 380 """ 381 382 data.fact_ti_djw_dti = (1.0 - data.w_ti_sqrd) * data.fact_ti**2

383 384 385 386 # Original {S2, te}. 387 #################### 388

389 -def calc_S2_te_djw_comps(data, params):

390 """Spectral density gradient component function. 391 392 Calculate the components of the spectral density gradient for the original model-free formula 393 with the parameters {S2, te}. 394 395 Replicated calculations are 396 397 (te + ti)^2 - (w.te.ti)^2 398 fact_djw_dte = ti^2 -----------------------------. 399 ((te + ti)^2 + (w.te.ti)^2)^2 400 """ 401 402 data.fact_djw_dte = data.ti**2 * (data.te_ti_sqrd - data.w_te_ti_sqrd) * data.inv_te_denom**2

403 404 405 406 # Original {S2, te} with diffusion parameters. 407 ############################################## 408

409 -def calc_diff_S2_te_djw_comps(data, params):

410 """Spectral density gradient component function. 411 412 Calculate the components of the spectral density gradient for the original model-free formula 413 with the parameters {S2, te} together with diffusion tensor parameters. 414 415 Replicated calculations are 416 417 1 - (w.ti)^2 418 fact_ti_djw_dti = ----------------, 419 (1 + (w.ti)^2)^2 420 421 422 (te + ti)^2 - (w.te.ti)^2 423 fact_djw = -----------------------------, 424 ((te + ti)^2 + (w.te.ti)^2)^2 425 426 427 (te + ti)^2 - (w.te.ti)^2 428 fact_te_djw_dti = te^2 -----------------------------, 429 ((te + ti)^2 + (w.te.ti)^2)^2 430 431 432 (te + ti)^2 - (w.te.ti)^2 433 fact_djw_dte = ti^2 -----------------------------. 434 ((te + ti)^2 + (w.te.ti)^2)^2 435 """ 436 437 fact_djw = (data.te_ti_sqrd - data.w_te_ti_sqrd) * data.inv_te_denom**2 438 data.fact_ti_djw_dti = (1.0 - data.w_ti_sqrd) * data.fact_ti**2 439 data.fact_te_djw_dti = params[data.te_i]**2 * fact_djw 440 data.fact_djw_dte = data.ti**2 * fact_djw

441 442 443 444 # Extended {S2f, S2, ts}. 445 ######################### 446

447 -def calc_S2f_S2_ts_djw_comps(data, params):

448 """Spectral density gradient component function. 449 450 Calculate the components of the spectral density gradient for the extended model-free formula 451 with the parameters {S2f, S2, ts}. 452 453 Replicated calculations are 454 455 456 (ts + ti)^2 - (w.ts.ti)^2 457 fact_djw_dts = ti^2 -----------------------------. 458 ((ts + ti)^2 + (w.ts.ti)^2)^2 459 """ 460 461 data.fact_djw_dts = data.ti**2 * (data.ts_ti_sqrd - data.w_ts_ti_sqrd) * data.inv_ts_denom**2

462 463 464 465 # Extended {S2f, S2, ts} with diffusion parameters. 466 ################################################### 467

468 -def calc_diff_S2f_S2_ts_djw_comps(data, params):

469 """Spectral density gradient component function. 470 471 Calculate the components of the spectral density gradient for the extended model-free formula 472 with the parameters {S2f, S2, ts} together with diffusion tensor parameters. 473 474 Replicated calculations are 475 476 1 - (w.ti)^2 477 fact_ti_djw_dti = ----------------, 478 (1 + (w.ti)^2)^2 479 480 481 (ts + ti)^2 - (w.ts.ti)^2 482 fact_djw = -----------------------------, 483 ((ts + ti)^2 + (w.ts.ti)^2)^2 484 485 486 (ts + ti)^2 - (w.ts.ti)^2 487 fact_ts_djw_dti = ts^2 -----------------------------, 488 ((ts + ti)^2 + (w.ts.ti)^2)^2 489 490 491 (ts + ti)^2 - (w.ts.ti)^2 492 fact_djw_dts = ti^2 -----------------------------. 493 ((ts + ti)^2 + (w.ts.ti)^2)^2 494 """ 495 496 497 fact_djw = (data.ts_ti_sqrd - data.w_ts_ti_sqrd) * data.inv_ts_denom**2 498 data.fact_ti_djw_dti = (1.0 - data.w_ti_sqrd) * data.fact_ti**2 499 data.fact_ts_djw_dti = params[data.ts_i]**2 * fact_djw 500 data.fact_djw_dts = data.ti**2 * fact_djw

501 502 503 504 # Extended {S2f, tf, S2, ts}. 505 ############################# 506

507 -def calc_S2f_tf_S2_ts_djw_comps(data, params):

508 """Spectral density gradient component function. 509 510 Calculate the components of the spectral density gradient for the extended model-free formula 511 with the parameters {S2f, tf, S2, ts}. 512 513 Replicated calculations are 514 515 (tf + ti)^2 - (w.tf.ti)^2 516 fact_djw_dtf = ti^2 -----------------------------, 517 ((tf + ti)^2 + (w.tf.ti)^2)^2 518 519 520 (ts + ti)^2 - (w.ts.ti)^2 521 fact_djw_dts = ti^2 -----------------------------. 522 ((ts + ti)^2 + (w.ts.ti)^2)^2 523 """ 524 525 data.fact_djw_dtf = data.ti**2 * (data.tf_ti_sqrd - data.w_tf_ti_sqrd) * data.inv_tf_denom**2 526 data.fact_djw_dts = data.ti**2 * (data.ts_ti_sqrd - data.w_ts_ti_sqrd) * data.inv_ts_denom**2

527 528 529 530 # Extended {S2f, tf, S2, ts} with diffusion parameters. 531 ####################################################### 532

533 -def calc_diff_S2f_tf_S2_ts_djw_comps(data, params):

534 """Spectral density gradient component function. 535 536 Calculate the components of the spectral density gradient for the extended model-free formula 537 with the parameters {S2f, tf, S2, ts} together with diffusion tensor parameters. 538 539 Replicated calculations are 540 541 1 - (w.ti)^2 542 fact_ti_djw_dti = ----------------, 543 (1 + (w.ti)^2)^2 544 545 546 (tf + ti)^2 - (w.tf.ti)^2 547 fact_tf_djw = -----------------------------, 548 ((tf + ti)^2 + (w.tf.ti)^2)^2 549 550 551 (ts + ti)^2 - (w.ts.ti)^2 552 fact_ts_djw = -----------------------------, 553 ((ts + ti)^2 + (w.ts.ti)^2)^2 554 555 556 (tf + ti)^2 - (w.tf.ti)^2 557 fact_tf_djw_dti = tf^2 -----------------------------, 558 ((tf + ti)^2 + (w.tf.ti)^2)^2 559 560 561 (ts + ti)^2 - (w.ts.ti)^2 562 fact_ts_djw_dti = ts^2 -----------------------------, 563 ((ts + ti)^2 + (w.ts.ti)^2)^2 564 565 566 (tf + ti)^2 - (w.tf.ti)^2 567 fact_djw_dtf = ti^2 -----------------------------, 568 ((tf + ti)^2 + (w.tf.ti)^2)^2 569 570 571 (ts + ti)^2 - (w.ts.ti)^2 572 fact_djw_dts = ti^2 -----------------------------. 573 ((ts + ti)^2 + (w.ts.ti)^2)^2 574 """ 575 576 # ti. 577 data.fact_ti_djw_dti = (1.0 - data.w_ti_sqrd) * data.fact_ti**2 578 579 # tf. 580 fact_tf_djw = (data.tf_ti_sqrd - data.w_tf_ti_sqrd) * data.inv_tf_denom**2 581 data.fact_tf_djw_dti = params[data.tf_i]**2 * fact_tf_djw 582 data.fact_djw_dtf = data.ti**2 * fact_tf_djw 583 584 # ts. 585 fact_ts_djw = (data.ts_ti_sqrd - data.w_ts_ti_sqrd) * data.inv_ts_denom**2 586 data.fact_ts_djw_dti = params[data.ts_i]**2 * fact_ts_djw 587 data.fact_djw_dts = data.ti**2 * fact_ts_djw

588 589 590 591 # Extended 2 {S2f, S2s, ts}. 592 ############################ 593

594 -def calc_S2f_S2s_ts_djw_comps(data, params):

595 """Spectral density gradient component function. 596 597 Calculate the components of the spectral density gradient for the extended model-free formula 598 with the parameters {S2f, S2s, ts}. 599 600 Replicated calculations are 601 602 (ts + ti)^2 - (w.ts.ti)^2 603 fact_djw_dts = ti^2 -----------------------------. 604 ((ts + ti)^2 + (w.ts.ti)^2)^2 605 """ 606 607 data.fact_djw_dts = data.ti**2 * (data.ts_ti_sqrd - data.w_ts_ti_sqrd) * data.inv_ts_denom**2

608 609 610 611 # Extended 2 {S2f, S2s, ts} with diffusion parameters. 612 ###################################################### 613

614 -def calc_diff_S2f_S2s_ts_djw_comps(data, params):

615 """Spectral density gradient component function. 616 617 Calculate the components of the spectral density gradient for the extended model-free formula 618 with the parameters {S2f, S2s, ts} together with diffusion tensor parameters. 619 620 Replicated calculations are 621 622 1 - (w.ti)^2 623 fact_ti_djw_dti = ----------------, 624 (1 + (w.ti)^2)^2 625 626 627 (ts + ti)^2 - (w.ts.ti)^2 628 fact_djw = -----------------------------, 629 ((ts + ti)^2 + (w.ts.ti)^2)^2 630 631 632 (ts + ti)^2 - (w.ts.ti)^2 633 fact_ts_djw_dti = ts^2 -----------------------------, 634 ((ts + ti)^2 + (w.ts.ti)^2)^2 635 636 637 (ts + ti)^2 - (w.ts.ti)^2 638 fact_djw_dts = ti^2 -----------------------------. 639 ((ts + ti)^2 + (w.ts.ti)^2)^2 640 """ 641 642 fact_djw = (data.ts_ti_sqrd - data.w_ts_ti_sqrd) * data.inv_ts_denom**2 643 data.fact_ti_djw_dti = (1.0 - data.w_ti_sqrd) * data.fact_ti**2 644 data.fact_ts_djw_dti = params[data.ts_i]**2 * fact_djw 645 data.fact_djw_dts = data.ti**2 * fact_djw

646 647 648 649 # Extended 2 {S2f, tf, S2s, ts}. 650 ################################ 651

652 -def calc_S2f_tf_S2s_ts_djw_comps(data, params):

653 """Spectral density gradient component function. 654 655 Calculate the components of the spectral density gradient for the extended model-free formula 656 with the parameters {S2f, tf, S2s, ts}. 657 658 Replicated calculations are 659 660 (tf + ti)^2 - (w.tf.ti)^2 661 fact_djw_dtf = ti^2 -----------------------------, 662 ((tf + ti)^2 + (w.tf.ti)^2)^2 663 664 665 (ts + ti)^2 - (w.ts.ti)^2 666 fact_djw_dts = ti^2 -----------------------------. 667 ((ts + ti)^2 + (w.ts.ti)^2)^2 668 """ 669 670 data.fact_djw_dtf = data.ti**2 * (data.tf_ti_sqrd - data.w_tf_ti_sqrd) * data.inv_tf_denom**2 671 data.fact_djw_dts = data.ti**2 * (data.ts_ti_sqrd - data.w_ts_ti_sqrd) * data.inv_ts_denom**2

672 673 674 675 # Extended 2 {S2f, tf, S2s, ts} with diffusion parameters. 676 ########################################################## 677

678 -def calc_diff_S2f_tf_S2s_ts_djw_comps(data, params):

679 """Spectral density gradient component function. 680 681 Calculate the components of the spectral density gradient for the extended model-free formula 682 with the parameters {S2f, tf, S2, ts} together with diffusion tensor parameters. 683 684 Replicated calculations are 685 686 1 - (w.ti)^2 687 fact_ti_djw_dti = ----------------, 688 (1 + (w.ti)^2)^2 689 690 691 (tf + ti)^2 - (w.tf.ti)^2 692 fact_tf_djw = -----------------------------, 693 ((tf + ti)^2 + (w.tf.ti)^2)^2 694 695 696 (ts + ti)^2 - (w.ts.ti)^2 697 fact_ts_djw = -----------------------------, 698 ((ts + ti)^2 + (w.ts.ti)^2)^2 699 700 701 (tf + ti)^2 - (w.tf.ti)^2 702 fact_tf_djw_dti = tf^2 -----------------------------, 703 ((tf + ti)^2 + (w.tf.ti)^2)^2 704 705 706 (ts + ti)^2 - (w.ts.ti)^2 707 fact_ts_djw_dti = ts^2 -----------------------------, 708 ((ts + ti)^2 + (w.ts.ti)^2)^2 709 710 711 (tf + ti)^2 - (w.tf.ti)^2 712 fact_djw_dtf = ti^2 -----------------------------, 713 ((tf + ti)^2 + (w.tf.ti)^2)^2 714 715 716 (ts + ti)^2 - (w.ts.ti)^2 717 fact_djw_dts = ti^2 -----------------------------. 718 ((ts + ti)^2 + (w.ts.ti)^2)^2 719 """ 720 721 # ti. 722 data.fact_ti_djw_dti = (1.0 - data.w_ti_sqrd) * data.fact_ti**2 723 724 # tf. 725 fact_tf_djw = (data.tf_ti_sqrd - data.w_tf_ti_sqrd) * data.inv_tf_denom**2 726 data.fact_tf_djw_dti = params[data.tf_i]**2 * fact_tf_djw 727 data.fact_djw_dtf = data.ti**2 * fact_tf_djw 728 729 # ts. 730 fact_ts_djw = (data.ts_ti_sqrd - data.w_ts_ti_sqrd) * data.inv_ts_denom**2 731 data.fact_ts_djw_dti = params[data.ts_i]**2 * fact_ts_djw 732 data.fact_djw_dts = data.ti**2 * fact_ts_djw

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Source Code for Module maths_fns.jw_mf_comps