forked from arichar6/raycon
-
Notifications
You must be signed in to change notification settings - Fork 0
/
Copy pathdisp_eig.m
755 lines (659 loc) · 33.3 KB
/
disp_eig.m
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
554
555
556
557
558
559
560
561
562
563
564
565
566
567
568
569
570
571
572
573
574
575
576
577
578
579
580
581
582
583
584
585
586
587
588
589
590
591
592
593
594
595
596
597
598
599
600
601
602
603
604
605
606
607
608
609
610
611
612
613
614
615
616
617
618
619
620
621
622
623
624
625
626
627
628
629
630
631
632
633
634
635
636
637
638
639
640
641
642
643
644
645
646
647
648
649
650
651
652
653
654
655
656
657
658
659
660
661
662
663
664
665
666
667
668
669
670
671
672
673
674
675
676
677
678
679
680
681
682
683
684
685
686
687
688
689
690
691
692
693
694
695
696
697
698
699
700
701
702
703
704
705
706
707
708
709
710
711
712
713
714
715
716
717
718
719
720
721
722
723
724
725
726
727
728
729
730
731
732
733
734
735
736
737
738
739
740
741
742
743
744
745
746
747
748
749
750
751
752
753
754
755
function out= disp_eig(yv,oper)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% DISPERSION -- Plasma dispersion characteristics
% Part of the RAYcON package
%
% yin Position in phase space [r,z,k_r,k_z]
% oper
% 'Msw' Dispersion relation approximated 1x1
% 'Dsp' Disperison relation according to MODEL
% 'Ant' Antenna IC
% 'Trj' RHS for trajectory
% 'Sgn' Sign of dt/dsigma, set by dUdom
% 'Pol' Polarization vector, eigenvector of D matrix
%
% ToDo Optimize computing
% - change Dij(c) vectors into D(c,i,j) matrix
% - remove terms that are zero using emacs search/substitute
%
% A. JAUN, Alfven Laboratory, KTH, 100 44 Stockholm, Sweden
% A.N. KAUFMAN, Lawrence Berkeley Laboratory, Berkeley, CA 94720, USA
% E.R. TRACY, College of William & Mary, Williamsburg, VA 23187-8795, USA
%
% (C) Version 7.0, 14-Aug-2006. All Rights Reserved.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
global cnst
global plasma rays
i = complex(0,1);
isCmplx=0;
%
% ----- Vectorized input quantities ------------------------------------------
%
if numel(yv) == 4
yin = yv.';
sym_mat = eye(4);
eval2nd = 0;
else
yin = yv;
% now assume one ray, and tracing position and symplectic matrix
sym_mat = reshape(yin(5:20),4,4);
yin = yin(1:4).';
eval2nd = 1;
end
if strcmp(oper,'Trj')
eval1st = 1;
else
eval1st = 0;
eval2nd = 0;
end
%
% ----- Adjusted position & wave vector in cylindrical coordinates -----------
zro=zeros(size(yin(:,1))); one=ones(size(zro));
if (rays.odeDim==4||rays.odeDim>=9)
rr=yin(:,1); zz=yin(:,2); kr=yin(:,3); kz=yin(:,4);
kf=one.*plasma.kant(2); % fi=omega*tim/(kf*rr);
end
rho=sqrt((rr-plasma.r0).^2+zz.^2);
theta=atan2(plasma.r0-rr,-zz)+pi/2;
%
%
% ===== Local plasma parameters ==============================================
%
% ----- Magnetic field and topology ------------------------------------------
[b,dbds,dbdt,bp,sflx,dsdr,dsdz,dtdr,dtdz, ...
ener,enef,enez,eber,ebef,ebez,eper,epef,epez, ...
dbds2,dbdst,dbdt2, ...
dsdr2,dsdrz,dsdz2,dsdr3,dsdr2z,dsdrz2,dsdz3, ...
dtdr2,dtdrz,dtdz2,...
denerdr, denefdr, denezdr,...
denerdz, denefdz, denezdz,...
deberdr, debefdr, debezdr,...
deberdz, debefdz, debezdz,...
deperdr, depefdr, depezdr,...
deperdz, depefdz, depezdz,... % end of first derivs
denerdr2, denerdrz, denerdz2,...
denefdr2, denefdrz, denefdz2,...
denezdr2, denezdrz, denezdz2,...
deberdr2, deberdrz, deberdz2,...
debefdr2, debefdrz, debefdz2,...
debezdr2, debezdrz, debezdz2,...
deperdr2, deperdrz, deperdz2,...
depefdr2, depefdrz, depefdz2,...
depezdr2, depezdrz, depezdz2]=magnetic(rho,theta);
%
% ----- Wave vector and refraction index -------------------------------------
coom=cnst.c/plasma.omega; coomsq=coom^2; om2=plasma.omega^2;
kn=kr.*ener +kf.*enef +kz.*enez; Nn=coom*kn; Nn2=Nn.^2;
kb=kr.*eber +kf.*ebef +kz.*ebez; Nb=coom*kb; Nb2=Nb.^2;
kp=kr.*eper +kf.*epef +kz.*epez; Np=coom*kp; Np2=Np.^2; %N2=Nn2+Nb2+Np2;
Nr = coom*kr;
Nf = coom*kf;
Nz = coom*kz;
%
% ----- Parabolic profiles and logarithmic derivatives -----------------------
nspec=size(plasma.amass,2);
p=(1-sflx.^2*plasma.na);
dLNpds = zeros(length(sflx),nspec);
dLNpds2 = dLNpds;
n = dLNpds;
T = dLNpds;
for k=1:nspec
dLNpds(:,k) =-2*sflx.*plasma.na(k).*plasma.nb(k)./p(:,k);
dLNpds2(:,k)=dLNpds(:,k)./sflx-dLNpds(:,k).^2./plasma.nb(k);
n(:,k)=plasma.n0(k).*p(:,k).^plasma.nb(k); % Density
T(:,k)=plasma.t0(k).*(1-sflx.^2*plasma.ta(k)).^plasma.nb(k); % Temperature
end
dLNnds =dLNpds;
dLNnds2=dLNpds2;
%
% ----- Plasma parameters and logarithmic derivatives ------------------------
omp2= ones(size(theta))*((plasma.acharge'.*cnst.e).^2./...
(plasma.amass'.*cnst.mp*cnst.eps0))'.*n;
omp = sqrt(omp2); % Plasma frequencies
omc = b*((plasma.acharge'.*cnst.e)./(plasma.amass'.*cnst.mp))'; % Cyclotron frequencies
omc2= omc.^2;
caoc2 = 1./sum(omp2./omc2, 2); % (c_Alfven/c_light)^2
dLNomp2ds=dLNnds; % Logarithmic derivatives
one=ones(size(plasma.amass'))';
dLNomcds =(dbds./b)*one;
dLNomcdt =(dbdt./b)*one;
% second derivatives
dLNomcds2=(dbds2./b-(dbds.^2 )./(b.^2))*one;
dLNomcdst=(dbdst./b-(dbds.*dbdt)./(b.^2))*one;
dLNomcdt2=(dbdt2./b-(dbdt.^2 )./(b.^2))*one;
switch plasma.MODEL(1:6)
%
% +++++ Cold plasma model ++++++++++++++++++++++++++++++++++++++++++++++++++++
%
case {'cld2x2'}
% --- Elementary functions and derivatives ---------------------------------
if (isCmplx && ~eval1st)
iomceps=0.003*i*om2;
else
iomceps=0;
end
omc2Mom2= omc2-om2-2*iomceps; % S,D,P
Si=omp2./omc2Mom2; S=1+sum( Si ,2);
Di=(omc/plasma.omega).*Si; D= sum( Di ,2);
% Pi=omp2/om2; P=1-sum( Pi ,2);
% 1st derivatives S,D,P
zro=zeros(size(S));
dLNSids =dLNnds-2.*omc2./omc2Mom2.*dLNomcds; dSds = sum( Si.*dLNSids ,2);
dLNSidt = -2.*omc2./omc2Mom2.*dLNomcdt; dSdt = sum( Si.*dLNSidt ,2);
dLNSidom =2*plasma.omega./omc2Mom2; dSdom= sum( Si.*dLNSidom,2);
dLNDids =dLNSids + dLNomcds; dDds = sum( Di.*dLNDids ,2);
dLNDidt =dLNSidt + dLNomcdt; dDdt = sum( Di.*dLNDidt ,2);
dLNDidom =(3*om2-omc2)./(plasma.omega*omc2Mom2); dDdom= sum( Di.*dLNDidom,2);
% % These are only needed in the 3x3 model
% zroi=zeros(size(Si));
% dLNPids =dLNnds;
% dPds =-sum( Pi.*dLNPids ,2);
% dLNPidt =zroi;
% dPdt = zro;
% dLNPidom=2/plasma.omega;
% dPdom= sum( Pi.*dLNPidom,2);
% second derivatives
if eval2nd
omOM=om2 ./omc2Mom2;
ocOM=omc2./omc2Mom2;
dLNSids2= 2*ocOM.*(2*omOM.*dLNomcds.*dLNomcds -dLNomcds2)+dLNnds2;
dLNSidst= 2*ocOM.*(2*omOM.*dLNomcds.*dLNomcdt -dLNomcdst);
dLNSidt2= 2*ocOM.*(2*omOM.*dLNomcdt.*dLNomcdt -dLNomcdt2);
dLNDids2= dLNomcds2 +dLNSids2;
dLNDidst= dLNomcdst +dLNSidst;
dLNDidt2= dLNomcdt2 +dLNSidt2;
dSids2 = Si.*(dLNSids2 +dLNSids.*dLNSids); dSds2=sum( dSids2 ,2);
dSidst = Si.*(dLNSidst +dLNSids.*dLNSidt); dSdst=sum( dSidst ,2);
dSidt2 = Si.*(dLNSidt2 +dLNSidt.*dLNSidt); dSdt2=sum( dSidt2 ,2);
dDids2 = Di.*(dLNDids2 +dLNDids.*dLNDids); dDds2=sum( dDids2 ,2);
dDidst = Di.*(dLNDidst +dLNDids.*dLNDidt); dDdst=sum( dDidst ,2);
dDidt2 = Di.*(dLNDidt2 +dLNDidt.*dLNDidt); dDdt2=sum( dDidt2 ,2);
% % These are only needed in the 3x3 model
% dLNPids2= dLNnds2; dLNPidst= zroi; dLNPidt2= zroi;
% dPids2= Pi.*(dLNPids2 +dLNPids.*dLNPids); dPds2=sum( dPids2 ,2);
% dPidst= Pi.*(dLNPidst +dLNPids.*dLNPidt); dPdst=sum( dPidst ,2);
% dPidt2= Pi.*(dLNPidt2 +dLNPidt.*dLNPidt); dPdt2=sum( dPidt2 ,2);
end
% --- Tensor elements and derivatives --------------------------------------
D11 = Nb2+Np2-S;
D12 =-Nn.*Nb-D*i;
D22 = Nn2+Np2-S;
cD12 = conj(D12);
% Dispersion tensor
DD=[D11 D12 cD12 D22];
% We will assume that the input yv is size [1,4], then reshape works
DD=reshape(DD,2,2);
[evects,evals]=eig(DD); % Two values
evs = diag(evals);
ind = find(abs(evs)==min(abs(evs)));
U = evs(ind); % The eval nearest zero is the one we want
pol = evects(:,ind); % get the column with the right evector
% Conversion monitor
mon2 = abs(D11 + D22); % trace DD
if eval1st
dD11ds =-dSds; dD11dt =-dSdt;
dD12ds =-dDds*i; dD12dt =-dDdt*i;
dD22ds =-dSds; dD22dt =-dSdt;
dD11dkn= zro; dD11dkb= Nb*coom*2; dD11dkp= Np*coom*2;
dD12dkn=-Nb*coom; dD12dkb=-Nn*coom; dD12dkp= zro;
dD22dkn= Nn*coom*2; dD22dkb= zro; dD22dkp= Np*coom*2;
dD11dkr=dD11dkn.*ener +dD11dkb.*eber +dD11dkp.*eper;
dD11dkz=dD11dkn.*enez +dD11dkb.*ebez +dD11dkp.*epez;
dD12dkr=dD12dkn.*ener +dD12dkb.*eber +dD12dkp.*eper;
dD12dkz=dD12dkn.*enez +dD12dkb.*ebez +dD12dkp.*epez;
dD22dkr=dD22dkn.*ener +dD22dkb.*eber +dD22dkp.*eper;
dD22dkz=dD22dkn.*enez +dD22dkb.*ebez +dD22dkp.*epez;
dD11dom=-2/plasma.omega*(Nb2+Np2)-dSdom;
dD12dom=-2/plasma.omega*(Nn.*Nb) -dDdom*i;
dD22dom=-2/plasma.omega*(Nn2+Np2)-dSdom;
% Sub-blocks of (dispersion matrix - eigenval*id)
subsum = D11+D22-2*U;
% Convert to r, z coordinates.
% Note that there are extra correction terms due to field
% curvature. They are derivatives of the trasformation elements:
% (d/dr)(ebef) etc.
dNndr = Nr.*denerdr + Nf.*denefdr + Nz.*denezdr;
dNndz = Nr.*denerdz + Nf.*denefdz + Nz.*denezdz;
dNbdr = Nr.*deberdr + Nf.*debefdr + Nz.*debezdr;
dNbdz = Nr.*deberdz + Nf.*debefdz + Nz.*debezdz;
dNpdr = Nr.*deperdr + Nf.*depefdr + Nz.*depezdr;
dNpdz = Nr.*deperdz + Nf.*depefdz + Nz.*depezdz;
dD11dr = dD11ds.*dsdr + dD11dt.*dtdr + 2*(Nb.*dNbdr + Np.*dNpdr);
dD11dz = dD11ds.*dsdz + dD11dt.*dtdz + 2*(Nb.*dNbdz + Np.*dNpdz);
dD12dr = dD12ds.*dsdr + dD12dt.*dtdr - Nn.*dNbdr - Nb.*dNndr;
dD12dz = dD12ds.*dsdz + dD12dt.*dtdz - Nn.*dNbdz - Nb.*dNndz;
dD22dr = dD22ds.*dsdr + dD22dt.*dtdr + 2*(Nn.*dNndr + Np.*dNpdr);
dD22dz = dD22ds.*dsdz + dD22dt.*dtdz + 2*(Nn.*dNndz + Np.*dNpdz);
% Gather things together to do all derivatives at the same time
dD11_vec = [dD11dom dD11dr dD11dz dD11dkr dD11dkz];
dD12_vec = [dD12dom dD12dr dD12dz dD12dkr dD12dkz];
dD22_vec = [dD22dom dD22dr dD22dz dD22dkr dD22dkz];
% Derivatives of the eigenvalue
%dU_vec = [dUdom dUds dUdt dUdkn dUdkb dUdkp];
dU_vec = (dD11_vec.*(D22-U) +dD22_vec.*(D11-U))./subsum...
- 2*real(cD12.*dD12_vec)./subsum;
end
% terms needed for second derivative of hamiltonian
if eval2nd
dSdr2 = dsdr2.*dSds + dtdr2.*dSdt ...
+ dsdr.*(dsdr.*dSds2 + dtdr.*dSdst) + dtdr.*(dsdr.*dSdst + dtdr.*dSdt2);
dSdrz = dsdrz.*dSds + dtdrz.*dSdt ...
+ dsdz.*(dsdr.*dSds2 + dtdr.*dSdst) + dtdz.*(dsdr.*dSdst + dtdr.*dSdt2);
dSdz2 = dsdz2.*dSds + dtdz2.*dSdt ...
+ dsdz.*(dsdz.*dSds2 + dtdz.*dSdst) + dtdz.*(dsdz.*dSdst + dtdz.*dSdt2);
dDdr2 = dsdr2.*dDds + dtdr2.*dDdt ...
+ dsdr.*(dsdr.*dDds2 + dtdr.*dDdst) + dtdr.*(dsdr.*dDdst + dtdr.*dDdt2);
dDdrz = dsdrz.*dDds + dtdrz.*dDdt ...
+ dsdz.*(dsdr.*dDds2 + dtdr.*dDdst) + dtdz.*(dsdr.*dDdst + dtdr.*dDdt2);
dDdz2 = dsdz2.*dDds + dtdz2.*dDdt ...
+ dsdz.*(dsdz.*dDds2 + dtdz.*dDdst) + dtdz.*(dsdz.*dDdst + dtdz.*dDdt2);
dNndr2 = Nr.*denerdr2 + Nf.*denefdr2 + Nz.*denezdr2;
dNndrz = Nr.*denerdrz + Nf.*denefdrz + Nz.*denezdrz;
dNndz2 = Nr.*denerdz2 + Nf.*denefdz2 + Nz.*denezdz2;
dNbdr2 = Nr.*deberdr2 + Nf.*debefdr2 + Nz.*debezdr2;
dNbdrz = Nr.*deberdrz + Nf.*debefdrz + Nz.*debezdrz;
dNbdz2 = Nr.*deberdz2 + Nf.*debefdz2 + Nz.*debezdz2;
dNpdr2 = Nr.*deperdr2 + Nf.*depefdr2 + Nz.*depezdr2;
dNpdrz = Nr.*deperdrz + Nf.*depefdrz + Nz.*depezdrz;
dNpdz2 = Nr.*deperdz2 + Nf.*depefdz2 + Nz.*depezdz2;
dD11dr2=2*(dNbdr.*dNbdr + Nb.*dNbdr2 + dNpdr.*dNpdr + Np.*dNpdr2) - dSdr2;
dD11drz=2*(dNbdr.*dNbdz + Nb.*dNbdrz + dNpdr.*dNpdz + Np.*dNpdrz) - dSdrz;
dD11dz2=2*(dNbdz.*dNbdz + Nb.*dNbdz2 + dNpdz.*dNpdz + Np.*dNpdz2) - dSdz2;
dD12dr2=-(Nn.*dNbdr2 + 2*dNndr.*dNbdr + Nb.*dNndr2)-i*dDdr2;
dD12drz=-(Nn.*dNbdrz + dNndr.*dNbdz + dNndz.*dNbdr + Nb.*dNndrz)-i*dDdrz;
dD12dz2=-(Nn.*dNbdz2 + 2*dNndz.*dNbdz + Nb.*dNndz2)-i*dDdz2;
dD22dr2=2*(dNndr.*dNndr + Nn.*dNndr2 + dNpdr.*dNpdr + Np.*dNpdr2) - dSdr2;
dD22drz=2*(dNndr.*dNndz + Nn.*dNndrz + dNpdr.*dNpdz + Np.*dNpdrz) - dSdrz;
dD22dz2=2*(dNndz.*dNndz + Nn.*dNndz2 + dNpdz.*dNpdz + Np.*dNpdz2) - dSdz2;
dD11drkr = 2*coom*(eber.*dNbdr + Nb.*deberdr + eper.*dNpdr + Np.*deperdr);
dD11drkz = 2*coom*(ebez.*dNbdr + Nb.*debezdr + epez.*dNpdr + Np.*depezdr);
dD11dzkr = 2*coom*(eber.*dNbdz + Nb.*deberdz + eper.*dNpdz + Np.*deperdz);
dD11dzkz = 2*coom*(ebez.*dNbdz + Nb.*debezdz + epez.*dNpdz + Np.*depezdz);
dD12drkr = -coom*(denerdr.*Nb + dNndr.*eber + ener.*dNbdr + Nn.*deberdr);
dD12drkz = -coom*(denezdr.*Nb + dNndr.*ebez + enez.*dNbdr + Nn.*debezdr);
dD12dzkr = -coom*(denerdz.*Nb + dNndz.*eber + ener.*dNbdz + Nn.*deberdz);
dD12dzkz = -coom*(denezdz.*Nb + dNndz.*ebez + enez.*dNbdz + Nn.*debezdz);
dD22drkr = 2*coom*(ener.*dNndr + Nn.*denerdr + eper.*dNpdr + Np.*deperdr);
dD22drkz = 2*coom*(enez.*dNndr + Nn.*denezdr + epez.*dNpdr + Np.*depezdr);
dD22dzkr = 2*coom*(ener.*dNndz + Nn.*denerdz + eper.*dNpdz + Np.*deperdz);
dD22dzkz = 2*coom*(enez.*dNndz + Nn.*denezdz + epez.*dNpdz + Np.*depezdz);
dD11dkr2 = 2*coomsq*(eber.^2 + eper.^2);
dD11dkz2 = 2*coomsq*(ebez.^2 + epez.^2);
dD11dkrkz= 2*coomsq*(eber.*ebez + eper.*epez);
dD12dkr2 = -coomsq*(ener.*eber + ener.*eber);
dD12dkz2 = -coomsq*(ener.*ebez + enez.*eber);
dD12dkrkz= -coomsq*(enez.*ebez + enez.*ebez);
dD22dkr2 = 2*coomsq*(ener.^2 + eper.^2);
dD22dkz2 = 2*coomsq*(enez.^2 + epez.^2);
dD22dkrkz= 2*coomsq*(ener.*enez + eper.*epez);
% dD11_mat = [dr dz dkr dkz].'*[dr dz dkr dkz];
dD11_mat = [dD11dr2 dD11drz dD11drkr dD11drkz;
dD11drz dD11dz2 dD11dzkr dD11dzkz;
dD11drkr dD11dzkr dD11dkr2 dD11dkrkz;
dD11drkz dD11dzkz dD11dkrkz dD11dkz2 ];
dD12_mat = [dD12dr2 dD12drz dD12drkr dD12drkz;
dD12drz dD12dz2 dD12dzkr dD12dzkz;
dD12drkr dD12dzkr dD12dkr2 dD12dkrkz;
dD12drkz dD12dzkz dD12dkrkz dD12dkz2 ];
dD22_mat = [dD22dr2 dD22drz dD22drkr dD22drkz;
dD22drz dD22dz2 dD22dzkr dD22dzkz;
dD22drkr dD22dzkr dD22dkr2 dD22dkrkz;
dD22drkz dD22dzkz dD22dkrkz dD22dkz2 ];
% for the second order derivatives, we don't need dom
dD11_vec = [dD11dr dD11dz dD11dkr dD11dkz];
dD12_vec = [dD12dr dD12dz dD12dkr dD12dkz];
dD22_vec = [dD22dr dD22dz dD22dkr dD22dkz];
dUdom = dU_vec(1);
dU_vec = dU_vec(2:end);
dU_mat = (1/subsum)*(...
dD11_mat*(D22-U) + dD22_mat*(D11-U)...
+(dD11_vec - dU_vec).'*(dD22_vec - dU_vec)...
+(dD22_vec - dU_vec).'*(dD11_vec - dU_vec)...
-2*real(cD12*dD12_mat + dD12_vec'*dD12_vec));
dU_vec = [dUdom dU_vec];
end
case {'cld3x3'}
% --- Elementary functions and derivatives ---------------------------------
if (isCmplx && ~eval1st)
iomceps=0.003*i*om2;
else
iomceps=0;
end
omc2Mom2= omc2-om2-2*iomceps; % S,D,P
Si=omp2./omc2Mom2; S=1+sum( Si ,2);
Di=(omc/plasma.omega).*Si; D= sum( Di ,2);
Pi=omp2/om2; P=1-sum( Pi ,2);
% 1st derivatives S,D,P
zroi=zeros(size(Si));
zro=zeros(size(S));
dLNSids =dLNnds-2.*omc2./omc2Mom2.*dLNomcds; dSds = sum( Si.*dLNSids ,2);
dLNSidt = -2.*omc2./omc2Mom2.*dLNomcdt; dSdt = sum( Si.*dLNSidt ,2);
dLNSidom =2*plasma.omega./omc2Mom2; dSdom= sum( Si.*dLNSidom,2);
dLNDids =dLNSids + dLNomcds; dDds = sum( Di.*dLNDids ,2);
dLNDidt =dLNSidt + dLNomcdt; dDdt = sum( Di.*dLNDidt ,2);
dLNDidom =(3*om2-omc2)./(plasma.omega*omc2Mom2); dDdom= sum( Di.*dLNDidom,2);
dLNPids = dLNnds;
dPds = -sum( Pi.*dLNPids ,2);
dLNPidt = zroi;
dPdt = zro;
dLNPidom = -2/plasma.omega;
dPdom = -sum( Pi.*dLNPidom,2);
% second derivatives
if eval2nd
omOM = om2 ./omc2Mom2;
ocOM = omc2./omc2Mom2;
dLNSids2 = 2*ocOM.*(2*omOM.*dLNomcds.*dLNomcds -dLNomcds2)+dLNnds2;
dLNSidst = 2*ocOM.*(2*omOM.*dLNomcds.*dLNomcdt -dLNomcdst);
dLNSidt2 = 2*ocOM.*(2*omOM.*dLNomcdt.*dLNomcdt -dLNomcdt2);
dLNDids2 = dLNomcds2 +dLNSids2;
dLNDidst = dLNomcdst +dLNSidst;
dLNDidt2 = dLNomcdt2 +dLNSidt2;
dSids2 = Si.*(dLNSids2 +dLNSids.*dLNSids); dSds2 = sum( dSids2 ,2);
dSidst = Si.*(dLNSidst +dLNSids.*dLNSidt); dSdst = sum( dSidst ,2);
dSidt2 = Si.*(dLNSidt2 +dLNSidt.*dLNSidt); dSdt2 = sum( dSidt2 ,2);
dDids2 = Di.*(dLNDids2 +dLNDids.*dLNDids); dDds2 = sum( dDids2 ,2);
dDidst = Di.*(dLNDidst +dLNDids.*dLNDidt); dDdst = sum( dDidst ,2);
dDidt2 = Di.*(dLNDidt2 +dLNDidt.*dLNDidt); dDdt2 = sum( dDidt2 ,2);
dLNPids2 = dLNnds2; dLNPidst= zroi; dLNPidt2 = zroi;
dPids2 = Pi.*(dLNPids2 +dLNPids.*dLNPids); dPds2 = -sum( dPids2 ,2);
dPidst = Pi.*(dLNPidst +dLNPids.*dLNPidt); dPdst = -sum( dPidst ,2);
dPidt2 = Pi.*(dLNPidt2 +dLNPidt.*dLNPidt); dPdt2 = -sum( dPidt2 ,2);
end
% --- Tensor elements and derivatives --------------------------------------
D11 = Nb2+Np2-S;
D12 =-Nn.*Nb-D*i;
D13 =-Nn.*Np;
D22 = Nn2+Np2-S;
D23 =-Nb.*Np;
D33 = Nn2+Nb2-P;
cD12 = conj(D12);
cD13 = conj(D13);
cD23 = conj(D23);
% Dispersion tensor
DD=[D11 D12 D13 cD12 D22 D23 cD13 cD23 D33];
% We will assume that the input yv is size [1,4], then reshape works
DD=reshape(DD,3,3);
% Convert to double to make sure that double precision LAPACK routines
% are used.
[evects,evals]=eig(double(DD)); % Three values
evs = diag(evals);
ind = find(abs(evs)==min(abs(evs)));
U = evs(ind); % The eval nearest zero is the one we want
pol = evects(:,ind); % get the column with the right evector
% Conversion monitor
mon2 = abs(det(DD([2 3],[2 3])) + ...
det(DD([1 3],[1 3])) + ...
det(DD([1 2],[1 2])) );
if eval1st
dD11ds =-dSds; dD11dt =-dSdt;
dD12ds =-dDds*i; dD12dt =-dDdt*i;
dD13ds = zro; dD13dt = zro;
dD22ds =-dSds; dD22dt =-dSdt;
dD23ds = zro; dD23dt = zro;
dD33ds =-dPds; dD33dt =-dPdt;
dD11dkn= zro; dD11dkb= Nb*coom*2; dD11dkp= Np*coom*2;
dD12dkn=-Nb*coom; dD12dkb=-Nn*coom; dD12dkp= zro;
dD13dkn=-Np*coom; dD13dkb= zro; dD13dkp=-Nn*coom;
dD22dkn= Nn*coom*2; dD22dkb= zro; dD22dkp= Np*coom*2;
dD23dkn= zro; dD23dkb=-Np*coom; dD23dkp=-Nb*coom;
dD33dkn= Nn*coom*2; dD33dkb= Nb*coom*2; dD33dkp= zro;
dD11dkr=dD11dkn.*ener +dD11dkb.*eber +dD11dkp.*eper;
dD11dkz=dD11dkn.*enez +dD11dkb.*ebez +dD11dkp.*epez;
dD12dkr=dD12dkn.*ener +dD12dkb.*eber +dD12dkp.*eper;
dD12dkz=dD12dkn.*enez +dD12dkb.*ebez +dD12dkp.*epez;
dD13dkr=dD13dkn.*ener +dD13dkb.*eber +dD13dkp.*eper;
dD13dkz=dD13dkn.*enez +dD13dkb.*ebez +dD13dkp.*epez;
dD22dkr=dD22dkn.*ener +dD22dkb.*eber +dD22dkp.*eper;
dD22dkz=dD22dkn.*enez +dD22dkb.*ebez +dD22dkp.*epez;
dD23dkr=dD23dkn.*ener +dD23dkb.*eber +dD23dkp.*eper;
dD23dkz=dD23dkn.*enez +dD23dkb.*ebez +dD23dkp.*epez;
dD33dkr=dD33dkn.*ener +dD33dkb.*eber +dD33dkp.*eper;
dD33dkz=dD33dkn.*enez +dD33dkb.*ebez +dD33dkp.*epez;
dD11dom=-2/plasma.omega*(Nb2+Np2)-dSdom;
dD12dom=-2/plasma.omega*(Nn.*Nb) -dDdom*i;
dD13dom=-2/plasma.omega*(Nn.*Np);
dD22dom=-2/plasma.omega*(Nn2+Np2)-dSdom;
dD23dom=-2/plasma.omega*(Nb.*Np);
dD33dom=-2/plasma.omega*(Nn2+Nb2)-dPdom;
DD2=DD-eye(3)*U;
% Sub-blocks of (dispersion matrix - eigenval*id)
U1 = det(DD2([2 3],[2 3]));
U2 = det(DD2([1 3],[1 3]));
U3 = det(DD2([1 2],[1 2]));
subsum = U1+U2+U3;
% Convert to r, z coordinates.
% Note that there are extra correction terms due to field
% curvature. They are derivatives of the trasformation elements:
% (d/dr)(ebef) etc.
dNndr = Nr.*denerdr + Nf.*denefdr + Nz.*denezdr;
dNndz = Nr.*denerdz + Nf.*denefdz + Nz.*denezdz;
dNbdr = Nr.*deberdr + Nf.*debefdr + Nz.*debezdr;
dNbdz = Nr.*deberdz + Nf.*debefdz + Nz.*debezdz;
dNpdr = Nr.*deperdr + Nf.*depefdr + Nz.*depezdr;
dNpdz = Nr.*deperdz + Nf.*depefdz + Nz.*depezdz;
dD11dr = dD11ds.*dsdr + dD11dt.*dtdr + 2*(Nb.*dNbdr + Np.*dNpdr);
dD11dz = dD11ds.*dsdz + dD11dt.*dtdz + 2*(Nb.*dNbdz + Np.*dNpdz);
dD12dr = dD12ds.*dsdr + dD12dt.*dtdr - Nn.*dNbdr - Nb.*dNndr;
dD12dz = dD12ds.*dsdz + dD12dt.*dtdz - Nn.*dNbdz - Nb.*dNndz;
dD13dr = dD13ds.*dsdr + dD13dt.*dtdr - dNndr.*Np - Nn.*dNpdr;
dD13dz = dD13ds.*dsdz + dD13dt.*dtdz - dNndz.*Np - Nn.*dNpdz;
dD22dr = dD22ds.*dsdr + dD22dt.*dtdr + 2*(Nn.*dNndr + Np.*dNpdr);
dD22dz = dD22ds.*dsdz + dD22dt.*dtdz + 2*(Nn.*dNndz + Np.*dNpdz);
dD23dr = dD23ds.*dsdr + dD23dt.*dtdr - dNbdr.*Np - Nb.*dNpdr;
dD23dz = dD23ds.*dsdz + dD23dt.*dtdz - dNbdz.*Np - Nb.*dNpdz;
dD33dr = dD33ds.*dsdr + dD33dt.*dtdr + 2*(Nn.*dNndr + Nb.*dNbdr);
dD33dz = dD33ds.*dsdz + dD33dt.*dtdz + 2*(Nn.*dNndz + Nb.*dNbdz);
% Gather things together to do all derivatives at the same time
dD11_vec = [dD11dom dD11dr dD11dz dD11dkr dD11dkz];
dD12_vec = [dD12dom dD12dr dD12dz dD12dkr dD12dkz];
dD13_vec = [dD13dom dD13dr dD13dz dD13dkr dD13dkz];
dD22_vec = [dD22dom dD22dr dD22dz dD22dkr dD22dkz];
dD23_vec = [dD23dom dD23dr dD23dz dD23dkr dD23dkz];
dD33_vec = [dD33dom dD33dr dD33dz dD33dkr dD33dkz];
% Derivatives of the eigenvalue
%dU_vec = [dUdom dUdr dUdz dUdkr dUdkz];
dU_vec = (dD11_vec.*U1 - (D11-U).*2.*real(cD23.*dD23_vec) ...
+dD22_vec.*U2 - (D22-U).*2.*real(cD13.*dD13_vec) ...
+dD33_vec.*U3 - (D33-U).*2.*real(cD12.*dD12_vec) )./subsum...
+ 2*real(D12.*D23.*conj(dD13_vec)...
+D12.*dD23_vec.*cD13...
+dD12_vec.*D23.*cD13)./subsum;
end
% terms needed for second derivative of hamiltonian
if eval2nd
dSdr2 = dsdr2.*dSds + dtdr2.*dSdt ...
+ dsdr.*(dsdr.*dSds2 + dtdr.*dSdst) + dtdr.*(dsdr.*dSdst + dtdr.*dSdt2);
dSdrz = dsdrz.*dSds + dtdrz.*dSdt ...
+ dsdz.*(dsdr.*dSds2 + dtdr.*dSdst) + dtdz.*(dsdr.*dSdst + dtdr.*dSdt2);
dSdz2 = dsdz2.*dSds + dtdz2.*dSdt ...
+ dsdz.*(dsdz.*dSds2 + dtdz.*dSdst) + dtdz.*(dsdz.*dSdst + dtdz.*dSdt2);
dDdr2 = dsdr2.*dDds + dtdr2.*dDdt ...
+ dsdr.*(dsdr.*dDds2 + dtdr.*dDdst) + dtdr.*(dsdr.*dDdst + dtdr.*dDdt2);
dDdrz = dsdrz.*dDds + dtdrz.*dDdt ...
+ dsdz.*(dsdr.*dDds2 + dtdr.*dDdst) + dtdz.*(dsdr.*dDdst + dtdr.*dDdt2);
dDdz2 = dsdz2.*dDds + dtdz2.*dDdt ...
+ dsdz.*(dsdz.*dDds2 + dtdz.*dDdst) + dtdz.*(dsdz.*dDdst + dtdz.*dDdt2);
dPdr2 = dsdr2.*dPds + dtdr2.*dPdt ...
+ dsdr.*(dsdr.*dPds2 + dtdr.*dPdst) + dtdr.*(dsdr.*dPdst + dtdr.*dPdt2);
dPdrz = dsdrz.*dPds + dtdrz.*dPdt ...
+ dsdz.*(dsdr.*dPds2 + dtdr.*dPdst) + dtdz.*(dsdr.*dPdst + dtdr.*dPdt2);
dPdz2 = dsdz2.*dPds + dtdz2.*dPdt ...
+ dsdz.*(dsdz.*dPds2 + dtdz.*dPdst) + dtdz.*(dsdz.*dPdst + dtdz.*dPdt2);
dNndr2 = Nr.*denerdr2 + Nf.*denefdr2 + Nz.*denezdr2;
dNndrz = Nr.*denerdrz + Nf.*denefdrz + Nz.*denezdrz;
dNndz2 = Nr.*denerdz2 + Nf.*denefdz2 + Nz.*denezdz2;
dNbdr2 = Nr.*deberdr2 + Nf.*debefdr2 + Nz.*debezdr2;
dNbdrz = Nr.*deberdrz + Nf.*debefdrz + Nz.*debezdrz;
dNbdz2 = Nr.*deberdz2 + Nf.*debefdz2 + Nz.*debezdz2;
dNpdr2 = Nr.*deperdr2 + Nf.*depefdr2 + Nz.*depezdr2;
dNpdrz = Nr.*deperdrz + Nf.*depefdrz + Nz.*depezdrz;
dNpdz2 = Nr.*deperdz2 + Nf.*depefdz2 + Nz.*depezdz2;
dD11dr2=2*(dNbdr.*dNbdr + Nb.*dNbdr2 + dNpdr.*dNpdr + Np.*dNpdr2) - dSdr2;
dD11drz=2*(dNbdr.*dNbdz + Nb.*dNbdrz + dNpdr.*dNpdz + Np.*dNpdrz) - dSdrz;
dD11dz2=2*(dNbdz.*dNbdz + Nb.*dNbdz2 + dNpdz.*dNpdz + Np.*dNpdz2) - dSdz2;
dD12dr2=-(Nn.*dNbdr2 + 2*dNndr.*dNbdr + Nb.*dNndr2)-i*dDdr2;
dD12drz=-(Nn.*dNbdrz + dNndr.*dNbdz + dNndz.*dNbdr + Nb.*dNndrz)-i*dDdrz;
dD12dz2=-(Nn.*dNbdz2 + 2*dNndz.*dNbdz + Nb.*dNndz2)-i*dDdz2;
dD13dr2=-(Nn.*dNpdr2 + 2*dNndr.*dNpdr + Np.*dNndr2);
dD13drz=-(Nn.*dNpdrz + dNndr.*dNpdz + dNndz.*dNpdr + Np.*dNndrz);
dD13dz2=-(Nn.*dNpdz2 + 2*dNndz.*dNpdz + Np.*dNndz2);
dD22dr2=2*(dNndr.*dNndr + Nn.*dNndr2 + dNpdr.*dNpdr + Np.*dNpdr2) - dSdr2;
dD22drz=2*(dNndr.*dNndz + Nn.*dNndrz + dNpdr.*dNpdz + Np.*dNpdrz) - dSdrz;
dD22dz2=2*(dNndz.*dNndz + Nn.*dNndz2 + dNpdz.*dNpdz + Np.*dNpdz2) - dSdz2;
dD23dr2=-(Nb.*dNpdr2 + 2*dNbdr.*dNpdr + Np.*dNbdr2);
dD23drz=-(Nb.*dNpdrz + dNbdr.*dNpdz + dNbdz.*dNpdr + Np.*dNbdrz);
dD23dz2=-(Nb.*dNpdz2 + 2*dNbdz.*dNpdz + Np.*dNbdz2);
dD33dr2=2*(dNndr.*dNndr + Nn.*dNndr2 + dNbdr.*dNbdr + Nb.*dNbdr2) - dPdr2;
dD33drz=2*(dNndr.*dNndz + Nn.*dNndrz + dNbdr.*dNbdz + Nb.*dNbdrz) - dPdrz;
dD33dz2=2*(dNndz.*dNndz + Nn.*dNndz2 + dNbdz.*dNbdz + Nb.*dNbdz2) - dPdz2;
dD11drkr = 2*coom*(eber.*dNbdr + Nb.*deberdr + eper.*dNpdr + Np.*deperdr);
dD11drkz = 2*coom*(ebez.*dNbdr + Nb.*debezdr + epez.*dNpdr + Np.*depezdr);
dD11dzkr = 2*coom*(eber.*dNbdz + Nb.*deberdz + eper.*dNpdz + Np.*deperdz);
dD11dzkz = 2*coom*(ebez.*dNbdz + Nb.*debezdz + epez.*dNpdz + Np.*depezdz);
dD12drkr = -coom*(denerdr.*Nb + dNndr.*eber + ener.*dNbdr + Nn.*deberdr);
dD12drkz = -coom*(denezdr.*Nb + dNndr.*ebez + enez.*dNbdr + Nn.*debezdr);
dD12dzkr = -coom*(denerdz.*Nb + dNndz.*eber + ener.*dNbdz + Nn.*deberdz);
dD12dzkz = -coom*(denezdz.*Nb + dNndz.*ebez + enez.*dNbdz + Nn.*debezdz);
dD13drkr = -coom*(denerdr.*Np + dNndr.*eper + ener.*dNpdr + Nn.*deperdr);
dD13drkz = -coom*(denezdr.*Np + dNndr.*epez + enez.*dNpdr + Nn.*depezdr);
dD13dzkr = -coom*(denerdz.*Np + dNndz.*eper + ener.*dNpdz + Nn.*deperdz);
dD13dzkz = -coom*(denezdz.*Np + dNndz.*epez + enez.*dNpdz + Nn.*depezdz);
dD22drkr = 2*coom*(ener.*dNndr + Nn.*denerdr + eper.*dNpdr + Np.*deperdr);
dD22drkz = 2*coom*(enez.*dNndr + Nn.*denezdr + epez.*dNpdr + Np.*depezdr);
dD22dzkr = 2*coom*(ener.*dNndz + Nn.*denerdz + eper.*dNpdz + Np.*deperdz);
dD22dzkz = 2*coom*(enez.*dNndz + Nn.*denezdz + epez.*dNpdz + Np.*depezdz);
dD23drkr = -coom*(deperdr.*Nb + dNpdr.*eber + eper.*dNbdr + Np.*deberdr);
dD23drkz = -coom*(depezdr.*Nb + dNpdr.*ebez + epez.*dNbdr + Np.*debezdr);
dD23dzkr = -coom*(deperdz.*Nb + dNpdz.*eber + eper.*dNbdz + Np.*deberdz);
dD23dzkz = -coom*(depezdz.*Nb + dNpdz.*ebez + epez.*dNbdz + Np.*debezdz);
dD33drkr = 2*coom*(ener.*dNndr + Nn.*denerdr + eber.*dNbdr + Nb.*deberdr);
dD33drkz = 2*coom*(enez.*dNndr + Nn.*denezdr + ebez.*dNbdr + Nb.*debezdr);
dD33dzkr = 2*coom*(ener.*dNndz + Nn.*denerdz + eber.*dNbdz + Nb.*deberdz);
dD33dzkz = 2*coom*(enez.*dNndz + Nn.*denezdz + ebez.*dNbdz + Nb.*debezdz);
dD11dkr2 = 2*coomsq*(eber.^2 + eper.^2);
dD11dkz2 = 2*coomsq*(ebez.^2 + epez.^2);
dD11dkrkz= 2*coomsq*(eber.*ebez + eper.*epez);
dD12dkr2 = -coomsq*(ener.*eber + ener.*eber);
dD12dkz2 = -coomsq*(ener.*ebez + enez.*eber);
dD12dkrkz= -coomsq*(enez.*ebez + enez.*ebez);
dD13dkr2 = -coomsq*(ener.*eper + ener.*eper);
dD13dkz2 = -coomsq*(ener.*epez + enez.*eper);
dD13dkrkz= -coomsq*(enez.*epez + enez.*epez);
dD22dkr2 = 2*coomsq*(ener.^2 + eper.^2);
dD22dkz2 = 2*coomsq*(enez.^2 + epez.^2);
dD22dkrkz= 2*coomsq*(ener.*enez + eper.*epez);
dD23dkr2 = -coomsq*(eper.*eber + eper.*eber);
dD23dkz2 = -coomsq*(eper.*ebez + epez.*eber);
dD23dkrkz= -coomsq*(epez.*ebez + epez.*ebez);
dD33dkr2 = 2*coomsq*(ener.^2 + eber.^2);
dD33dkz2 = 2*coomsq*(enez.^2 + ebez.^2);
dD33dkrkz= 2*coomsq*(ener.*enez + eber.*ebez);
% dD11_mat = [ds dt dkr dkz].'*[ds dt dkr dkz];
dD11_mat = [dD11dr2 dD11drz dD11drkr dD11drkz;
dD11drz dD11dz2 dD11dzkr dD11dzkz;
dD11drkr dD11dzkr dD11dkr2 dD11dkrkz;
dD11drkz dD11dzkz dD11dkrkz dD11dkz2 ];
dD12_mat = [dD12dr2 dD12drz dD12drkr dD12drkz;
dD12drz dD12dz2 dD12dzkr dD12dzkz;
dD12drkr dD12dzkr dD12dkr2 dD12dkrkz;
dD12drkz dD12dzkz dD12dkrkz dD12dkz2 ];
dD13_mat = [dD13dr2 dD13drz dD13drkr dD13drkz;
dD13drz dD13dz2 dD13dzkr dD13dzkz;
dD13drkr dD13dzkr dD13dkr2 dD13dkrkz;
dD13drkz dD13dzkz dD13dkrkz dD13dkz2 ];
dD22_mat = [dD22dr2 dD22drz dD22drkr dD22drkz;
dD22drz dD22dz2 dD22dzkr dD22dzkz;
dD22drkr dD22dzkr dD22dkr2 dD22dkrkz;
dD22drkz dD22dzkz dD22dkrkz dD22dkz2 ];
dD23_mat = [dD23dr2 dD23drz dD23drkr dD23drkz;
dD23drz dD23dz2 dD23dzkr dD23dzkz;
dD23drkr dD23dzkr dD23dkr2 dD23dkrkz;
dD23drkz dD23dzkz dD23dkrkz dD23dkz2 ];
dD33_mat = [dD33dr2 dD33drz dD33drkr dD33drkz;
dD33drz dD33dz2 dD33dzkr dD33dzkz;
dD33drkr dD33dzkr dD33dkr2 dD33dkrkz;
dD33drkz dD33dzkz dD33dkrkz dD33dkz2 ];
% for the second order derivatives, we don't need dom
dD11_vec = [dD11dr dD11dz dD11dkr dD11dkz];
dD12_vec = [dD12dr dD12dz dD12dkr dD12dkz];
dD13_vec = [dD13dr dD13dz dD13dkr dD13dkz];
dD22_vec = [dD22dr dD22dz dD22dkr dD22dkz];
dD23_vec = [dD23dr dD23dz dD23dkr dD23dkz];
dD33_vec = [dD33dr dD33dz dD33dkr dD33dkz];
dUdom = dU_vec(1);
dU_vec = dU_vec(2:end);
dU1_vec = (dD22_vec - dU_vec).*(D33 - U)...
+ (dD33_vec - dU_vec).*(D22 - U)...
-2*real(cD23*dD23_vec);
dU2_vec = (dD11_vec - dU_vec).*(D33 - U)...
+ (dD33_vec - dU_vec).*(D11 - U)...
-2*real(cD13*dD13_vec);
dU3_vec = (dD22_vec - dU_vec).*(D11 - U)...
+ (dD11_vec - dU_vec).*(D22 - U)...
-2*real(cD12*dD12_vec);
dB_mat = 2*real( D23.*dD12_vec.'*conj(dD13_vec) ...
+ D12.*dD23_vec.'*conj(dD13_vec) ...
+ D12.*D23.*conj(dD13_mat) ...
+ dD12_vec.'*dD23_vec.*cD13 ...
+ D12.*dD23_mat.*cD13 ...
+ D12.*conj(dD13_vec).'*dD23_vec ...
+ dD12_mat.*D23.*cD13 ...
+ dD23_vec.'*dD12_vec.*cD13 ...
+ conj(dD13_vec).'*dD12_vec.*D23 );
dU_mat = (1/subsum).*( ...
dD11_mat*U1 + dD22_mat*U2 + dD33_mat*U3 ...
+dU1_vec.'*(dD11_vec-dU_vec) ...
+dU2_vec.'*(dD22_vec-dU_vec) ...
+dU3_vec.'*(dD33_vec-dU_vec) ...
-2*(dD11_vec-dU_vec).'*real(cD23*dD23_vec) ...
-2*(dD22_vec-dU_vec).'*real(cD13*dD13_vec) ...
-2*(dD33_vec-dU_vec).'*real(cD12*dD12_vec) ...
-(D11-U)*2*real(cD23*dD23_mat +dD23_vec'*dD23_vec) ...% ' not .' => conj
-(D22-U)*2*real(cD13*dD13_mat +dD13_vec'*dD13_vec) ...
-(D33-U)*2*real(cD12*dD12_mat +dD12_vec'*dD12_vec) ...
+dB_mat);
test = dU_mat;
if nnz( (test - test.') > 1E-10)
disp(test - test.');
error('disp_eig:cld3x3:symmetric','Matrix not symmetric');
end
dU_vec = [dUdom dU_vec];
end
% +++++ Warm plasma model, etc ++++++++++++++++++++++++++++++++++++++++++++++
otherwise
error(['Unknown model ''' plasma.MODEL(1:6) '''.']);
end;
%
% ===== Variable output dimensions ===========================================
%
switch oper
case {'Mon'} % --- Conversion monitors
mon1=0;
tmp=real(cat(1,mon1',mon2'));
out=reshape(tmp,1,numel(tmp));
case {'Dsp','Msw'} % --- Dispersion relations
out = real(U);
case {'Ten'} % --- Dispersion tensor
out = reshape(DD,9,1);
case {'Trj'} % --- ODE trajectory
% dUdom = dU_vec(1);
% dUdomM=(1./dUdom)'; % fn of t, not sigma
dUdomM = 1; % --- Solve as a fn of sigma, not t
J = [zeros(2,2) eye(2); -eye(2) zeros(2,2)]; % I think the sign here is different
% dU = [dUdr dUdz dUdkr dUdkz].';
dU = reshape(dU_vec(2:5),4,1);
dz = (J*dU);
if eval2nd
d2U = dU_mat;
ds = reshape(J*d2U*sym_mat,16,1);
out = real(dUdomM*([dz; ds ]));
else
out = real(dUdomM*dz);
end
% case {'Sgn'}
% out = sign(dUdom); % return the sign of the derivative of time wrt ray parameter
case {'Pol'}
out = pol;
otherwise
error(['Unknown oper ''' oper '''.']);
end