请教用Fortran编写程序求解微分方程组

chunshui2003

通过在论坛里学习了这么久，使我对于分岔有了一些初步的认识，并且可以在建立微分方程组后用matlab编写出求解方程及绘制具体图形。但是随着时间的推移，我越发觉得matlab的优点是后处理，前期计算非常缓慢，尤其是遇到循环，速度更是让人难以忍受（之前的一个敏感性分析，算了1000次，2个小时）。在做分岔图的时候也是如此。所以想采用科学计算能力更强的fortran进行前期计算，现在也正在学，但渴望提高快一些。

我希望论坛里用fortran并且搞分岔稳定方面的朋友能够教我一下，我不是一个聪明人，但还算努力，苦于没有老师，研究这个方面的师兄们也早就不在了。

请大家提供一个duffing系统或者Rossler系统或者其他的任何一个微分方程组求解，保存数据的fortran语言例子，让我好好学习一下。

Chen系统的就可以，a，b，c参数任意可


dx/dt=a*(y-x)
dy/dt=(c-a)*x+c*y-x*z
dz/dt=x*y-b*z



如果搞清楚了，我还会同大家分享经验（我真的希望能够这样），谢谢！

yangxiaokang

怎么突然想改弦更张了，fortran语言我不熟悉，等做出来了给我们也介绍下经验:victory:

lixiaoju

你是想求解微分方程吗？Fortran有现成的龙格库塔积分程序，如果需要可以回复，我只是不知道你现在还需要不，呵呵

chunshui2003

谢谢楼上，我找到了一本算法集，但是现在暂时不需要用fortran计算了。等再进行微分方程求解时，再向你请教。

cailiang

这个就是用四阶龙格库塔方法积分微分方程，徐世良有本书，专门有这方面的编程程序，你可以找找啊。fortran用起来确实怪方便的。

chunshui2003

谢谢楼上的建议！

笨与呆无关

lixiaoju 发表于 2010-7-2 09:43

                               
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你是想求解微分方程吗？Fortran有现成的龙格库塔积分程序，如果需要可以回复，我只是不知道你现在还需要不， ...

你好，我想要有关一阶微分方程组及求解的Fortran程序。你有么

gghhjj

笨与呆无关 发表于 2013-10-30 15:32
你好，我想要有关一阶微分方程组及求解的Fortran程序。你有么

一般fortran的函数库中就有各种龙哥库塔法的函数可以调用，实在没有的话可以参考下面的程序

经典四阶龙哥库塔法

1.       SUBROUTINE runge4(N,X,Y,K,H,get_dydx)                                            
   
2. C-----THIS SUBROUTINE ADVANCES THE SOLUTION TO THE SYSTEM OF N ODE'S  
   
3. C-----DY(i)/DX = f(X,Y(1),Y(2),,.....Y(N)), i=1,...,N      
   
4. C-----USING THE FOURTH ORDER RUNGE-KUTTA METHOD.                        
   
5. C-----N = Number of differential equations in the system. N<=10
   
6. C-----X = INITIAL VALUE OF THE INDEPENDENT VARIABLE; IT IS INCREASED         
   
7. C-----    BY THE SUBROUTINE.                                                  
   
8. C-----Y = INITIAL VALUE OF THE DEPENDENT VARIABLE(S); THE ROUTINE            
   
9. C-----    OVERWRITES THIS WITH THE NEW VALUE(S).   
   
10. C-----H = time step
    
11. C-----K = number of steps to advance the equations
    
12. C-----get_dydx:  subroutine  get_dydx(X,Y,DYDX)
    
13. 
    
14.       IMPLICIT REAL*8(A-H,O-Z)                     
    
15.       EXTERNAL get_dydx      
    
16.       DIMENSION Y(10),YS(10),YSS(10),YSSS(10),T1(10),T2(10),                  
    
17.      1          T3(10),T4(10)                                                
    
18. C-----THE MAIN LOOP                     
    
19. C      print*, X, Y(1), Y(2), Y(3), Y(4), Y(5)
    
20. 
    
21.       DO 1 I=1,K                                                              
    
22. C-----TEMPORARY ARRAYS ARE NEEDED FOR THE FUNCTIONS TO SAVE THEM              
    
23. C-----FOR THE FINAL CORRECTOR STEP.                                          
    
24. C-----FIRST (HALF STEP) PREDICTOR                                             
    
25.       call get_dydx(X,Y,T1)
    
26.       DO 2 J=1,N                                                              
    
27.       YS(J) = Y(J) + .5 * H * T1(J)
    
28.     2 CONTINUE                                                               
    
29. C-----SECOND STEP (HALF STEP CORRECTOR)                                       
    
30.       X = X + .5 * H                                                         
    
31.       call get_dydx(X,YS,T2)
    
32.       DO 3 J=1,N                                                              
    
33.       YSS(J) = Y(J) + .5 * H * T2(J)                                          
    
34.     3 CONTINUE                                                               
    
35. C-----THIRD STEP (FULL STEP MIDPOINT PREDICTOR)                              
    
36.       call get_dydx(X,YSS,T3)
    
37.       DO 4 J=1,N                                                              
    
38.       YSSS(J) = Y(J) + H * T3(J)                                             
    
39.     4 CONTINUE                                                               
    
40. C-----FINAL STEP (SIMPSON'S RULE CORRECTOR)                                   
    
41.       X = X + .5 * H                                                         
    
42.       call get_dydx(X,YSSS,T4)
    
43.       DO 5 J=1,N                                                              
    
44.       Y(J) = Y(J) + (H/6.)*(T1(J) + 2.*(T2(J)+T3(J)) + T4(J))                 
    
45.     5 CONTINUE                                                               
    
46.     1 CONTINUE                                                               
    
47.       RETURN                                                                  
    
48.       END     

复制代码

gghhjj

四五阶龙哥库塔法

1.       SUBROUTINE RKF45(F,NEQN,Y,T,TOUT,RELERR,ABSERR,IFLAG,WORK,IWORK)
   
2. C
   
3. C     FEHLBERG FOURTH-FIFTH ORDER RUNGE-KUTTA METHOD
   
4. C
   
5. C     WRITTEN BY H.A.WATTS AND L.F.SHAMPINE
   
6. C                   SANDIA LABORATORIES
   
7. C                  ALBUQUERQUE,NEW MEXICO
   
8. C
   
9. C    RKF45 IS PRIMARILY DESIGNED TO SOLVE NON-STIFF AND MILDLY STIFF
   
10. C    DIFFERENTIAL EQUATIONS WHEN DERIVATIVE EVALUATIONS ARE INEXPENSIVE.
    
11. C    RKF45 SHOULD GENERALLY NOT BE USED WHEN THE USER IS DEMANDING
    
12. C    HIGH ACCURACY.
    
13. C
    
14. C ABSTRACT
    
15. C
    
16. C    SUBROUTINE  RKF45  INTEGRATES A SYSTEM OF NEQN FIRST ORDER
    
17. C    ORDINARY DIFFERENTIAL EQUATIONS OF THE FORM
    
18. C             DY(I)/DT = F(T,Y(1),Y(2),...,Y(NEQN))
    
19. C              WHERE THE Y(I) ARE GIVEN AT T .
    
20. C    TYPICALLY THE SUBROUTINE IS USED TO INTEGRATE FROM T TO TOUT BUT IT
    
21. C    CAN BE USED AS A ONE-STEP INTEGRATOR TO ADVANCE THE SOLUTION A
    
22. C    SINGLE STEP IN THE DIRECTION OF TOUT.  ON RETURN THE PARAMETERS IN
    
23. C    THE CALL LIST ARE SET FOR CONTINUING THE INTEGRATION. THE USER HAS
    
24. C    ONLY TO CALL RKF45 AGAIN (AND PERHAPS DEFINE A NEW VALUE FOR TOUT).
    
25. C    ACTUALLY, RKF45 IS AN INTERFACING ROUTINE WHICH CALLS SUBROUTINE
    
26. C    RKFS FOR THE SOLUTION.  RKFS IN TURN CALLS SUBROUTINE  FEHL WHICH
    
27. C    COMPUTES AN APPROXIMATE SOLUTION OVER ONE STEP.
    
28. C
    
29. C    RKF45  USES THE RUNGE-KUTTA-FEHLBERG (4,5)  METHOD DESCRIBED
    
30. C    IN THE REFERENCE
    
31. C    E.FEHLBERG , LOW-ORDER CLASSICAL RUNGE-KUTTA FORMULAS WITH STEPSIZE
    
32. C                 CONTROL , NASA TR R-315
    
33. C
    
34. C    THE PERFORMANCE OF RKF45 IS ILLUSTRATED IN THE REFERENCE
    
35. C    L.F.SHAMPINE,H.A.WATTS,S.DAVENPORT, SOLVING NON-STIFF ORDINARY
    
36. C                 DIFFERENTIAL EQUATIONS-THE STATE OF THE ART ,
    
37. C                 SANDIA LABORATORIES REPORT SAND75-0182 ,
    
38. C                 TO APPEAR IN SIAM REVIEW.
    
39. C
    
40. C
    
41. C    THE PARAMETERS REPRESENT-
    
42. C      F -- SUBROUTINE F(T,Y,YP) TO EVALUATE DERIVATIVES YP(I)=DY(I)/DT
    
43. C      NEQN -- NUMBER OF EQUATIONS TO BE INTEGRATED
    
44. C      Y(*) -- SOLUTION VECTOR AT T
    
45. C      T -- INDEPENDENT VARIABLE
    
46. C      TOUT -- OUTPUT POINT AT WHICH SOLUTION IS DESIRED
    
47. C      RELERR,ABSERR -- RELATIVE AND ABSOLUTE ERROR TOLERANCES FOR LOCAL
    
48. C            ERROR TEST. AT EACH STEP THE CODE REQUIRES THAT
    
49. C                 ABS(LOCAL ERROR) .LE. RELERR*ABS(Y) + ABSERR
    
50. C            FOR EACH COMPONENT OF THE LOCAL ERROR AND SOLUTION VECTORS
    
51. C      IFLAG -- INDICATOR FOR STATUS OF INTEGRATION
    
52. C      WORK(*) -- ARRAY TO HOLD INFORMATION INTERNAL TO RKF45 WHICH IS
    
53. C            NECESSARY FOR SUBSEQUENT CALLS. MUST BE DIMENSIONED
    
54. C            AT LEAST  3+6*NEQN
    
55. C      IWORK(*) -- INTEGER ARRAY USED TO HOLD INFORMATION INTERNAL TO
    
56. C            RKF45 WHICH IS NECESSARY FOR SUBSEQUENT CALLS. MUST BE
    
57. C            DIMENSIONED AT LEAST  5
    
58. C
    
59. C
    
60. C!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
    
61. C
    
62. C        IMPORTANT!!!!  SOME OF THE VARIABLES IN RKF45 NEED TO BE
    
63. C        DOUBLE PRECISION. THEY ARE:
    
64. C               
    
65. C                THE ARRAY "Y"
    
66. C                T
    
67. C                TOUT
    
68. C                RELERR
    
69. C                ABSERR
    
70. C                THE ARRAY "WORK"
    
71. C
    
72. C
    
73. C  FIRST CALL TO RKF45
    
74. C
    
75. C    THE USER MUST PROVIDE STORAGE IN HIS CALLING PROGRAM FOR THE ARRAYS
    
76. C    IN THE CALL LIST  -      Y(NEQN) , WORK(3+6*NEQN) , IWORK(5)  ,
    
77. C    DECLARE F IN AN EXTERNAL STATEMENT, SUPPLY SUBROUTINE F(T,Y,YP) AND
    
78. C    INITIALIZE THE FOLLOWING PARAMETERS-
    
79. C
    
80. C      NEQN -- NUMBER OF EQUATIONS TO BE INTEGRATED.  (NEQN .GE. 1)
    
81. C      Y(*) -- VECTOR OF INITIAL CONDITIONS
    
82. C      T -- STARTING POINT OF INTEGRATION , MUST BE A VARIABLE
    
83. C      TOUT -- OUTPUT POINT AT WHICH SOLUTION IS DESIRED.
    
84. C            T=TOUT IS ALLOWED ON THE FIRST CALL ONLY, IN WHICH CASE
    
85. C            RKF45 RETURNS WITH IFLAG=2 IF CONTINUATION IS POSSIBLE.
    
86. C      RELERR,ABSERR -- RELATIVE AND ABSOLUTE LOCAL ERROR TOLERANCES
    
87. C            WHICH MUST BE NON-NEGATIVE. RELERR MUST BE A VARIABLE WHILE
    
88. C            ABSERR MAY BE A CONSTANT. THE CODE SHOULD NORMALLY NOT BE
    
89. C            USED WITH RELATIVE ERROR CONTROL SMALLER THAN ABOUT 1.E-8 .
    
90. C            TO AVOID LIMITING PRECISION DIFFICULTIES THE CODE REQUIRES
    
91. C            RELERR TO BE LARGER THAN AN INTERNALLY COMPUTED RELATIVE
    
92. C            ERROR PARAMETER WHICH IS MACHINE DEPENDENT. IN PARTICULAR,
    
93. C            PURE ABSOLUTE ERROR IS NOT PERMITTED. IF A SMALLER THAN
    
94. C            ALLOWABLE VALUE OF RELERR IS ATTEMPTED, RKF45 INCREASES
    
95. C            RELERR APPROPRIATELY AND RETURNS CONTROL TO THE USER BEFORE
    
96. C            CONTINUING THE INTEGRATION.
    
97. C      IFLAG -- +1,-1  INDICATOR TO INITIALIZE THE CODE FOR EACH NEW
    
98. C            PROBLEM. NORMAL INPUT IS +1. THE USER SHOULD SET IFLAG=-1
    
99. C            ONLY WHEN ONE-STEP INTEGRATOR CONTROL IS ESSENTIAL. IN THIS
    
100. C            CASE, RKF45 ATTEMPTS TO ADVANCE THE SOLUTION A SINGLE STEP
     
101. C            IN THE DIRECTION OF TOUT EACH TIME IT IS CALLED. SINCE THIS
     
102. C            MODE OF OPERATION RESULTS IN EXTRA COMPUTING OVERHEAD, IT
     
103. C            SHOULD BE AVOIDED UNLESS NEEDED.
     
104. C
     
105. C
     
106. C  OUTPUT FROM RKF45
     
107. C
     
108. C      Y(*) -- SOLUTION AT T
     
109. C      T -- LAST POINT REACHED IN INTEGRATION.
     
110. C      IFLAG = 2 -- INTEGRATION REACHED TOUT. INDICATES SUCCESSFUL RETUR
     
111. C                   AND IS THE NORMAL MODE FOR CONTINUING INTEGRATION.
     
112. C            =-2 -- A SINGLE SUCCESSFUL STEP IN THE DIRECTION OF TOUT
     
113. C                   HAS BEEN TAKEN. NORMAL MODE FOR CONTINUING
     
114. C                   INTEGRATION ONE STEP AT A TIME.
     
115. C            = 3 -- INTEGRATION WAS NOT COMPLETED BECAUSE RELATIVE ERROR
     
116. C                   TOLERANCE WAS TOO SMALL. RELERR HAS BEEN INCREASED
     
117. C                   APPROPRIATELY FOR CONTINUING.
     
118. C            = 4 -- INTEGRATION WAS NOT COMPLETED BECAUSE MORE THAN
     
119. C                   3000 DERIVATIVE EVALUATIONS WERE NEEDED. THIS
     
120. C                   IS APPROXIMATELY 500 STEPS.
     
121. C            = 5 -- INTEGRATION WAS NOT COMPLETED BECAUSE SOLUTION
     
122. C                   VANISHED MAKING A PURE RELATIVE ERROR TEST
     
123. C                   IMPOSSIBLE. MUST USE NON-ZERO ABSERR TO CONTINUE.
     
124. C                   USING THE ONE-STEP INTEGRATION MODE FOR ONE STEP
     
125. C                   IS A GOOD WAY TO PROCEED.
     
126. C            = 6 -- INTEGRATION WAS NOT COMPLETED BECAUSE REQUESTED
     
127. C                   ACCURACY COULD NOT BE ACHIEVED USING SMALLEST
     
128. C                   ALLOWABLE STEPSIZE. USER MUST INCREASE THE ERROR
     
129. C                   TOLERANCE BEFORE CONTINUED INTEGRATION CAN BE
     
130. C                   ATTEMPTED.
     
131. C            = 7 -- IT IS LIKELY THAT RKF45 IS INEFFICIENT FOR SOLVING
     
132. C                   THIS PROBLEM. TOO MUCH OUTPUT IS RESTRICTING THE
     
133. C                   NATURAL STEPSIZE CHOICE. USE THE ONE-STEP INTEGRATOR
     
134. C                   MODE.
     
135. C            = 8 -- INVALID INPUT PARAMETERS
     
136. C                   THIS INDICATOR OCCURS IF ANY OF THE FOLLOWING IS
     
137. C                   SATISFIED -   NEQN .LE. 0
     
138. C                                 T=TOUT  AND  IFLAG .NE. +1 OR -1
     
139. C                                 RELERR OR ABSERR .LT. 0.
     
140. C                                 IFLAG .EQ. 0  OR  .LT. -2  OR  .GT. 8
     
141. C      WORK(*),IWORK(*) -- INFORMATION WHICH IS USUALLY OF NO INTEREST
     
142. C                   TO THE USER BUT NECESSARY FOR SUBSEQUENT CALLS.
     
143. C                   WORK(1),...,WORK(NEQN) CONTAIN THE FIRST DERIVATIVES
     
144. C                   OF THE SOLUTION VECTOR Y AT T. WORK(NEQN+1) CONTAINS
     
145. C                   THE STEPSIZE H TO BE ATTEMPTED ON THE NEXT STEP.
     
146. C                   IWORK(1) CONTAINS THE DERIVATIVE EVALUATION COUNTER.
     
147. C
     
148. C
     
149. C  SUBSEQUENT CALLS TO RKF45
     
150. C
     
151. C    SUBROUTINE RKF45 RETURNS WITH ALL INFORMATION NEEDED TO CONTINUE
     
152. C    THE INTEGRATION. IF THE INTEGRATION REACHED TOUT, THE USER NEED ONL
     
153. C    DEFINE A NEW TOUT AND CALL RKF45 AGAIN. IN THE ONE-STEP INTEGRATOR
     
154. C    MODE (IFLAG=-2) THE USER MUST KEEP IN MIND THAT EACH STEP TAKEN IS
     
155. C    IN THE DIRECTION OF THE CURRENT TOUT. UPON REACHING TOUT (INDICATED
     
156. C    BY CHANGING IFLAG TO 2),THE USER MUST THEN DEFINE A NEW TOUT AND
     
157. C    RESET IFLAG TO -2 TO CONTINUE IN THE ONE-STEP INTEGRATOR MODE.
     
158. C
     
159. C    IF THE INTEGRATION WAS NOT COMPLETED BUT THE USER STILL WANTS TO
     
160. C    CONTINUE (IFLAG=3,4 CASES), HE JUST CALLS RKF45 AGAIN. WITH IFLAG=3
     
161. C    THE RELERR PARAMETER HAS BEEN ADJUSTED APPROPRIATELY FOR CONTINUING
     
162. C    THE INTEGRATION. IN THE CASE OF IFLAG=4 THE FUNCTION COUNTER WILL
     
163. C    BE RESET TO 0 AND ANOTHER 3000 FUNCTION EVALUATIONS ARE ALLOWED.
     
164. C
     
165. C    HOWEVER,IN THE CASE IFLAG=5, THE USER MUST FIRST ALTER THE ERROR
     
166. C    CRITERION TO USE A POSITIVE VALUE OF ABSERR BEFORE INTEGRATION CAN
     
167. C    PROCEED. IF HE DOES NOT,EXECUTION IS TERMINATED.
     
168. C
     
169. C    ALSO,IN THE CASE IFLAG=6, IT IS NECESSARY FOR THE USER TO RESET
     
170. C    IFLAG TO 2 (OR -2 WHEN THE ONE-STEP INTEGRATION MODE IS BEING USED)
     
171. C    AS WELL AS INCREASING EITHER ABSERR,RELERR OR BOTH BEFORE THE
     
172. C    INTEGRATION CAN BE CONTINUED. IF THIS IS NOT DONE, EXECUTION WILL
     
173. C    BE TERMINATED. THE OCCURRENCE OF IFLAG=6 INDICATES A TROUBLE SPOT
     
174. C    (SOLUTION IS CHANGING RAPIDLY,SINGULARITY MAY BE PRESENT) AND IT
     
175. C    OFTEN IS INADVISABLE TO CONTINUE.
     
176. C
     
177. C    IF IFLAG=7 IS ENCOUNTERED, THE USER SHOULD USE THE ONE-STEP
     
178. C    INTEGRATION MODE WITH THE STEPSIZE DETERMINED BY THE CODE OR
     
179. C    CONSIDER SWITCHING TO THE ADAMS CODES DE/STEP,INTRP. IF THE USER
     
180. C    INSISTS UPON CONTINUING THE INTEGRATION WITH RKF45, HE MUST RESET
     
181. C    IFLAG TO 2 BEFORE CALLING RKF45 AGAIN. OTHERWISE,EXECUTION WILL BE
     
182. C    TERMINATED.
     
183. C
     
184. C    IF IFLAG=8 IS OBTAINED, INTEGRATION CAN NOT BE CONTINUED UNLESS
     
185. C    THE INVALID INPUT PARAMETERS ARE CORRECTED.
     
186. C
     
187. C    IT SHOULD BE NOTED THAT THE ARRAYS WORK,IWORK CONTAIN INFORMATION
     
188. C    REQUIRED FOR SUBSEQUENT INTEGRATION. ACCORDINGLY, WORK AND IWORK
     
189. C    SHOULD NOT BE ALTERED.
     
190. C
     
191. C
     
192.       INTEGER NEQN,IFLAG,IWORK(5)
     
193.       DOUBLE PRECISION Y(NEQN),T,TOUT,RELERR,ABSERR,WORK(1)
     
194. C
     
195.       EXTERNAL F
     
196. C
     
197.       INTEGER K1,K2,K3,K4,K5,K6,K1M
     
198. C
     
199. C
     
200. C     COMPUTE INDICES FOR THE SPLITTING OF THE WORK ARRAY
     
201. C
     
202.       K1M=NEQN+1
     
203.       K1=K1M+1
     
204.       K2=K1+NEQN
     
205.       K3=K2+NEQN
     
206.       K4=K3+NEQN
     
207.       K5=K4+NEQN
     
208.       K6=K5+NEQN
     
209. C
     
210. C     THIS INTERFACING ROUTINE MERELY RELIEVES THE USER OF A LONG
     
211. C     CALLING LIST VIA THE SPLITTING APART OF TWO WORKING STORAGE
     
212. C     ARRAYS. IF THIS IS NOT COMPATIBLE WITH THE USERS COMPILER,
     
213. C     HE MUST USE RKFS DIRECTLY.
     
214. C
     
215.       CALL RKFS(F,NEQN,Y,T,TOUT,RELERR,ABSERR,IFLAG,WORK(1),WORK(K1M),
     
216.      1          WORK(K1),WORK(K2),WORK(K3),WORK(K4),WORK(K5),WORK(K6),
     
217.      2          WORK(K6+1),IWORK(1),IWORK(2),IWORK(3),IWORK(4),IWORK(5))
     
218. C
     
219.       RETURN
     
220.       END
     
221.       SUBROUTINE RKFS(F,NEQN,Y,T,TOUT,RELERR,ABSERR,IFLAG,YP,H,F1,F2,F3,
     
222.      1                F4,F5,SAVRE,SAVAE,NFE,KOP,INIT,JFLAG,KFLAG)
     
223. C
     
224. C     FEHLBERG FOURTH-FIFTH ORDER RUNGE-KUTTA METHOD
     
225. C
     
226. C
     
227. C     RKFS INTEGRATES A SYSTEM OF FIRST ORDER ORDINARY DIFFERENTIAL
     
228. C     EQUATIONS AS DESCRIBED IN THE COMMENTS FOR RKF45 .
     
229. C     THE ARRAYS YP,F1,F2,F3,F4,AND F5 (OF DIMENSION AT LEAST NEQN) AND
     
230. C     THE VARIABLES H,SAVRE,SAVAE,NFE,KOP,INIT,JFLAG,AND KFLAG ARE USED
     
231. C     INTERNALLY BY THE CODE AND APPEAR IN THE CALL LIST TO ELIMINATE
     
232. C     LOCAL RETENTION OF VARIABLES BETWEEN CALLS. ACCORDINGLY, THEY
     
233. C     SHOULD NOT BE ALTERED. ITEMS OF POSSIBLE INTEREST ARE
     
234. C         YP - DERIVATIVE OF SOLUTION VECTOR AT T
     
235. C         H  - AN APPROPRIATE STEPSIZE TO BE USED FOR THE NEXT STEP
     
236. C         NFE- COUNTER ON THE NUMBER OF DERIVATIVE FUNCTION EVALUATIONS
     
237. C
     
238. C
     
239.       LOGICAL HFAILD,OUTPUT
     
240. C
     
241.       INTEGER  NEQN,IFLAG,NFE,KOP,INIT,JFLAG,KFLAG
     
242.       DOUBLE PRECISION  Y(NEQN),T,TOUT,RELERR,ABSERR,H,YP(NEQN),
     
243.      1  F1(NEQN),F2(NEQN),F3(NEQN),F4(NEQN),F5(NEQN),SAVRE,
     
244.      2  SAVAE
     
245. C
     
246.       EXTERNAL F
     
247. C
     
248.       DOUBLE PRECISION  A,AE,DT,EE,EEOET,ESTTOL,ET,HMIN,REMIN,RER,S,
     
249.      1  SCALE,TOL,TOLN,U26,EPSP1,EPS,YPK
     
250. C
     
251.       INTEGER  K,MAXNFE,MFLAG
     
252. C
     
253.       DOUBLE PRECISION  DABS,DMAX1,DMIN1,DSIGN
     
254. C
     
255. C  REMIN IS THE MINIMUM ACCEPTABLE VALUE OF RELERR.  ATTEMPTS
     
256. C  TO OBTAIN HIGHER ACCURACY WITH THIS SUBROUTINE ARE USUALLY
     
257. C  VERY EXPENSIVE AND OFTEN UNSUCCESSFUL.
     
258. C
     
259.       DATA REMIN/1.D-12/
     
260. C
     
261. C
     
262. C     THE EXPENSE IS CONTROLLED BY RESTRICTING THE NUMBER
     
263. C     OF FUNCTION EVALUATIONS TO BE APPROXIMATELY MAXNFE.
     
264. C     AS SET, THIS CORRESPONDS TO ABOUT 500 STEPS.
     
265. C
     
266.       DATA MAXNFE/3000/
     
267. C
     
268. C
     
269. C     CHECK INPUT PARAMETERS
     
270. C
     
271. C
     
272.       IF (NEQN .LT. 1) GO TO 10
     
273.       IF ((RELERR .LT. 0.0D0)  .OR.  (ABSERR .LT. 0.0D0)) GO TO 10
     
274.       MFLAG=IABS(IFLAG)
     
275.       IF ((MFLAG .GE. 1)  .AND.  (MFLAG .LE. 8)) GO TO 20
     
276. C
     
277. C     INVALID INPUT
     
278.    10 IFLAG=8
     
279.       RETURN
     
280. C
     
281. C     IS THIS THE FIRST CALL
     
282.    20 IF (MFLAG .EQ. 1) GO TO 50
     
283. C
     
284. C     CHECK CONTINUATION POSSIBILITIES
     
285. C
     
286.       IF ((T .EQ. TOUT) .AND. (KFLAG .NE. 3)) GO TO 10
     
287.       IF (MFLAG .NE. 2) GO TO 25
     
288. C
     
289. C     IFLAG = +2 OR -2
     
290.       IF (KFLAG .EQ. 3) GO TO 45
     
291.       IF (INIT .EQ. 0) GO TO 45
     
292.       IF (KFLAG .EQ. 4) GO TO 40
     
293.       IF ((KFLAG .EQ. 5)  .AND.  (ABSERR .EQ. 0.0D0)) GO TO 30
     
294.       IF ((KFLAG .EQ. 6)  .AND.  (RELERR .LE. SAVRE)  .AND.
     
295.      1    (ABSERR .LE. SAVAE)) GO TO 30
     
296.       GO TO 50
     
297. C
     
298. C     IFLAG = 3,4,5,6,7 OR 8
     
299.    25 IF (IFLAG .EQ. 3) GO TO 45
     
300.       IF (IFLAG .EQ. 4) GO TO 40
     
301.       IF ((IFLAG .EQ. 5) .AND. (ABSERR .GT. 0.0D0)) GO TO 45
     
302. C
     
303. C     INTEGRATION CANNOT BE CONTINUED SINCE USER DID NOT RESPOND TO
     
304. C     THE INSTRUCTIONS PERTAINING TO IFLAG=5,6,7 OR 8
     
305.    30 STOP
     
306. C
     
307. C     RESET FUNCTION EVALUATION COUNTER
     
308.    40 NFE=0
     
309.       IF (MFLAG .EQ. 2) GO TO 50
     
310. C
     
311. C     RESET FLAG VALUE FROM PREVIOUS CALL
     
312.    45 IFLAG=JFLAG
     
313.       IF (KFLAG .EQ. 3) MFLAG=IABS(IFLAG)
     
314. C
     
315. C     SAVE INPUT IFLAG AND SET CONTINUATION FLAG VALUE FOR SUBSEQUENT
     
316. C     INPUT CHECKING
     
317.    50 JFLAG=IFLAG
     
318.       KFLAG=0
     
319. C
     
320. C     SAVE RELERR AND ABSERR FOR CHECKING INPUT ON SUBSEQUENT CALLS
     
321.       SAVRE=RELERR
     
322.       SAVAE=ABSERR
     
323. C
     
324. C  COMPUTE MACHINE EPSILON
     
325. C
     
326.       EPS = 1.0D0
     
327.    53 EPS = EPS/2.0D0
     
328.       EPSP1 = EPS + 1.0D0
     
329.       IF (EPSP1 .GT. 1.0D0) GO TO 53
     
330. C
     
331. C     RESTRICT RELATIVE ERROR TOLERANCE TO BE AT LEAST AS LARGE AS
     
332. C     2*EPS+REMIN TO AVOID LIMITING PRECISION DIFFICULTIES ARISING
     
333. C     FROM IMPOSSIBLE ACCURACY REQUESTS
     
334. C
     
335.       RER=2.0D0*EPS+REMIN
     
336.       IF (RELERR .GE. RER) GO TO 55
     
337. C
     
338. C     RELATIVE ERROR TOLERANCE TOO SMALL
     
339.       RELERR=RER
     
340.       IFLAG=3
     
341.       KFLAG=3
     
342.       RETURN
     
343. C
     
344.    55 U26=26.0D0*EPS
     
345. C
     
346.       DT=TOUT-T
     
347. C
     
348.       IF (MFLAG .EQ. 1) GO TO 60
     
349.       IF (INIT .EQ. 0) GO TO 65
     
350.       GO TO 80
     
351. C
     
352. C     INITIALIZATION --
     
353. C                       SET INITIALIZATION COMPLETION INDICATOR,INIT
     
354. C                       SET INDICATOR FOR TOO MANY OUTPUT POINTS,KOP
     
355. C                       EVALUATE INITIAL DERIVATIVES
     
356. C                       SET COUNTER FOR FUNCTION EVALUATIONS,NFE
     
357. C                       EVALUATE INITIAL DERIVATIVES
     
358. C                       SET COUNTER FOR FUNCTION EVALUATIONS,NFE
     
359. C                       ESTIMATE STARTING STEPSIZE
     
360. C
     
361.    60 INIT=0
     
362.       KOP=0
     
363. C
     
364.       A=T
     
365.       CALL F(A,Y,YP)
     
366.       NFE=1
     
367.       IF (T .NE. TOUT) GO TO 65
     
368.       IFLAG=2
     
369.       RETURN
     
370. C
     
371. C
     
372.    65 INIT=1
     
373.       H=DABS(DT)
     
374.       TOLN=0.
     
375.       DO 70 K=1,NEQN
     
376.         TOL=RELERR*DABS(Y(K))+ABSERR
     
377.         IF (TOL .LE. 0.) GO TO 70
     
378.         TOLN=TOL
     
379.         YPK=DABS(YP(K))
     
380.         IF (YPK*H**5 .GT. TOL) H=(TOL/YPK)**0.2D0
     
381.    70 CONTINUE
     
382.       IF (TOLN .LE. 0.0D0) H=0.0D0
     
383.       H=DMAX1(H,U26*DMAX1(DABS(T),DABS(DT)))
     
384.       JFLAG=ISIGN(2,IFLAG)
     
385. C
     
386. C
     
387. C     SET STEPSIZE FOR INTEGRATION IN THE DIRECTION FROM T TO TOUT
     
388. C
     
389.    80 H=DSIGN(H,DT)
     
390. C
     
391. C     TEST TO SEE IF RKF45 IS BEING SEVERELY IMPACTED BY TOO MANY
     
392. C     OUTPUT POINTS
     
393. C
     
394.       IF (DABS(H) .GE. 2.0D0*DABS(DT)) KOP=KOP+1
     
395.       IF (KOP .NE. 100) GO TO 85
     
396. C
     
397. C     UNNECESSARY FREQUENCY OF OUTPUT
     
398.       KOP=0
     
399.       IFLAG=7
     
400.       RETURN
     
401. C
     
402.    85 IF (DABS(DT) .GT. U26*DABS(T)) GO TO 95
     
403. C
     
404. C     IF TOO CLOSE TO OUTPUT POINT,EXTRAPOLATE AND RETURN
     
405. C
     
406.       DO 90 K=1,NEQN
     
407.    90   Y(K)=Y(K)+DT*YP(K)
     
408.       A=TOUT
     
409.       CALL F(A,Y,YP)
     
410.       NFE=NFE+1
     
411.       GO TO 300
     
412. C
     
413. C
     
414. C     INITIALIZE OUTPUT POINT INDICATOR
     
415. C
     
416.    95 OUTPUT= .FALSE.
     
417. C
     
418. C     TO AVOID PREMATURE UNDERFLOW IN THE ERROR TOLERANCE FUNCTION,
     
419. C     SCALE THE ERROR TOLERANCES
     
420. C
     
421.       SCALE=2.0D0/RELERR
     
422.       AE=SCALE*ABSERR
     
423. C
     
424. C
     
425. C     STEP BY STEP INTEGRATION
     
426. C
     
427.   100 HFAILD= .FALSE.
     
428. C
     
429. C     SET SMALLEST ALLOWABLE STEPSIZE
     
430. C
     
431.       HMIN=U26*DABS(T)
     
432. C
     
433. C     ADJUST STEPSIZE IF NECESSARY TO HIT THE OUTPUT POINT.
     
434. C     LOOK AHEAD TWO STEPS TO AVOID DRASTIC CHANGES IN THE STEPSIZE AND
     
435. C     THUS LESSEN THE IMPACT OF OUTPUT POINTS ON THE CODE.
     
436. C
     
437.       DT=TOUT-T
     
438.       IF (DABS(DT) .GE. 2.0D0*DABS(H)) GO TO 200
     
439.       IF (DABS(DT) .GT. DABS(H)) GO TO 150
     
440. C
     
441. C     THE NEXT SUCCESSFUL STEP WILL COMPLETE THE INTEGRATION TO THE
     
442. C     OUTPUT POINT
     
443. C
     
444.       OUTPUT= .TRUE.
     
445.       H=DT
     
446.       GO TO 200
     
447. C
     
448.   150 H=0.5D0*DT
     
449. C
     
450. C
     
451. C
     
452. C     CORE INTEGRATOR FOR TAKING A SINGLE STEP
     
453. C
     
454. C     THE TOLERANCES HAVE BEEN SCALED TO AVOID PREMATURE UNDERFLOW IN
     
455. C     COMPUTING THE ERROR TOLERANCE FUNCTION ET.
     
456. C     TO AVOID PROBLEMS WITH ZERO CROSSINGS,RELATIVE ERROR IS MEASURED
     
457. C     USING THE AVERAGE OF THE MAGNITUDES OF THE SOLUTION AT THE
     
458. C     BEGINNING AND END OF A STEP.
     
459. C     THE ERROR ESTIMATE FORMULA HAS BEEN GROUPED TO CONTROL LOSS OF
     
460. C     SIGNIFICANCE.
     
461. C     TO DISTINGUISH THE VARIOUS ARGUMENTS, H IS NOT PERMITTED
     
462. C     TO BECOME SMALLER THAN 26 UNITS OF ROUNDOFF IN T.
     
463. C     PRACTICAL LIMITS ON THE CHANGE IN THE STEPSIZE ARE ENFORCED TO
     
464. C     SMOOTH THE STEPSIZE SELECTION PROCESS AND TO AVOID EXCESSIVE
     
465. C     CHATTERING ON PROBLEMS HAVING DISCONTINUITIES.
     
466. C     TO PREVENT UNNECESSARY FAILURES, THE CODE USES 9/10 THE STEPSIZE
     
467. C     IT ESTIMATES WILL SUCCEED.
     
468. C     AFTER A STEP FAILURE, THE STEPSIZE IS NOT ALLOWED TO INCREASE FOR
     
469. C     THE NEXT ATTEMPTED STEP. THIS MAKES THE CODE MORE EFFICIENT ON
     
470. C     PROBLEMS HAVING DISCONTINUITIES AND MORE EFFECTIVE IN GENERAL
     
471. C     SINCE LOCAL EXTRAPOLATION IS BEING USED AND EXTRA CAUTION SEEMS
     
472. C     WARRANTED.
     
473. C
     
474. C
     
475. C     TEST NUMBER OF DERIVATIVE FUNCTION EVALUATIONS.
     
476. C     IF OKAY,TRY TO ADVANCE THE INTEGRATION FROM T TO T+H
     
477. C
     
478.   200 IF (NFE .LE. MAXNFE) GO TO 220
     
479. C
     
480. C     TOO MUCH WORK
     
481.       IFLAG=4
     
482.       KFLAG=4
     
483.       RETURN
     
484. C
     
485. C     ADVANCE AN APPROXIMATE SOLUTION OVER ONE STEP OF LENGTH H
     
486. C
     
487.   220 CALL FEHL(F,NEQN,Y,T,H,YP,F1,F2,F3,F4,F5,F1)
     
488.       NFE=NFE+5
     
489. C
     
490. C     COMPUTE AND TEST ALLOWABLE TOLERANCES VERSUS LOCAL ERROR ESTIMATES
     
491. C     AND REMOVE SCALING OF TOLERANCES. NOTE THAT RELATIVE ERROR IS
     
492. C     MEASURED WITH RESPECT TO THE AVERAGE OF THE MAGNITUDES OF THE
     
493. C     SOLUTION AT THE BEGINNING AND END OF THE STEP.
     
494. C
     
495.       EEOET=0.0D0
     
496.       DO 250 K=1,NEQN
     
497.         ET=DABS(Y(K))+DABS(F1(K))+AE
     
498.         IF (ET .GT. 0.0D0) GO TO 240
     
499. C
     
500. C       INAPPROPRIATE ERROR TOLERANCE
     
501.         IFLAG=5
     
502.         RETURN
     
503. C
     
504.   240   EE=DABS((-2090.0D0*YP(K)+(21970.0D0*F3(K)-15048.0D0*F4(K)))+
     
505.      1                        (22528.0D0*F2(K)-27360.0D0*F5(K)))
     
506.   250   EEOET=DMAX1(EEOET,EE/ET)
     
507. C
     
508.       ESTTOL=DABS(H)*EEOET*SCALE/752400.0D0
     
509. C
     
510.       IF (ESTTOL .LE. 1.0D0) GO TO 260
     
511. C
     
512. C
     
513. C     UNSUCCESSFUL STEP
     
514. C                       REDUCE THE STEPSIZE , TRY AGAIN
     
515. C                       THE DECREASE IS LIMITED TO A FACTOR OF 1/10
     
516. C
     
517.       HFAILD= .TRUE.
     
518.       OUTPUT= .FALSE.
     
519.       S=0.1D0
     
520.       IF (ESTTOL .LT. 59049.0D0) S=0.9D0/ESTTOL**0.2D0
     
521.       H=S*H
     
522.       IF (DABS(H) .GT. HMIN) GO TO 200
     
523. C
     
524. C     REQUESTED ERROR UNATTAINABLE AT SMALLEST ALLOWABLE STEPSIZE
     
525.       IFLAG=6
     
526.       KFLAG=6
     
527.       RETURN
     
528. C
     
529. C
     
530. C     SUCCESSFUL STEP
     
531. C                        STORE SOLUTION AT T+H
     
532. C                        AND EVALUATE DERIVATIVES THERE
     
533. C
     
534.   260 T=T+H
     
535.       DO 270 K=1,NEQN
     
536.   270   Y(K)=F1(K)
     
537.       A=T
     
538.       CALL F(A,Y,YP)
     
539.       NFE=NFE+1
     
540. C
     
541. C
     
542. C                       CHOOSE NEXT STEPSIZE
     
543. C                       THE INCREASE IS LIMITED TO A FACTOR OF 5
     
544. C                       IF STEP FAILURE HAS JUST OCCURRED, NEXT
     
545. C                          STEPSIZE IS NOT ALLOWED TO INCREASE
     
546. C
     
547.       S=5.0D0
     
548.       IF (ESTTOL .GT. 1.889568D-4) S=0.9D0/ESTTOL**0.2D0
     
549.       IF (HFAILD) S=DMIN1(S,1.0D0)
     
550.       H=DSIGN(DMAX1(S*DABS(H),HMIN),H)
     
551. C
     
552. C     END OF CORE INTEGRATOR
     
553. C
     
554. C
     
555. C     SHOULD WE TAKE ANOTHER STEP
     
556. C
     
557.       IF (OUTPUT) GO TO 300
     
558.       IF (IFLAG .GT. 0) GO TO 100
     
559. C
     
560. C
     
561. C     INTEGRATION SUCCESSFULLY COMPLETED
     
562. C
     
563. C     ONE-STEP MODE
     
564.       IFLAG=-2
     
565.       RETURN
     
566. C
     
567. C     INTERVAL MODE
     
568.   300 T=TOUT
     
569.       IFLAG=2
     
570.       RETURN
     
571. C
     
572.       END
     
573.       SUBROUTINE FEHL(F,NEQN,Y,T,H,YP,F1,F2,F3,F4,F5,S)
     
574. C
     
575. C     FEHLBERG FOURTH-FIFTH ORDER RUNGE-KUTTA METHOD
     
576. C
     
577. C    FEHL INTEGRATES A SYSTEM OF NEQN FIRST ORDER
     
578. C    ORDINARY DIFFERENTIAL EQUATIONS OF THE FORM
     
579. C             DY(I)/DT=F(T,Y(1),---,Y(NEQN))
     
580. C    WHERE THE INITIAL VALUES Y(I) AND THE INITIAL DERIVATIVES
     
581. C    YP(I) ARE SPECIFIED AT THE STARTING POINT T. FEHL ADVANCES
     
582. C    THE SOLUTION OVER THE FIXED STEP H AND RETURNS
     
583. C    THE FIFTH ORDER (SIXTH ORDER ACCURATE LOCALLY) SOLUTION
     
584. C    APPROXIMATION AT T+H IN ARRAY S(I).
     
585. C    F1,---,F5 ARE ARRAYS OF DIMENSION NEQN WHICH ARE NEEDED
     
586. C    FOR INTERNAL STORAGE.
     
587. C    THE FORMULAS HAVE BEEN GROUPED TO CONTROL LOSS OF SIGNIFICANCE.
     
588. C    FEHL SHOULD BE CALLED WITH AN H NOT SMALLER THAN 13 UNITS OF
     
589. C    ROUNDOFF IN T SO THAT THE VARIOUS INDEPENDENT ARGUMENTS CAN BE
     
590. C    DISTINGUISHED.
     
591. C
     
592. C
     
593.       external F
     
594.       INTEGER  NEQN
     
595.       DOUBLE PRECISION  Y(NEQN),T,H,YP(NEQN),F1(NEQN),F2(NEQN),
     
596.      1  F3(NEQN),F4(NEQN),F5(NEQN),S(NEQN)
     
597. C
     
598.       DOUBLE PRECISION  CH
     
599.       INTEGER  K
     
600. C
     
601.       CH=H/4.0D0
     
602.       DO 221 K=1,NEQN
     
603.   221   F5(K)=Y(K)+CH*YP(K)
     
604.       CALL F(T+CH,F5,F1)
     
605. C
     
606.       CH=3.0D0*H/32.0D0
     
607.       DO 222 K=1,NEQN
     
608.   222   F5(K)=Y(K)+CH*(YP(K)+3.0D0*F1(K))
     
609.       CALL F(T+3.0D0*H/8.0D0,F5,F2)
     
610. C
     
611.       CH=H/2197.0D0
     
612.       DO 223 K=1,NEQN
     
613.   223   F5(K)=Y(K)+CH*(1932.0D0*YP(K)+(7296.0D0*F2(K)-7200.0D0*F1(K)))
     
614.       CALL F(T+12.0D0*H/13.0D0,F5,F3)
     
615. C
     
616.       CH=H/4104.0D0
     
617.       DO 224 K=1,NEQN
     
618.   224   F5(K)=Y(K)+CH*((8341.0D0*YP(K)-845.0D0*F3(K))+
     
619.      1                            (29440.0D0*F2(K)-32832.0D0*F1(K)))
     
620.       CALL F(T+H,F5,F4)
     
621. C
     
622.       CH=H/20520.0D0
     
623.       DO 225 K=1,NEQN
     
624.   225   F1(K)=Y(K)+CH*((-6080.0D0*YP(K)+(9295.0D0*F3(K)-
     
625.      1         5643.0D0*F4(K)))+(41040.0D0*F1(K)-28352.0D0*F2(K)))
     
626.       CALL F(T+H/2.0D0,F1,F5)
     
627. C
     
628. C     COMPUTE APPROXIMATE SOLUTION AT T+H
     
629. C
     
630.       CH=H/7618050.0D0
     
631.       DO 230 K=1,NEQN
     
632.   230   S(K)=Y(K)+CH*((902880.0D0*YP(K)+(3855735.0D0*F3(K)-
     
633.      1        1371249.0D0*F4(K)))+(3953664.0D0*F2(K)+
     
634.      2        277020.0D0*F5(K)))
     
635. C
     
636.       RETURN
     
637.       END

复制代码

gghhjj

四五阶龙哥库塔法使用范例

1. C  SAMPLE PROGRAM FOR RKF45
   
2. C
   
3. C
   
4.       EXTERNAL ORBIT
   
5.       DOUBLE PRECISION T,Y(4),TOUT,RELERR,ABSERR
   
6.       DOUBLE PRECISION TFINAL,TPRINT,ECC,ALFA,ALFASQ,WORK(27)
   
7.       INTEGER IWORK(5), IFLAG, NEQN
   
8.       DOUBLE PRECISION DSQRT
   
9.       COMMON ALFASQ
   
10.       ECC = 0.25
    
11.       ALFA = 3.141592653589/4.0
    
12.       ALFASQ = ALFA*ALFA
    
13.       NEQN = 4
    
14.       T = 0.0
    
15.       Y(1) = 1.0 - ECC
    
16.       Y(2) = 0.0
    
17.       Y(3) = 0.0
    
18.       Y(4) = ALFA*DSQRT( (1.0 + ECC)/(1.0 - ECC) )
    
19.       RELERR = 1.0D-9
    
20.       ABSERR = 0.0
    
21.       TFINAL = 12.0
    
22.       TPRINT = 0.5
    
23.       IFLAG = 1
    
24.       TOUT = T
    
25.    10 CALL RKF45(ORBIT,NEQN,Y,T,TOUT,RELERR,ABSERR,IFLAG,WORK,IWORK)
    
26.       WRITE(1,11) T, Y(1), Y(2)
    
27.       GO TO (80,20,30,40,50,60,70,80), IFLAG
    
28.    20 TOUT = T + TPRINT
    
29.       IF (T .LT. TFINAL) GO TO 10
    
30.       STOP
    
31.    30 WRITE(1,31) RELERR,ABSERR
    
32.       GO TO 10
    
33.    40 WRITE(1,41)
    
34.       GO TO 10
    
35.    50 ABSERR = 1.0D-9
    
36.       WRITE(1,31) RELERR,ABSERR
    
37.       GO TO 10
    
38.    60 RELERR = 10.0*RELERR
    
39.       WRITE(1,31) RELERR,ABSERR
    
40.       IFLAG = 2
    
41.       GO TO 10
    
42.    70 WRITE(1,71)
    
43.       IFLAG = 2
    
44.       GO TO 10
    
45.    80 WRITE(1,81)
    
46.       STOP
    
47. C
    
48.    11 FORMAT(F5.1, 2F15.9)
    
49.    31 FORMAT(17H TOLERANCES RESET, 2E12.3)
    
50.    41 FORMAT(11H MANY STEPS)
    
51.    71 FORMAT(12H MUCH OUTPUT)
    
52.    81 FORMAT(14H IMPROPER CALL)
    
53.       END
    
54. 
    
55. 
    
56.       SUBROUTINE ORBIT (T, Y, YP)
    
57.       DOUBLE PRECISION T, Y(4), YP(4), R, ALFASQ
    
58.       DOUBLE PRECISION DSQRT
    
59.       COMMON ALFASQ
    
60.       R = Y(1)*Y(1) + Y(2)*Y(2)
    
61.       R = R*DSQRT(R)/ALFASQ
    
62.       YP(1) = Y(3)
    
63.       YP(2) = Y(4)
    
64.       YP(3) = -Y(1)/R
    
65.       YP(4) = -Y(2)/R
    
66.       RETURN
    
67.       END

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