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1 C Work performed under the auspices of the U.S. Department of Energy |
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2 C by Lawrence Livermore National Laboratory under contract number |
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3 C W-7405-Eng-48. |
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4 C |
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5 SUBROUTINE DNEDD(X,Y,YPRIME,NEQ,RES,JACD,PDUM,H,WT, |
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6 * JSTART,IDID,RPAR,IPAR,PHI,GAMMA,DUMSVR,DELTA,E, |
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7 * WM,IWM,CJ,CJOLD,CJLAST,S,UROUND,DUME,DUMS,DUMR, |
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8 * EPCON,JCALC,JFDUM,KP1,NONNEG,NTYPE,IERNLS) |
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9 C |
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10 C***BEGIN PROLOGUE DNEDD |
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11 C***REFER TO DDASPK |
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12 C***DATE WRITTEN 891219 (YYMMDD) |
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13 C***REVISION DATE 900926 (YYMMDD) |
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14 C |
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15 C |
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16 C----------------------------------------------------------------------- |
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17 C***DESCRIPTION |
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18 C |
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19 C DNEDD solves a nonlinear system of |
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20 C algebraic equations of the form |
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21 C G(X,Y,YPRIME) = 0 for the unknown Y. |
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22 C |
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23 C The method used is a modified Newton scheme. |
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24 C |
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25 C The parameters represent |
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26 C |
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27 C X -- Independent variable. |
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28 C Y -- Solution vector. |
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29 C YPRIME -- Derivative of solution vector. |
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30 C NEQ -- Number of unknowns. |
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31 C RES -- External user-supplied subroutine |
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32 C to evaluate the residual. See RES description |
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33 C in DDASPK prologue. |
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34 C JACD -- External user-supplied routine to evaluate the |
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35 C Jacobian. See JAC description for the case |
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36 C INFO(12) = 0 in the DDASPK prologue. |
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37 C PDUM -- Dummy argument. |
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38 C H -- Appropriate step size for next step. |
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39 C WT -- Vector of weights for error criterion. |
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40 C JSTART -- Indicates first call to this routine. |
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41 C If JSTART = 0, then this is the first call, |
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42 C otherwise it is not. |
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43 C IDID -- Completion flag, output by DNEDD. |
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44 C See IDID description in DDASPK prologue. |
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45 C RPAR,IPAR -- Real and integer arrays used for communication |
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46 C between the calling program and external user |
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47 C routines. They are not altered within DASPK. |
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48 C PHI -- Array of divided differences used by |
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49 C DNEDD. The length is NEQ*(K+1),where |
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50 C K is the maximum order. |
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51 C GAMMA -- Array used to predict Y and YPRIME. The length |
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52 C is MAXORD+1 where MAXORD is the maximum order. |
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53 C DUMSVR -- Dummy argument. |
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54 C DELTA -- Work vector for NLS of length NEQ. |
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55 C E -- Error accumulation vector for NLS of length NEQ. |
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56 C WM,IWM -- Real and integer arrays storing |
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57 C matrix information such as the matrix |
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58 C of partial derivatives, permutation |
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59 C vector, and various other information. |
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60 C CJ -- Parameter always proportional to 1/H. |
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61 C CJOLD -- Saves the value of CJ as of the last call to DMATD. |
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62 C Accounts for changes in CJ needed to |
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63 C decide whether to call DMATD. |
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64 C CJLAST -- Previous value of CJ. |
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65 C S -- A scalar determined by the approximate rate |
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66 C of convergence of the Newton iteration and used |
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67 C in the convergence test for the Newton iteration. |
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68 C |
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69 C If RATE is defined to be an estimate of the |
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70 C rate of convergence of the Newton iteration, |
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71 C then S = RATE/(1.D0-RATE). |
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72 C |
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73 C The closer RATE is to 0., the faster the Newton |
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74 C iteration is converging; the closer RATE is to 1., |
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75 C the slower the Newton iteration is converging. |
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76 C |
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77 C On the first Newton iteration with an up-dated |
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78 C preconditioner S = 100.D0, Thus the initial |
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79 C RATE of convergence is approximately 1. |
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80 C |
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81 C S is preserved from call to call so that the rate |
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82 C estimate from a previous step can be applied to |
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83 C the current step. |
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84 C UROUND -- Unit roundoff. |
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85 C DUME -- Dummy argument. |
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86 C DUMS -- Dummy argument. |
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87 C DUMR -- Dummy argument. |
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88 C EPCON -- Tolerance to test for convergence of the Newton |
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89 C iteration. |
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90 C JCALC -- Flag used to determine when to update |
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91 C the Jacobian matrix. In general: |
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92 C |
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93 C JCALC = -1 ==> Call the DMATD routine to update |
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94 C the Jacobian matrix. |
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95 C JCALC = 0 ==> Jacobian matrix is up-to-date. |
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96 C JCALC = 1 ==> Jacobian matrix is out-dated, |
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97 C but DMATD will not be called unless |
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98 C JCALC is set to -1. |
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99 C JFDUM -- Dummy argument. |
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100 C KP1 -- The current order(K) + 1; updated across calls. |
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101 C NONNEG -- Flag to determine nonnegativity constraints. |
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102 C NTYPE -- Identification code for the NLS routine. |
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103 C 0 ==> modified Newton; direct solver. |
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104 C IERNLS -- Error flag for nonlinear solver. |
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105 C 0 ==> nonlinear solver converged. |
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106 C 1 ==> recoverable error inside nonlinear solver. |
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107 C -1 ==> unrecoverable error inside nonlinear solver. |
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108 C |
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109 C All variables with "DUM" in their names are dummy variables |
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110 C which are not used in this routine. |
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111 C |
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112 C Following is a list and description of local variables which |
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113 C may not have an obvious usage. They are listed in roughly the |
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114 C order they occur in this subroutine. |
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115 C |
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116 C The following group of variables are passed as arguments to |
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117 C the Newton iteration solver. They are explained in greater detail |
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118 C in DNSD: |
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119 C TOLNEW, MULDEL, MAXIT, IERNEW |
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120 C |
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121 C IERTYP -- Flag which tells whether this subroutine is correct. |
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122 C 0 ==> correct subroutine. |
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123 C 1 ==> incorrect subroutine. |
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124 C |
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125 C----------------------------------------------------------------------- |
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126 C***ROUTINES CALLED |
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127 C DDWNRM, RES, DMATD, DNSD |
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128 C |
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129 C***END PROLOGUE DNEDD |
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130 C |
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131 C |
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132 IMPLICIT DOUBLE PRECISION(A-H,O-Z) |
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133 DIMENSION Y(*),YPRIME(*),WT(*) |
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134 DIMENSION DELTA(*),E(*) |
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135 DIMENSION WM(*),IWM(*), RPAR(*),IPAR(*) |
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136 DIMENSION PHI(NEQ,*),GAMMA(*) |
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137 EXTERNAL RES, JACD |
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138 C |
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139 PARAMETER (LNRE=12, LNJE=13) |
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140 C |
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141 SAVE MULDEL, MAXIT, XRATE |
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142 DATA MULDEL/1/, MAXIT/4/, XRATE/0.25D0/ |
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143 C |
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144 C Verify that this is the correct subroutine. |
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145 C |
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146 IERTYP = 0 |
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147 IF (NTYPE .NE. 0) THEN |
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148 IERTYP = 1 |
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149 GO TO 380 |
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150 ENDIF |
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151 C |
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152 C If this is the first step, perform initializations. |
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153 C |
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154 IF (JSTART .EQ. 0) THEN |
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155 CJOLD = CJ |
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156 JCALC = -1 |
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157 ENDIF |
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158 C |
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159 C Perform all other initializations. |
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160 C |
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161 IERNLS = 0 |
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162 C |
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163 C Decide whether new Jacobian is needed. |
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164 C |
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165 TEMP1 = (1.0D0 - XRATE)/(1.0D0 + XRATE) |
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166 TEMP2 = 1.0D0/TEMP1 |
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167 IF (CJ/CJOLD .LT. TEMP1 .OR. CJ/CJOLD .GT. TEMP2) JCALC = -1 |
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168 IF (CJ .NE. CJLAST) S = 100.D0 |
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169 C |
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170 C----------------------------------------------------------------------- |
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171 C Entry point for updating the Jacobian with current |
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172 C stepsize. |
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173 C----------------------------------------------------------------------- |
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174 300 CONTINUE |
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175 C |
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176 C Initialize all error flags to zero. |
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177 C |
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178 IERJ = 0 |
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179 IRES = 0 |
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180 IERNEW = 0 |
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181 C |
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182 C Predict the solution and derivative and compute the tolerance |
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183 C for the Newton iteration. |
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184 C |
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185 DO 310 I=1,NEQ |
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186 Y(I)=PHI(I,1) |
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187 310 YPRIME(I)=0.0D0 |
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188 DO 330 J=2,KP1 |
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189 DO 320 I=1,NEQ |
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190 Y(I)=Y(I)+PHI(I,J) |
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191 320 YPRIME(I)=YPRIME(I)+GAMMA(J)*PHI(I,J) |
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192 330 CONTINUE |
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193 PNORM = DDWNRM (NEQ,Y,WT,RPAR,IPAR) |
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194 TOLNEW = 100.D0*UROUND*PNORM |
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195 C |
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196 C Call RES to initialize DELTA. |
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197 C |
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198 IWM(LNRE)=IWM(LNRE)+1 |
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199 CALL RES(X,Y,YPRIME,CJ,DELTA,IRES,RPAR,IPAR) |
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200 IF (IRES .LT. 0) GO TO 380 |
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201 C |
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202 C If indicated, reevaluate the iteration matrix |
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203 C J = dG/dY + CJ*dG/dYPRIME (where G(X,Y,YPRIME)=0). |
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204 C Set JCALC to 0 as an indicator that this has been done. |
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205 C |
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206 IF(JCALC .EQ. -1) THEN |
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207 IWM(LNJE)=IWM(LNJE)+1 |
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208 JCALC=0 |
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209 CALL DMATD(NEQ,X,Y,YPRIME,DELTA,CJ,H,IERJ,WT,E,WM,IWM, |
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210 * RES,IRES,UROUND,JACD,RPAR,IPAR) |
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211 CJOLD=CJ |
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212 S = 100.D0 |
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213 IF (IRES .LT. 0) GO TO 380 |
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214 IF(IERJ .NE. 0)GO TO 380 |
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215 ENDIF |
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216 C |
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217 C Call the nonlinear Newton solver. |
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218 C |
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219 TEMP1 = 2.0D0/(1.0D0 + CJ/CJOLD) |
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220 CALL DNSD(X,Y,YPRIME,NEQ,RES,PDUM,WT,RPAR,IPAR,DUMSVR, |
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221 * DELTA,E,WM,IWM,CJ,DUMS,DUMR,DUME,EPCON,S,TEMP1, |
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222 * TOLNEW,MULDEL,MAXIT,IRES,IDUM,IERNEW) |
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223 C |
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224 IF (IERNEW .GT. 0 .AND. JCALC .NE. 0) THEN |
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225 C |
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226 C The Newton iteration had a recoverable failure with an old |
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227 C iteration matrix. Retry the step with a new iteration matrix. |
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228 C |
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229 JCALC = -1 |
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230 GO TO 300 |
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231 ENDIF |
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232 C |
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233 IF (IERNEW .NE. 0) GO TO 380 |
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234 C |
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235 C The Newton iteration has converged. If nonnegativity of |
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236 C solution is required, set the solution nonnegative, if the |
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237 C perturbation to do it is small enough. If the change is too |
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238 C large, then consider the corrector iteration to have failed. |
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239 C |
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240 375 IF(NONNEG .EQ. 0) GO TO 390 |
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241 DO 377 I = 1,NEQ |
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242 377 DELTA(I) = MIN(Y(I),0.0D0) |
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243 DELNRM = DDWNRM(NEQ,DELTA,WT,RPAR,IPAR) |
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244 IF(DELNRM .GT. EPCON) GO TO 380 |
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245 DO 378 I = 1,NEQ |
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246 378 E(I) = E(I) - DELTA(I) |
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247 GO TO 390 |
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248 C |
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249 C |
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250 C Exits from nonlinear solver. |
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251 C No convergence with current iteration |
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252 C matrix, or singular iteration matrix. |
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253 C Compute IERNLS and IDID accordingly. |
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254 C |
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255 380 CONTINUE |
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256 IF (IRES .LE. -2 .OR. IERTYP .NE. 0) THEN |
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257 IERNLS = -1 |
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258 IF (IRES .LE. -2) IDID = -11 |
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259 IF (IERTYP .NE. 0) IDID = -15 |
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260 ELSE |
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261 IERNLS = 1 |
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262 IF (IRES .LT. 0) IDID = -10 |
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263 IF (IERJ .NE. 0) IDID = -8 |
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264 ENDIF |
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265 C |
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266 390 JCALC = 1 |
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267 RETURN |
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268 C |
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269 C------END OF SUBROUTINE DNEDD------------------------------------------ |
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270 END |
