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1 SUBROUTINE ZBIRY(ZR, ZI, ID, KODE, BIR, BII, IERR) |
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2 C***BEGIN PROLOGUE ZBIRY |
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3 C***DATE WRITTEN 830501 (YYMMDD) |
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4 C***REVISION DATE 890801 (YYMMDD) |
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5 C***CATEGORY NO. B5K |
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6 C***KEYWORDS AIRY FUNCTION,BESSEL FUNCTIONS OF ORDER ONE THIRD |
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7 C***AUTHOR AMOS, DONALD E., SANDIA NATIONAL LABORATORIES |
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8 C***PURPOSE TO COMPUTE AIRY FUNCTIONS BI(Z) AND DBI(Z) FOR COMPLEX Z |
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9 C***DESCRIPTION |
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10 C |
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11 C ***A DOUBLE PRECISION ROUTINE*** |
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12 C ON KODE=1, CBIRY COMPUTES THE COMPLEX AIRY FUNCTION BI(Z) OR |
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13 C ITS DERIVATIVE DBI(Z)/DZ ON ID=0 OR ID=1 RESPECTIVELY. ON |
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14 C KODE=2, A SCALING OPTION CEXP(-AXZTA)*BI(Z) OR CEXP(-AXZTA)* |
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15 C DBI(Z)/DZ IS PROVIDED TO REMOVE THE EXPONENTIAL BEHAVIOR IN |
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16 C BOTH THE LEFT AND RIGHT HALF PLANES WHERE |
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17 C ZTA=(2/3)*Z*CSQRT(Z)=CMPLX(XZTA,YZTA) AND AXZTA=ABS(XZTA). |
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18 C DEFINTIONS AND NOTATION ARE FOUND IN THE NBS HANDBOOK OF |
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19 C MATHEMATICAL FUNCTIONS (REF. 1). |
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20 C |
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21 C INPUT ZR,ZI ARE DOUBLE PRECISION |
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22 C ZR,ZI - Z=CMPLX(ZR,ZI) |
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23 C ID - ORDER OF DERIVATIVE, ID=0 OR ID=1 |
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24 C KODE - A PARAMETER TO INDICATE THE SCALING OPTION |
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25 C KODE= 1 RETURNS |
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26 C BI=BI(Z) ON ID=0 OR |
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27 C BI=DBI(Z)/DZ ON ID=1 |
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28 C = 2 RETURNS |
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29 C BI=CEXP(-AXZTA)*BI(Z) ON ID=0 OR |
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30 C BI=CEXP(-AXZTA)*DBI(Z)/DZ ON ID=1 WHERE |
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31 C ZTA=(2/3)*Z*CSQRT(Z)=CMPLX(XZTA,YZTA) |
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32 C AND AXZTA=ABS(XZTA) |
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33 C |
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34 C OUTPUT BIR,BII ARE DOUBLE PRECISION |
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35 C BIR,BII- COMPLEX ANSWER DEPENDING ON THE CHOICES FOR ID AND |
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36 C KODE |
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37 C IERR - ERROR FLAG |
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38 C IERR=0, NORMAL RETURN - COMPUTATION COMPLETED |
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39 C IERR=1, INPUT ERROR - NO COMPUTATION |
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40 C IERR=2, OVERFLOW - NO COMPUTATION, REAL(Z) |
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41 C TOO LARGE ON KODE=1 |
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42 C IERR=3, CABS(Z) LARGE - COMPUTATION COMPLETED |
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43 C LOSSES OF SIGNIFCANCE BY ARGUMENT REDUCTION |
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44 C PRODUCE LESS THAN HALF OF MACHINE ACCURACY |
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45 C IERR=4, CABS(Z) TOO LARGE - NO COMPUTATION |
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46 C COMPLETE LOSS OF ACCURACY BY ARGUMENT |
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47 C REDUCTION |
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48 C IERR=5, ERROR - NO COMPUTATION, |
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49 C ALGORITHM TERMINATION CONDITION NOT MET |
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50 C |
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51 C***LONG DESCRIPTION |
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52 C |
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53 C BI AND DBI ARE COMPUTED FOR CABS(Z).GT.1.0 FROM THE I BESSEL |
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54 C FUNCTIONS BY |
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55 C |
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56 C BI(Z)=C*SQRT(Z)*( I(-1/3,ZTA) + I(1/3,ZTA) ) |
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57 C DBI(Z)=C * Z * ( I(-2/3,ZTA) + I(2/3,ZTA) ) |
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58 C C=1.0/SQRT(3.0) |
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59 C ZTA=(2/3)*Z**(3/2) |
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60 C |
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61 C WITH THE POWER SERIES FOR CABS(Z).LE.1.0. |
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62 C |
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63 C IN MOST COMPLEX VARIABLE COMPUTATION, ONE MUST EVALUATE ELE- |
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64 C MENTARY FUNCTIONS. WHEN THE MAGNITUDE OF Z IS LARGE, LOSSES |
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65 C OF SIGNIFICANCE BY ARGUMENT REDUCTION OCCUR. CONSEQUENTLY, IF |
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66 C THE MAGNITUDE OF ZETA=(2/3)*Z**1.5 EXCEEDS U1=SQRT(0.5/UR), |
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67 C THEN LOSSES EXCEEDING HALF PRECISION ARE LIKELY AND AN ERROR |
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68 C FLAG IERR=3 IS TRIGGERED WHERE UR=DMAX1(D1MACH(4),1.0D-18) IS |
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69 C DOUBLE PRECISION UNIT ROUNDOFF LIMITED TO 18 DIGITS PRECISION. |
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70 C ALSO, IF THE MAGNITUDE OF ZETA IS LARGER THAN U2=0.5/UR, THEN |
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71 C ALL SIGNIFICANCE IS LOST AND IERR=4. IN ORDER TO USE THE INT |
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72 C FUNCTION, ZETA MUST BE FURTHER RESTRICTED NOT TO EXCEED THE |
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73 C LARGEST INTEGER, U3=I1MACH(9). THUS, THE MAGNITUDE OF ZETA |
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74 C MUST BE RESTRICTED BY MIN(U2,U3). ON 32 BIT MACHINES, U1,U2, |
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75 C AND U3 ARE APPROXIMATELY 2.0E+3, 4.2E+6, 2.1E+9 IN SINGLE |
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76 C PRECISION ARITHMETIC AND 1.3E+8, 1.8E+16, 2.1E+9 IN DOUBLE |
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77 C PRECISION ARITHMETIC RESPECTIVELY. THIS MAKES U2 AND U3 LIMIT- |
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78 C ING IN THEIR RESPECTIVE ARITHMETICS. THIS MEANS THAT THE MAG- |
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79 C NITUDE OF Z CANNOT EXCEED 3.1E+4 IN SINGLE AND 2.1E+6 IN |
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80 C DOUBLE PRECISION ARITHMETIC. THIS ALSO MEANS THAT ONE CAN |
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81 C EXPECT TO RETAIN, IN THE WORST CASES ON 32 BIT MACHINES, |
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82 C NO DIGITS IN SINGLE PRECISION AND ONLY 7 DIGITS IN DOUBLE |
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83 C PRECISION ARITHMETIC. SIMILAR CONSIDERATIONS HOLD FOR OTHER |
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84 C MACHINES. |
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85 C |
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86 C THE APPROXIMATE RELATIVE ERROR IN THE MAGNITUDE OF A COMPLEX |
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87 C BESSEL FUNCTION CAN BE EXPRESSED BY P*10**S WHERE P=MAX(UNIT |
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88 C ROUNDOFF,1.0E-18) IS THE NOMINAL PRECISION AND 10**S REPRE- |
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89 C SENTS THE INCREASE IN ERROR DUE TO ARGUMENT REDUCTION IN THE |
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90 C ELEMENTARY FUNCTIONS. HERE, S=MAX(1,ABS(LOG10(CABS(Z))), |
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91 C ABS(LOG10(FNU))) APPROXIMATELY (I.E. S=MAX(1,ABS(EXPONENT OF |
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92 C CABS(Z),ABS(EXPONENT OF FNU)) ). HOWEVER, THE PHASE ANGLE MAY |
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93 C HAVE ONLY ABSOLUTE ACCURACY. THIS IS MOST LIKELY TO OCCUR WHEN |
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94 C ONE COMPONENT (IN ABSOLUTE VALUE) IS LARGER THAN THE OTHER BY |
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95 C SEVERAL ORDERS OF MAGNITUDE. IF ONE COMPONENT IS 10**K LARGER |
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96 C THAN THE OTHER, THEN ONE CAN EXPECT ONLY MAX(ABS(LOG10(P))-K, |
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97 C 0) SIGNIFICANT DIGITS; OR, STATED ANOTHER WAY, WHEN K EXCEEDS |
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98 C THE EXPONENT OF P, NO SIGNIFICANT DIGITS REMAIN IN THE SMALLER |
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99 C COMPONENT. HOWEVER, THE PHASE ANGLE RETAINS ABSOLUTE ACCURACY |
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100 C BECAUSE, IN COMPLEX ARITHMETIC WITH PRECISION P, THE SMALLER |
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101 C COMPONENT WILL NOT (AS A RULE) DECREASE BELOW P TIMES THE |
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102 C MAGNITUDE OF THE LARGER COMPONENT. IN THESE EXTREME CASES, |
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103 C THE PRINCIPAL PHASE ANGLE IS ON THE ORDER OF +P, -P, PI/2-P, |
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104 C OR -PI/2+P. |
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105 C |
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106 C***REFERENCES HANDBOOK OF MATHEMATICAL FUNCTIONS BY M. ABRAMOWITZ |
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107 C AND I. A. STEGUN, NBS AMS SERIES 55, U.S. DEPT. OF |
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108 C COMMERCE, 1955. |
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109 C |
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110 C COMPUTATION OF BESSEL FUNCTIONS OF COMPLEX ARGUMENT |
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111 C AND LARGE ORDER BY D. E. AMOS, SAND83-0643, MAY, 1983 |
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112 C |
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113 C A SUBROUTINE PACKAGE FOR BESSEL FUNCTIONS OF A COMPLEX |
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114 C ARGUMENT AND NONNEGATIVE ORDER BY D. E. AMOS, SAND85- |
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115 C 1018, MAY, 1985 |
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116 C |
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117 C A PORTABLE PACKAGE FOR BESSEL FUNCTIONS OF A COMPLEX |
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118 C ARGUMENT AND NONNEGATIVE ORDER BY D. E. AMOS, TRANS. |
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119 C MATH. SOFTWARE, 1986 |
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120 C |
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121 C***ROUTINES CALLED ZBINU,XZABS,ZDIV,XZSQRT,D1MACH,I1MACH |
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122 C***END PROLOGUE ZBIRY |
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123 C COMPLEX BI,CONE,CSQ,CY,S1,S2,TRM1,TRM2,Z,ZTA,Z3 |
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124 DOUBLE PRECISION AA, AD, AK, ALIM, ATRM, AZ, AZ3, BB, BII, BIR, |
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125 * BK, CC, CK, COEF, CONEI, CONER, CSQI, CSQR, CYI, CYR, C1, C2, |
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126 * DIG, DK, D1, D2, EAA, ELIM, FID, FMR, FNU, FNUL, PI, RL, R1M5, |
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127 * SFAC, STI, STR, S1I, S1R, S2I, S2R, TOL, TRM1I, TRM1R, TRM2I, |
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128 * TRM2R, TTH, ZI, ZR, ZTAI, ZTAR, Z3I, Z3R, D1MACH, XZABS |
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129 INTEGER ID, IERR, K, KODE, K1, K2, NZ, I1MACH |
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130 DIMENSION CYR(2), CYI(2) |
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131 DATA TTH, C1, C2, COEF, PI /6.66666666666666667D-01, |
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132 * 6.14926627446000736D-01,4.48288357353826359D-01, |
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133 * 5.77350269189625765D-01,3.14159265358979324D+00/ |
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134 DATA CONER, CONEI /1.0D0,0.0D0/ |
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135 C***FIRST EXECUTABLE STATEMENT ZBIRY |
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136 IERR = 0 |
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137 NZ=0 |
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138 IF (ID.LT.0 .OR. ID.GT.1) IERR=1 |
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139 IF (KODE.LT.1 .OR. KODE.GT.2) IERR=1 |
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140 IF (IERR.NE.0) RETURN |
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141 AZ = XZABS(ZR,ZI) |
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142 TOL = DMAX1(D1MACH(4),1.0D-18) |
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143 FID = DBLE(FLOAT(ID)) |
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144 IF (AZ.GT.1.0E0) GO TO 70 |
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145 C----------------------------------------------------------------------- |
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146 C POWER SERIES FOR CABS(Z).LE.1. |
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147 C----------------------------------------------------------------------- |
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148 S1R = CONER |
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149 S1I = CONEI |
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150 S2R = CONER |
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151 S2I = CONEI |
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152 IF (AZ.LT.TOL) GO TO 130 |
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153 AA = AZ*AZ |
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154 IF (AA.LT.TOL/AZ) GO TO 40 |
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155 TRM1R = CONER |
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156 TRM1I = CONEI |
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157 TRM2R = CONER |
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158 TRM2I = CONEI |
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159 ATRM = 1.0D0 |
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160 STR = ZR*ZR - ZI*ZI |
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161 STI = ZR*ZI + ZI*ZR |
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162 Z3R = STR*ZR - STI*ZI |
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163 Z3I = STR*ZI + STI*ZR |
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164 AZ3 = AZ*AA |
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165 AK = 2.0D0 + FID |
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166 BK = 3.0D0 - FID - FID |
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167 CK = 4.0D0 - FID |
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168 DK = 3.0D0 + FID + FID |
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169 D1 = AK*DK |
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170 D2 = BK*CK |
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171 AD = DMIN1(D1,D2) |
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172 AK = 24.0D0 + 9.0D0*FID |
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173 BK = 30.0D0 - 9.0D0*FID |
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174 DO 30 K=1,25 |
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175 STR = (TRM1R*Z3R-TRM1I*Z3I)/D1 |
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176 TRM1I = (TRM1R*Z3I+TRM1I*Z3R)/D1 |
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177 TRM1R = STR |
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178 S1R = S1R + TRM1R |
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179 S1I = S1I + TRM1I |
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180 STR = (TRM2R*Z3R-TRM2I*Z3I)/D2 |
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181 TRM2I = (TRM2R*Z3I+TRM2I*Z3R)/D2 |
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182 TRM2R = STR |
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183 S2R = S2R + TRM2R |
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184 S2I = S2I + TRM2I |
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185 ATRM = ATRM*AZ3/AD |
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186 D1 = D1 + AK |
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187 D2 = D2 + BK |
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188 AD = DMIN1(D1,D2) |
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189 IF (ATRM.LT.TOL*AD) GO TO 40 |
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190 AK = AK + 18.0D0 |
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191 BK = BK + 18.0D0 |
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192 30 CONTINUE |
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193 40 CONTINUE |
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194 IF (ID.EQ.1) GO TO 50 |
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195 BIR = C1*S1R + C2*(ZR*S2R-ZI*S2I) |
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196 BII = C1*S1I + C2*(ZR*S2I+ZI*S2R) |
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197 IF (KODE.EQ.1) RETURN |
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198 CALL XZSQRT(ZR, ZI, STR, STI) |
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199 ZTAR = TTH*(ZR*STR-ZI*STI) |
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200 ZTAI = TTH*(ZR*STI+ZI*STR) |
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201 AA = ZTAR |
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202 AA = -DABS(AA) |
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203 EAA = DEXP(AA) |
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204 BIR = BIR*EAA |
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205 BII = BII*EAA |
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206 RETURN |
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207 50 CONTINUE |
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208 BIR = S2R*C2 |
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209 BII = S2I*C2 |
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210 IF (AZ.LE.TOL) GO TO 60 |
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211 CC = C1/(1.0D0+FID) |
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212 STR = S1R*ZR - S1I*ZI |
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213 STI = S1R*ZI + S1I*ZR |
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214 BIR = BIR + CC*(STR*ZR-STI*ZI) |
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215 BII = BII + CC*(STR*ZI+STI*ZR) |
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216 60 CONTINUE |
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217 IF (KODE.EQ.1) RETURN |
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218 CALL XZSQRT(ZR, ZI, STR, STI) |
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219 ZTAR = TTH*(ZR*STR-ZI*STI) |
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220 ZTAI = TTH*(ZR*STI+ZI*STR) |
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221 AA = ZTAR |
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222 AA = -DABS(AA) |
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223 EAA = DEXP(AA) |
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224 BIR = BIR*EAA |
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225 BII = BII*EAA |
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226 RETURN |
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227 C----------------------------------------------------------------------- |
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228 C CASE FOR CABS(Z).GT.1.0 |
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229 C----------------------------------------------------------------------- |
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230 70 CONTINUE |
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231 FNU = (1.0D0+FID)/3.0D0 |
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232 C----------------------------------------------------------------------- |
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233 C SET PARAMETERS RELATED TO MACHINE CONSTANTS. |
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234 C TOL IS THE APPROXIMATE UNIT ROUNDOFF LIMITED TO 1.0E-18. |
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235 C ELIM IS THE APPROXIMATE EXPONENTIAL OVER- AND UNDERFLOW LIMIT. |
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236 C EXP(-ELIM).LT.EXP(-ALIM)=EXP(-ELIM)/TOL AND |
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237 C EXP(ELIM).GT.EXP(ALIM)=EXP(ELIM)*TOL ARE INTERVALS NEAR |
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238 C UNDERFLOW AND OVERFLOW LIMITS WHERE SCALED ARITHMETIC IS DONE. |
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239 C RL IS THE LOWER BOUNDARY OF THE ASYMPTOTIC EXPANSION FOR LARGE Z. |
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240 C DIG = NUMBER OF BASE 10 DIGITS IN TOL = 10**(-DIG). |
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241 C FNUL IS THE LOWER BOUNDARY OF THE ASYMPTOTIC SERIES FOR LARGE FNU. |
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242 C----------------------------------------------------------------------- |
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243 K1 = I1MACH(15) |
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244 K2 = I1MACH(16) |
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245 R1M5 = D1MACH(5) |
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246 K = MIN0(IABS(K1),IABS(K2)) |
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247 ELIM = 2.303D0*(DBLE(FLOAT(K))*R1M5-3.0D0) |
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248 K1 = I1MACH(14) - 1 |
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249 AA = R1M5*DBLE(FLOAT(K1)) |
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250 DIG = DMIN1(AA,18.0D0) |
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251 AA = AA*2.303D0 |
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252 ALIM = ELIM + DMAX1(-AA,-41.45D0) |
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253 RL = 1.2D0*DIG + 3.0D0 |
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254 FNUL = 10.0D0 + 6.0D0*(DIG-3.0D0) |
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255 C----------------------------------------------------------------------- |
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256 C TEST FOR RANGE |
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257 C----------------------------------------------------------------------- |
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258 AA=0.5D0/TOL |
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259 BB=DBLE(FLOAT(I1MACH(9)))*0.5D0 |
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260 AA=DMIN1(AA,BB) |
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261 AA=AA**TTH |
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262 IF (AZ.GT.AA) GO TO 260 |
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263 AA=DSQRT(AA) |
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264 IF (AZ.GT.AA) IERR=3 |
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265 CALL XZSQRT(ZR, ZI, CSQR, CSQI) |
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266 ZTAR = TTH*(ZR*CSQR-ZI*CSQI) |
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267 ZTAI = TTH*(ZR*CSQI+ZI*CSQR) |
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268 C----------------------------------------------------------------------- |
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269 C RE(ZTA).LE.0 WHEN RE(Z).LT.0, ESPECIALLY WHEN IM(Z) IS SMALL |
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270 C----------------------------------------------------------------------- |
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271 SFAC = 1.0D0 |
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272 AK = ZTAI |
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273 IF (ZR.GE.0.0D0) GO TO 80 |
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274 BK = ZTAR |
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275 CK = -DABS(BK) |
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276 ZTAR = CK |
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277 ZTAI = AK |
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278 80 CONTINUE |
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279 IF (ZI.NE.0.0D0 .OR. ZR.GT.0.0D0) GO TO 90 |
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280 ZTAR = 0.0D0 |
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281 ZTAI = AK |
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282 90 CONTINUE |
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283 AA = ZTAR |
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284 IF (KODE.EQ.2) GO TO 100 |
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285 C----------------------------------------------------------------------- |
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286 C OVERFLOW TEST |
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287 C----------------------------------------------------------------------- |
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288 BB = DABS(AA) |
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289 IF (BB.LT.ALIM) GO TO 100 |
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290 BB = BB + 0.25D0*DLOG(AZ) |
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291 SFAC = TOL |
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292 IF (BB.GT.ELIM) GO TO 190 |
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293 100 CONTINUE |
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294 FMR = 0.0D0 |
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295 IF (AA.GE.0.0D0 .AND. ZR.GT.0.0D0) GO TO 110 |
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296 FMR = PI |
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297 IF (ZI.LT.0.0D0) FMR = -PI |
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298 ZTAR = -ZTAR |
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299 ZTAI = -ZTAI |
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300 110 CONTINUE |
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301 C----------------------------------------------------------------------- |
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302 C AA=FACTOR FOR ANALYTIC CONTINUATION OF I(FNU,ZTA) |
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303 C KODE=2 RETURNS EXP(-ABS(XZTA))*I(FNU,ZTA) FROM CBESI |
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304 C----------------------------------------------------------------------- |
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305 CALL ZBINU(ZTAR, ZTAI, FNU, KODE, 1, CYR, CYI, NZ, RL, FNUL, TOL, |
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306 * ELIM, ALIM) |
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307 IF (NZ.LT.0) GO TO 200 |
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308 AA = FMR*FNU |
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309 Z3R = SFAC |
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310 STR = DCOS(AA) |
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311 STI = DSIN(AA) |
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312 S1R = (STR*CYR(1)-STI*CYI(1))*Z3R |
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313 S1I = (STR*CYI(1)+STI*CYR(1))*Z3R |
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314 FNU = (2.0D0-FID)/3.0D0 |
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315 CALL ZBINU(ZTAR, ZTAI, FNU, KODE, 2, CYR, CYI, NZ, RL, FNUL, TOL, |
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316 * ELIM, ALIM) |
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317 CYR(1) = CYR(1)*Z3R |
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318 CYI(1) = CYI(1)*Z3R |
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319 CYR(2) = CYR(2)*Z3R |
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320 CYI(2) = CYI(2)*Z3R |
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321 C----------------------------------------------------------------------- |
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322 C BACKWARD RECUR ONE STEP FOR ORDERS -1/3 OR -2/3 |
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323 C----------------------------------------------------------------------- |
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324 CALL ZDIV(CYR(1), CYI(1), ZTAR, ZTAI, STR, STI) |
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325 S2R = (FNU+FNU)*STR + CYR(2) |
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326 S2I = (FNU+FNU)*STI + CYI(2) |
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327 AA = FMR*(FNU-1.0D0) |
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328 STR = DCOS(AA) |
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329 STI = DSIN(AA) |
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330 S1R = COEF*(S1R+S2R*STR-S2I*STI) |
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331 S1I = COEF*(S1I+S2R*STI+S2I*STR) |
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332 IF (ID.EQ.1) GO TO 120 |
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333 STR = CSQR*S1R - CSQI*S1I |
|
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334 S1I = CSQR*S1I + CSQI*S1R |
|
|
335 S1R = STR |
|
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336 BIR = S1R/SFAC |
|
|
337 BII = S1I/SFAC |
|
|
338 RETURN |
|
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339 120 CONTINUE |
|
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340 STR = ZR*S1R - ZI*S1I |
|
|
341 S1I = ZR*S1I + ZI*S1R |
|
|
342 S1R = STR |
|
|
343 BIR = S1R/SFAC |
|
|
344 BII = S1I/SFAC |
|
|
345 RETURN |
|
|
346 130 CONTINUE |
|
|
347 AA = C1*(1.0D0-FID) + FID*C2 |
|
|
348 BIR = AA |
|
|
349 BII = 0.0D0 |
|
|
350 RETURN |
|
|
351 190 CONTINUE |
|
|
352 IERR=2 |
|
|
353 NZ=0 |
|
|
354 RETURN |
|
|
355 200 CONTINUE |
|
|
356 IF(NZ.EQ.(-1)) GO TO 190 |
|
|
357 NZ=0 |
|
|
358 IERR=5 |
|
|
359 RETURN |
|
|
360 260 CONTINUE |
|
|
361 IERR=4 |
|
|
362 NZ=0 |
|
|
363 RETURN |
|
|
364 END |
