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1 SUBROUTINE ZAIRY(ZR, ZI, ID, KODE, AIR, AII, NZ, IERR) |
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2 C***BEGIN PROLOGUE ZAIRY |
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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 AI(Z) AND DAI(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, ZAIRY COMPUTES THE COMPLEX AIRY FUNCTION AI(Z) OR |
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13 C ITS DERIVATIVE DAI(Z)/DZ ON ID=0 OR ID=1 RESPECTIVELY. ON |
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14 C KODE=2, A SCALING OPTION CEXP(ZTA)*AI(Z) OR CEXP(ZTA)* |
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15 C DAI(Z)/DZ IS PROVIDED TO REMOVE THE EXPONENTIAL DECAY IN |
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16 C -PI/3.LT.ARG(Z).LT.PI/3 AND THE EXPONENTIAL GROWTH IN |
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17 C PI/3.LT.ABS(ARG(Z)).LT.PI WHERE ZTA=(2/3)*Z*CSQRT(Z). |
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18 C |
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19 C WHILE THE AIRY FUNCTIONS AI(Z) AND DAI(Z)/DZ ARE ANALYTIC IN |
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20 C THE WHOLE Z PLANE, THE CORRESPONDING SCALED FUNCTIONS DEFINED |
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21 C FOR KODE=2 HAVE A CUT ALONG THE NEGATIVE REAL AXIS. |
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22 C DEFINTIONS AND NOTATION ARE FOUND IN THE NBS HANDBOOK OF |
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23 C MATHEMATICAL FUNCTIONS (REF. 1). |
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24 C |
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25 C INPUT ZR,ZI ARE DOUBLE PRECISION |
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26 C ZR,ZI - Z=CMPLX(ZR,ZI) |
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27 C ID - ORDER OF DERIVATIVE, ID=0 OR ID=1 |
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28 C KODE - A PARAMETER TO INDICATE THE SCALING OPTION |
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29 C KODE= 1 RETURNS |
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30 C AI=AI(Z) ON ID=0 OR |
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31 C AI=DAI(Z)/DZ ON ID=1 |
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32 C = 2 RETURNS |
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33 C AI=CEXP(ZTA)*AI(Z) ON ID=0 OR |
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34 C AI=CEXP(ZTA)*DAI(Z)/DZ ON ID=1 WHERE |
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35 C ZTA=(2/3)*Z*CSQRT(Z) |
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36 C |
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37 C OUTPUT AIR,AII ARE DOUBLE PRECISION |
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38 C AIR,AII- COMPLEX ANSWER DEPENDING ON THE CHOICES FOR ID AND |
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39 C KODE |
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40 C NZ - UNDERFLOW INDICATOR |
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41 C NZ= 0 , NORMAL RETURN |
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42 C NZ= 1 , AI=CMPLX(0.0D0,0.0D0) DUE TO UNDERFLOW IN |
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43 C -PI/3.LT.ARG(Z).LT.PI/3 ON KODE=1 |
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44 C IERR - ERROR FLAG |
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45 C IERR=0, NORMAL RETURN - COMPUTATION COMPLETED |
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46 C IERR=1, INPUT ERROR - NO COMPUTATION |
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47 C IERR=2, OVERFLOW - NO COMPUTATION, REAL(ZTA) |
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48 C TOO LARGE ON KODE=1 |
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49 C IERR=3, CABS(Z) LARGE - COMPUTATION COMPLETED |
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50 C LOSSES OF SIGNIFCANCE BY ARGUMENT REDUCTION |
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51 C PRODUCE LESS THAN HALF OF MACHINE ACCURACY |
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52 C IERR=4, CABS(Z) TOO LARGE - NO COMPUTATION |
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53 C COMPLETE LOSS OF ACCURACY BY ARGUMENT |
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54 C REDUCTION |
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55 C IERR=5, ERROR - NO COMPUTATION, |
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56 C ALGORITHM TERMINATION CONDITION NOT MET |
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57 C |
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58 C***LONG DESCRIPTION |
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59 C |
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60 C AI AND DAI ARE COMPUTED FOR CABS(Z).GT.1.0 FROM THE K BESSEL |
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61 C FUNCTIONS BY |
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62 C |
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63 C AI(Z)=C*SQRT(Z)*K(1/3,ZTA) , DAI(Z)=-C*Z*K(2/3,ZTA) |
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64 C C=1.0/(PI*SQRT(3.0)) |
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65 C ZTA=(2/3)*Z**(3/2) |
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66 C |
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67 C WITH THE POWER SERIES FOR CABS(Z).LE.1.0. |
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68 C |
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69 C IN MOST COMPLEX VARIABLE COMPUTATION, ONE MUST EVALUATE ELE- |
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70 C MENTARY FUNCTIONS. WHEN THE MAGNITUDE OF Z IS LARGE, LOSSES |
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71 C OF SIGNIFICANCE BY ARGUMENT REDUCTION OCCUR. CONSEQUENTLY, IF |
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72 C THE MAGNITUDE OF ZETA=(2/3)*Z**1.5 EXCEEDS U1=SQRT(0.5/UR), |
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73 C THEN LOSSES EXCEEDING HALF PRECISION ARE LIKELY AND AN ERROR |
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74 C FLAG IERR=3 IS TRIGGERED WHERE UR=DMAX1(D1MACH(4),1.0D-18) IS |
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75 C DOUBLE PRECISION UNIT ROUNDOFF LIMITED TO 18 DIGITS PRECISION. |
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76 C ALSO, IF THE MAGNITUDE OF ZETA IS LARGER THAN U2=0.5/UR, THEN |
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77 C ALL SIGNIFICANCE IS LOST AND IERR=4. IN ORDER TO USE THE INT |
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78 C FUNCTION, ZETA MUST BE FURTHER RESTRICTED NOT TO EXCEED THE |
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79 C LARGEST INTEGER, U3=I1MACH(9). THUS, THE MAGNITUDE OF ZETA |
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80 C MUST BE RESTRICTED BY MIN(U2,U3). ON 32 BIT MACHINES, U1,U2, |
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81 C AND U3 ARE APPROXIMATELY 2.0E+3, 4.2E+6, 2.1E+9 IN SINGLE |
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82 C PRECISION ARITHMETIC AND 1.3E+8, 1.8E+16, 2.1E+9 IN DOUBLE |
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83 C PRECISION ARITHMETIC RESPECTIVELY. THIS MAKES U2 AND U3 LIMIT- |
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84 C ING IN THEIR RESPECTIVE ARITHMETICS. THIS MEANS THAT THE MAG- |
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85 C NITUDE OF Z CANNOT EXCEED 3.1E+4 IN SINGLE AND 2.1E+6 IN |
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86 C DOUBLE PRECISION ARITHMETIC. THIS ALSO MEANS THAT ONE CAN |
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87 C EXPECT TO RETAIN, IN THE WORST CASES ON 32 BIT MACHINES, |
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88 C NO DIGITS IN SINGLE PRECISION AND ONLY 7 DIGITS IN DOUBLE |
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89 C PRECISION ARITHMETIC. SIMILAR CONSIDERATIONS HOLD FOR OTHER |
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90 C MACHINES. |
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91 C |
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92 C THE APPROXIMATE RELATIVE ERROR IN THE MAGNITUDE OF A COMPLEX |
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93 C BESSEL FUNCTION CAN BE EXPRESSED BY P*10**S WHERE P=MAX(UNIT |
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94 C ROUNDOFF,1.0E-18) IS THE NOMINAL PRECISION AND 10**S REPRE- |
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95 C SENTS THE INCREASE IN ERROR DUE TO ARGUMENT REDUCTION IN THE |
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96 C ELEMENTARY FUNCTIONS. HERE, S=MAX(1,ABS(LOG10(CABS(Z))), |
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97 C ABS(LOG10(FNU))) APPROXIMATELY (I.E. S=MAX(1,ABS(EXPONENT OF |
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98 C CABS(Z),ABS(EXPONENT OF FNU)) ). HOWEVER, THE PHASE ANGLE MAY |
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99 C HAVE ONLY ABSOLUTE ACCURACY. THIS IS MOST LIKELY TO OCCUR WHEN |
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100 C ONE COMPONENT (IN ABSOLUTE VALUE) IS LARGER THAN THE OTHER BY |
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101 C SEVERAL ORDERS OF MAGNITUDE. IF ONE COMPONENT IS 10**K LARGER |
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102 C THAN THE OTHER, THEN ONE CAN EXPECT ONLY MAX(ABS(LOG10(P))-K, |
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103 C 0) SIGNIFICANT DIGITS; OR, STATED ANOTHER WAY, WHEN K EXCEEDS |
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104 C THE EXPONENT OF P, NO SIGNIFICANT DIGITS REMAIN IN THE SMALLER |
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105 C COMPONENT. HOWEVER, THE PHASE ANGLE RETAINS ABSOLUTE ACCURACY |
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106 C BECAUSE, IN COMPLEX ARITHMETIC WITH PRECISION P, THE SMALLER |
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107 C COMPONENT WILL NOT (AS A RULE) DECREASE BELOW P TIMES THE |
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108 C MAGNITUDE OF THE LARGER COMPONENT. IN THESE EXTREME CASES, |
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109 C THE PRINCIPAL PHASE ANGLE IS ON THE ORDER OF +P, -P, PI/2-P, |
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110 C OR -PI/2+P. |
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111 C |
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112 C***REFERENCES HANDBOOK OF MATHEMATICAL FUNCTIONS BY M. ABRAMOWITZ |
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113 C AND I. A. STEGUN, NBS AMS SERIES 55, U.S. DEPT. OF |
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114 C COMMERCE, 1955. |
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115 C |
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116 C COMPUTATION OF BESSEL FUNCTIONS OF COMPLEX ARGUMENT |
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117 C AND LARGE ORDER BY D. E. AMOS, SAND83-0643, MAY, 1983 |
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118 C |
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119 C A SUBROUTINE PACKAGE FOR BESSEL FUNCTIONS OF A COMPLEX |
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120 C ARGUMENT AND NONNEGATIVE ORDER BY D. E. AMOS, SAND85- |
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121 C 1018, MAY, 1985 |
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122 C |
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123 C A PORTABLE PACKAGE FOR BESSEL FUNCTIONS OF A COMPLEX |
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124 C ARGUMENT AND NONNEGATIVE ORDER BY D. E. AMOS, TRANS. |
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125 C MATH. SOFTWARE, 1986 |
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126 C |
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127 C***ROUTINES CALLED ZACAI,ZBKNU,XZEXP,XZSQRT,I1MACH,D1MACH |
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128 C***END PROLOGUE ZAIRY |
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129 C COMPLEX AI,CONE,CSQ,CY,S1,S2,TRM1,TRM2,Z,ZTA,Z3 |
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130 DOUBLE PRECISION AA, AD, AII, AIR, AK, ALIM, ATRM, AZ, AZ3, BK, |
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131 * CC, CK, COEF, CONEI, CONER, CSQI, CSQR, CYI, CYR, C1, C2, DIG, |
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132 * DK, D1, D2, ELIM, FID, FNU, PTR, RL, R1M5, SFAC, STI, STR, |
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133 * S1I, S1R, S2I, S2R, TOL, TRM1I, TRM1R, TRM2I, TRM2R, TTH, ZEROI, |
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134 * ZEROR, ZI, ZR, ZTAI, ZTAR, Z3I, Z3R, D1MACH, XZABS, ALAZ, BB |
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135 INTEGER ID, IERR, IFLAG, K, KODE, K1, K2, MR, NN, NZ, I1MACH |
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136 DIMENSION CYR(1), CYI(1) |
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137 DATA TTH, C1, C2, COEF /6.66666666666666667D-01, |
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138 * 3.55028053887817240D-01,2.58819403792806799D-01, |
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139 * 1.83776298473930683D-01/ |
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140 DATA ZEROR, ZEROI, CONER, CONEI /0.0D0,0.0D0,1.0D0,0.0D0/ |
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141 C***FIRST EXECUTABLE STATEMENT ZAIRY |
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142 IERR = 0 |
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143 NZ=0 |
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144 IF (ID.LT.0 .OR. ID.GT.1) IERR=1 |
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145 IF (KODE.LT.1 .OR. KODE.GT.2) IERR=1 |
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146 IF (IERR.NE.0) RETURN |
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147 AZ = XZABS(ZR,ZI) |
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148 TOL = DMAX1(D1MACH(4),1.0D-18) |
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149 FID = DBLE(FLOAT(ID)) |
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150 IF (AZ.GT.1.0D0) GO TO 70 |
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151 C----------------------------------------------------------------------- |
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152 C POWER SERIES FOR CABS(Z).LE.1. |
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153 C----------------------------------------------------------------------- |
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154 S1R = CONER |
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155 S1I = CONEI |
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156 S2R = CONER |
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157 S2I = CONEI |
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158 IF (AZ.LT.TOL) GO TO 170 |
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159 AA = AZ*AZ |
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160 IF (AA.LT.TOL/AZ) GO TO 40 |
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161 TRM1R = CONER |
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162 TRM1I = CONEI |
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163 TRM2R = CONER |
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164 TRM2I = CONEI |
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165 ATRM = 1.0D0 |
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166 STR = ZR*ZR - ZI*ZI |
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167 STI = ZR*ZI + ZI*ZR |
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168 Z3R = STR*ZR - STI*ZI |
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169 Z3I = STR*ZI + STI*ZR |
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170 AZ3 = AZ*AA |
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171 AK = 2.0D0 + FID |
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172 BK = 3.0D0 - FID - FID |
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173 CK = 4.0D0 - FID |
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174 DK = 3.0D0 + FID + FID |
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175 D1 = AK*DK |
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176 D2 = BK*CK |
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177 AD = DMIN1(D1,D2) |
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178 AK = 24.0D0 + 9.0D0*FID |
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179 BK = 30.0D0 - 9.0D0*FID |
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180 DO 30 K=1,25 |
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181 STR = (TRM1R*Z3R-TRM1I*Z3I)/D1 |
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182 TRM1I = (TRM1R*Z3I+TRM1I*Z3R)/D1 |
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183 TRM1R = STR |
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184 S1R = S1R + TRM1R |
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185 S1I = S1I + TRM1I |
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186 STR = (TRM2R*Z3R-TRM2I*Z3I)/D2 |
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187 TRM2I = (TRM2R*Z3I+TRM2I*Z3R)/D2 |
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188 TRM2R = STR |
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189 S2R = S2R + TRM2R |
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190 S2I = S2I + TRM2I |
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191 ATRM = ATRM*AZ3/AD |
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192 D1 = D1 + AK |
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193 D2 = D2 + BK |
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194 AD = DMIN1(D1,D2) |
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195 IF (ATRM.LT.TOL*AD) GO TO 40 |
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196 AK = AK + 18.0D0 |
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197 BK = BK + 18.0D0 |
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198 30 CONTINUE |
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199 40 CONTINUE |
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200 IF (ID.EQ.1) GO TO 50 |
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201 AIR = S1R*C1 - C2*(ZR*S2R-ZI*S2I) |
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202 AII = S1I*C1 - C2*(ZR*S2I+ZI*S2R) |
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203 IF (KODE.EQ.1) RETURN |
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204 CALL XZSQRT(ZR, ZI, STR, STI) |
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205 ZTAR = TTH*(ZR*STR-ZI*STI) |
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206 ZTAI = TTH*(ZR*STI+ZI*STR) |
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207 CALL XZEXP(ZTAR, ZTAI, STR, STI) |
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208 PTR = AIR*STR - AII*STI |
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209 AII = AIR*STI + AII*STR |
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210 AIR = PTR |
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211 RETURN |
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212 50 CONTINUE |
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213 AIR = -S2R*C2 |
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214 AII = -S2I*C2 |
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215 IF (AZ.LE.TOL) GO TO 60 |
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216 STR = ZR*S1R - ZI*S1I |
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217 STI = ZR*S1I + ZI*S1R |
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218 CC = C1/(1.0D0+FID) |
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219 AIR = AIR + CC*(STR*ZR-STI*ZI) |
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220 AII = AII + CC*(STR*ZI+STI*ZR) |
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221 60 CONTINUE |
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222 IF (KODE.EQ.1) RETURN |
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223 CALL XZSQRT(ZR, ZI, STR, STI) |
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224 ZTAR = TTH*(ZR*STR-ZI*STI) |
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225 ZTAI = TTH*(ZR*STI+ZI*STR) |
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226 CALL XZEXP(ZTAR, ZTAI, STR, STI) |
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227 PTR = STR*AIR - STI*AII |
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228 AII = STR*AII + STI*AIR |
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229 AIR = PTR |
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230 RETURN |
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231 C----------------------------------------------------------------------- |
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232 C CASE FOR CABS(Z).GT.1.0 |
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233 C----------------------------------------------------------------------- |
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234 70 CONTINUE |
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235 FNU = (1.0D0+FID)/3.0D0 |
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236 C----------------------------------------------------------------------- |
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237 C SET PARAMETERS RELATED TO MACHINE CONSTANTS. |
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238 C TOL IS THE APPROXIMATE UNIT ROUNDOFF LIMITED TO 1.0D-18. |
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239 C ELIM IS THE APPROXIMATE EXPONENTIAL OVER- AND UNDERFLOW LIMIT. |
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240 C EXP(-ELIM).LT.EXP(-ALIM)=EXP(-ELIM)/TOL AND |
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241 C EXP(ELIM).GT.EXP(ALIM)=EXP(ELIM)*TOL ARE INTERVALS NEAR |
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242 C UNDERFLOW AND OVERFLOW LIMITS WHERE SCALED ARITHMETIC IS DONE. |
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243 C RL IS THE LOWER BOUNDARY OF THE ASYMPTOTIC EXPANSION FOR LARGE Z. |
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244 C DIG = NUMBER OF BASE 10 DIGITS IN TOL = 10**(-DIG). |
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245 C----------------------------------------------------------------------- |
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246 K1 = I1MACH(15) |
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247 K2 = I1MACH(16) |
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248 R1M5 = D1MACH(5) |
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249 K = MIN0(IABS(K1),IABS(K2)) |
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250 ELIM = 2.303D0*(DBLE(FLOAT(K))*R1M5-3.0D0) |
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251 K1 = I1MACH(14) - 1 |
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252 AA = R1M5*DBLE(FLOAT(K1)) |
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253 DIG = DMIN1(AA,18.0D0) |
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254 AA = AA*2.303D0 |
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255 ALIM = ELIM + DMAX1(-AA,-41.45D0) |
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256 RL = 1.2D0*DIG + 3.0D0 |
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257 ALAZ = DLOG(AZ) |
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258 C-------------------------------------------------------------------------- |
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259 C TEST FOR PROPER RANGE |
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260 C----------------------------------------------------------------------- |
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261 AA=0.5D0/TOL |
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262 BB=DBLE(FLOAT(I1MACH(9)))*0.5D0 |
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263 AA=DMIN1(AA,BB) |
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264 AA=AA**TTH |
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265 IF (AZ.GT.AA) GO TO 260 |
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266 AA=DSQRT(AA) |
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267 IF (AZ.GT.AA) IERR=3 |
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268 CALL XZSQRT(ZR, ZI, CSQR, CSQI) |
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269 ZTAR = TTH*(ZR*CSQR-ZI*CSQI) |
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270 ZTAI = TTH*(ZR*CSQI+ZI*CSQR) |
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271 C----------------------------------------------------------------------- |
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272 C RE(ZTA).LE.0 WHEN RE(Z).LT.0, ESPECIALLY WHEN IM(Z) IS SMALL |
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273 C----------------------------------------------------------------------- |
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274 IFLAG = 0 |
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275 SFAC = 1.0D0 |
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276 AK = ZTAI |
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277 IF (ZR.GE.0.0D0) GO TO 80 |
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278 BK = ZTAR |
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279 CK = -DABS(BK) |
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280 ZTAR = CK |
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281 ZTAI = AK |
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282 80 CONTINUE |
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283 IF (ZI.NE.0.0D0) GO TO 90 |
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284 IF (ZR.GT.0.0D0) GO TO 90 |
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285 ZTAR = 0.0D0 |
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286 ZTAI = AK |
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287 90 CONTINUE |
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288 AA = ZTAR |
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289 IF (AA.GE.0.0D0 .AND. ZR.GT.0.0D0) GO TO 110 |
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290 IF (KODE.EQ.2) GO TO 100 |
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291 C----------------------------------------------------------------------- |
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292 C OVERFLOW TEST |
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293 C----------------------------------------------------------------------- |
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294 IF (AA.GT.(-ALIM)) GO TO 100 |
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295 AA = -AA + 0.25D0*ALAZ |
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296 IFLAG = 1 |
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297 SFAC = TOL |
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298 IF (AA.GT.ELIM) GO TO 270 |
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299 100 CONTINUE |
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300 C----------------------------------------------------------------------- |
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301 C CBKNU AND CACON RETURN EXP(ZTA)*K(FNU,ZTA) ON KODE=2 |
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302 C----------------------------------------------------------------------- |
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303 MR = 1 |
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304 IF (ZI.LT.0.0D0) MR = -1 |
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305 CALL ZACAI(ZTAR, ZTAI, FNU, KODE, MR, 1, CYR, CYI, NN, RL, TOL, |
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306 * ELIM, ALIM) |
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307 IF (NN.LT.0) GO TO 280 |
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308 NZ = NZ + NN |
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309 GO TO 130 |
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310 110 CONTINUE |
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311 IF (KODE.EQ.2) GO TO 120 |
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312 C----------------------------------------------------------------------- |
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313 C UNDERFLOW TEST |
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314 C----------------------------------------------------------------------- |
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315 IF (AA.LT.ALIM) GO TO 120 |
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316 AA = -AA - 0.25D0*ALAZ |
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317 IFLAG = 2 |
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318 SFAC = 1.0D0/TOL |
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319 IF (AA.LT.(-ELIM)) GO TO 210 |
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320 120 CONTINUE |
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321 CALL ZBKNU(ZTAR, ZTAI, FNU, KODE, 1, CYR, CYI, NZ, TOL, ELIM, |
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322 * ALIM) |
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323 130 CONTINUE |
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324 S1R = CYR(1)*COEF |
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325 S1I = CYI(1)*COEF |
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326 IF (IFLAG.NE.0) GO TO 150 |
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327 IF (ID.EQ.1) GO TO 140 |
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328 AIR = CSQR*S1R - CSQI*S1I |
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329 AII = CSQR*S1I + CSQI*S1R |
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330 RETURN |
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331 140 CONTINUE |
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332 AIR = -(ZR*S1R-ZI*S1I) |
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333 AII = -(ZR*S1I+ZI*S1R) |
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334 RETURN |
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335 150 CONTINUE |
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336 S1R = S1R*SFAC |
|
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337 S1I = S1I*SFAC |
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338 IF (ID.EQ.1) GO TO 160 |
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339 STR = S1R*CSQR - S1I*CSQI |
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340 S1I = S1R*CSQI + S1I*CSQR |
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341 S1R = STR |
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342 AIR = S1R/SFAC |
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343 AII = S1I/SFAC |
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344 RETURN |
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345 160 CONTINUE |
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346 STR = -(S1R*ZR-S1I*ZI) |
|
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347 S1I = -(S1R*ZI+S1I*ZR) |
|
|
348 S1R = STR |
|
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349 AIR = S1R/SFAC |
|
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350 AII = S1I/SFAC |
|
|
351 RETURN |
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352 170 CONTINUE |
|
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353 AA = 1.0D+3*D1MACH(1) |
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354 S1R = ZEROR |
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355 S1I = ZEROI |
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356 IF (ID.EQ.1) GO TO 190 |
|
|
357 IF (AZ.LE.AA) GO TO 180 |
|
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358 S1R = C2*ZR |
|
|
359 S1I = C2*ZI |
|
|
360 180 CONTINUE |
|
|
361 AIR = C1 - S1R |
|
|
362 AII = -S1I |
|
|
363 RETURN |
|
|
364 190 CONTINUE |
|
|
365 AIR = -C2 |
|
|
366 AII = 0.0D0 |
|
|
367 AA = DSQRT(AA) |
|
|
368 IF (AZ.LE.AA) GO TO 200 |
|
|
369 S1R = 0.5D0*(ZR*ZR-ZI*ZI) |
|
|
370 S1I = ZR*ZI |
|
|
371 200 CONTINUE |
|
|
372 AIR = AIR + C1*S1R |
|
|
373 AII = AII + C1*S1I |
|
|
374 RETURN |
|
|
375 210 CONTINUE |
|
|
376 NZ = 1 |
|
|
377 AIR = ZEROR |
|
|
378 AII = ZEROI |
|
|
379 RETURN |
|
|
380 270 CONTINUE |
|
|
381 NZ = 0 |
|
|
382 IERR=2 |
|
|
383 RETURN |
|
|
384 280 CONTINUE |
|
|
385 IF(NN.EQ.(-1)) GO TO 270 |
|
|
386 NZ=0 |
|
|
387 IERR=5 |
|
|
388 RETURN |
|
|
389 260 CONTINUE |
|
|
390 IERR=4 |
|
|
391 NZ=0 |
|
|
392 RETURN |
|
|
393 END |
