Computer Evaluation of Mathematical Functions

PREFACE ......Page 6
CONTENTS ......Page 8
1.1. INTRODUCTION......Page 12
1.2.  DESIGN  REQUIREMENTS FOR FUNCTION  EVALUATION  ROUTINES ......Page 13
1.3. ABSOLUTE AND RELATIVE ERROR ......Page 16
1.4. TRUNCATION  ERROR AND ROUNDING ERROR ......Page 19
1.5. INSTABILITY AND THE PROPAGATION  OF ROUNDING ERROR ......Page 23
1.6. INVESTIGATING THE ACCURACY OF COMPUTED RESULTS ......Page 27
2.1.  NEWTON\'S METHOD FOR  SQUARE-ROOT  EVALUATION ......Page 33
2.2.  STARTING APPROXIMATIONS  FOR NEWTON\'S METHOD ......Page 35
2.3. DETERMINING THE NUMBER OF ITERATIONS ......Page 39
2.4. CUBE-ROOT  EVALUATION ......Page 42
3.1.  THE NEED FOR  RANGE REDUCTION ......Page 49
3.2.  RANGE-REDUCTION TECHNIQUES ......Page 51
3.3. REDUCING THE APPROXIMATION RANGE  TO A  VERY SMALL INTERVAL ......Page 54
3.4. PROPAGATION OF ROUNDING ERROR ......Page 56
4.1.  POLYNOMIAL EVALUATION  BY NESTED MULTIPLICATION ......Page 62
4.2.  ECONOMICAL EVALUATION  METHODS ......Page 64
4.3. ECONOMICAL EVALUATION OF FOURTH-DEGREE POLYNOMIALS ......Page 66
4.4. ECONOMICAL EVALUATION OF  FIFTH-DEGREE POLYNOMIALS ......Page 67
4.5. ECONOMICAL EVALUATION OF SIXTH-DEGREE POLYNOMIALS ......Page 68
4.6. ACCURACY OF  COMPUTED RESULTS ......Page 70
5.1.  OPTIMAL POLYNOMIAL APPROXIMATIONS ......Page 75
5.2.  CHEBYSHEV\'S THEOREM ON POLYNOMIAL APPROXIMATIONS ......Page 77
5.3. EXAMPLES OF MINIMAX POLYNOMIAL APPROXIMATIONS ......Page 78
5.4. MINIMAX-RELATIVE-ERROR APPROXIMATIONS  WHEN F(x) HAS A ZERO IN [a,b]......Page 81
5.5. MINIMAX POLYNOMIAL APPROXIMATIONS  TO EVEN OR ODD FUNCTIONS ......Page 83
5.6. STANDARD ERROR CURVES ......Page 84
5.7. A  LINEAR APPROXIMATION  TO  THE SQUARE-ROOT FUNCTION ......Page 86
5.8. NEAR-MINIMAX POLYNOMIAL APPROXIMATIONS ......Page 87
5.9. BOUNDS FOR  MINIMAX ERROR ......Page 89
5.10. MINIMAX APPROXIMATIONS WITH CONSTRAINTS ......Page 90
5.11. REMEZ\' METHOD FOR POLYNOMIAL APPROXIMATIONS ......Page 94
6.1. CHEBYSHEV POLYNOM IALS j FUNDAM ENTAL PROPERTIES ......Page 106
6.2.  ZEROS, CRITICAL POI NTS, DISCRETE ORTHOGONALITY PROPERTIES ......Page 108
6.3. MINIMAX PROPERTY OF CHEBYSHEV POLYNOMIALS ......Page 110
6.4. SHIFTED CHEBYSHEV POLYNOMIALS ......Page 111
6.5. CHEBYSHEV SERIES ......Page 113
6.6. DETERMINING NUMERICAL VALUES  OF CHEBYSHEV SERIES COEFFICIENTS ......Page 116
7.1.  INTRODUCTION ......Page 126
7.2.  TRUNCATION OF POWER SERIES ......Page 129
7.3. METHOD OF ECONOMIZATION ......Page 133
7.4. CHEBYSHEV INTERPOLATION ......Page 136
7.5. TRUNCATION  OF CHEBYSHEV SERIES ......Page 141
8.1.  EVALUATING A  RATIONAL APPROXIMATION  IN FRACTIONAL FORM ......Page 152
8.2. EVALUATING  A  RATIONAL APPROXIMATION  IN CONTINUED FRACTION FORM ......Page 153
8.3. EXAMPLES ON  EVALUATION  OF RATIONAL FUNCTIONS ......Page 158
8.4. ACCURACY OF  COMPUTED RESULTS ......Page 160
9.1.  OPTIMAL RATIONAL APPROXIMATIONS ......Page 165
9.2.  EXAMPLES  OF  MINIMAX RATIONAL APPROXIMATIONS ......Page 168
9.3. COMPARISON OF RATIONAL AND  POLYNOMIAL  APPROXIMATIONS ......Page 172
9.4. MINIMAX RATIONAL APPROXIMATIONS  TO EVEN AND ODD FUNCTIONS ......Page 173
9.5.  STANDARD ERROR CURVES ......Page 174
9.6. NEAR-MINIMAX RATIONAL  APPROXIMATIONS ......Page 175
9.7. MINIMAX RATIONAL APPROXIMATIONS  SUBJECT TO CONSTRAINTS ......Page 177
9.8. REMEZ\' METHOD FOR  RATIONAL APPROXIMATIONS ......Page 181
10.1.  INTRODUCTION ......Page 191
10.2.  PADE APPROXIMATIONS ......Page 192
10.3. MAEHLY\'S METHOD ......Page 196
10.4. TRUNCATING CONTINUED FRACTION EXPANSIONS ......Page 201
10.5. ECONOMIZATION OF CONTINUED FRACTIONS ......Page 207
11.1.  INTRODUCTION ......Page 217
11.2.  ILLUSTRATIVE DERIVATION  OF  AN ASYMPTOTIC EXPANSION ......Page 219
11.3. OBTAINING  APPROXIMATIONS BY TRUNCATING  ASYMPTOTIC EXPANSIONS ......Page 220
11.4. APPROXIMATIONS SIMILAR TO TRUNCATED  ASYMPTOTIC EXPANSIONS ......Page 223
APPENDIX ......Page 229
INDEX ......Page 230