Index 973 polynomial 728,730f.,748 for multiple precision multiplication 918 rational function 724f,731 number-theoretic transforms 509f relation to interpolation 105 operation count 504 for Romberg integration 140 optimal (Wiener)filtering 547f,565f see also Interpolation order of storage in 507 Extremization see Minimization partial differential equations 833,857ff. Parzen window 554 F-distribution probability function 226,229 periodicity of 503 periodogram 550ff.,574 F-test for differences of variances 617,619 power spectrum estimation 549ff FACR see Fourier analysis and cyclic reduc- tion (FACR) for quadrature 130 of real data in 2D and 3D 525ff. Facsimile standard 909 of real functions 510ff.,525ff. Factorial double (denoted "!!"253 related algorithms 509f. evaluation of 165 Sande-Tukey algorithm 509 sine transform 514ff..859 relation to gamma function 213 routine for 214f. Singleton's algorithm 532 False position 354ff. square window 553 Family tree 345 treatment of end effects in convolution 540 FAS(full approximation storage algorithm) 882任. treatment of end effects in correlation 546 Fast Fourier transform (FFT)504ff.,889 Tukey's trick for frequency doubling 582 alternative algorithms 509f. use in smoothing data 650 applications 537f used for Lomb periodogram 581f. as approximation to continuous transform variance of power spectrum estimate 552, 503 556 Bartlett window 554 virtual memory machine 535f. bit reversal 505f.,532 Welch window 554 and Clenshaw-Curtis quadrature 196 Winograd algorithms 509 convolution 509,530,538ff.,918 see also Discrete Fourier transform (DFT); convolution of large data sets 543f Fourier transform;Spectral density Cooley-Tukey algorithm 509 Faure sequence 310 correlation 545f Fax (facsimile)Group 3 standard 909 cosine transform 196,517f,860f fcomplex (data type) 24,948 cosine transform,second form 519,861 Feasible vector 431 Danielson-Lanczos lemma 504f.532 FFT see Fast Fourier transform (FFT) data sets not a power of 2 509 Field,in data record 338 data smoothing 650 Figure-of-merit function 656 data windowing 553ff. Filon's method 590 decimation-in-frequency algorithm 509 Filter 558ff. decimation-in-time algorithm 509 acausal 559 discrete autocorrelation 546 bilinear transformation method 561 discrete convolution theorem 538ff. causal 559,650 discrete correlation theorem 545 characteristic polynomial 561 at double frequency 582 data smoothing 650 endpoint corrections 585f digital 558ff. extemal storage 532 DISPO 650 figures of merit for data windows 554 by fast Fourier transform(FFT)530, filtering 558ff. 558ff FIR filter 559f finite impulse response (FIR)538,559f. Fourier integrals 584ff. homogeneous modes of 561 Fourier integrals,infinite range 590f. infinite impulse response (IIR)559,573 Hamming window 554 Kalman 705 Hann window 554 linear 559ff. history 504 low-pass for smoothing 650 IIR filter 559 nonrecursive 559f. image processing 812,814 optimal (Wiener)542,547ff.,565f.,650 integrals using 130 quadrature mirror 592.600 inverse of cosine transform 518f. realizable 559,561 inverse of sine transform 517 recursive 559,573 large data sets 532 Remes exchange algorithm 560 leakage 551 Savitzky-Golay 189,650ff memory-local algorithm 535f stability of 561 multidimensional 521ff. in the time domain 558ff. for multiple precision arithmetic 915 Fine-to-coarse operator 873Index 973 polynomial 728, 730f., 748 rational function 724ff., 731 relation to interpolation 105 for Romberg integration 140 see also Interpolation Extremization see Minimization F-distribution probability function 226, 229 F-test for differences of variances 617, 619 FACR see Fourier analysis and cyclic reduc￾tion (FACR) Facsimile standard 909 Factorial double (denoted “!!") 253 evaluation of 165 relation to gamma function 213 routine for 214f. False position 354ff. Family tree 345 FAS (full approximation storage algorithm) 882ff. Fast Fourier transform (FFT) 504ff., 889 alternative algorithms 509f. applications 537ff. as approximation to continuous transform 503 Bartlett window 554 bit reversal 505f., 532 and Clenshaw-Curtis quadrature 196 convolution 509, 530, 538ff., 918 convolution of large data sets 543f. Cooley-Tukey algorithm 509 correlation 545f. cosine transform 196, 517ff., 860f. cosine transform, second form 519, 861 Danielson-Lanczos lemma 504f., 532 data sets not a power of 2 509 data smoothing 650 data windowing 553ff. decimation-in-frequency algorithm 509 decimation-in-time algorithm 509 discrete autocorrelation 546 discrete convolution theorem 538ff. discrete correlation theorem 545 at double frequency 582 endpoint corrections 585f. external storage 532 figures of merit for data windows 554 filtering 558ff. FIR filter 559f. Fourier integrals 584ff. Fourier integrals, infinite range 590f. Hamming window 554 Hann window 554 history 504 IIR filter 559 image processing 812, 814 integrals using 130 inverse of cosine transform 518f. inverse of sine transform 517 large data sets 532 leakage 551 memory-local algorithm 535f. multidimensional 521ff. for multiple precision arithmetic 915 for multiple precision multiplication 918 number-theoretic transforms 509f. operation count 504 optimal (Wiener) filtering 547ff., 565f. order of storage in 507 partial differential equations 833, 857ff. Parzen window 554 periodicity of 503 periodogram 550ff., 574 power spectrum estimation 549ff. for quadrature 130 of real data in 2D and 3D 525ff. of real functions 510ff., 525ff. related algorithms 509f. Sande-Tukey algorithm 509 sine transform 514ff., 859 Singleton’s algorithm 532 square window 553 treatment of end effects in convolution 540 treatment of end effects in correlation 546 Tukey’s trick for frequency doubling 582 use in smoothing data 650 used for Lomb periodogram 581f. variance of power spectrum estimate 552, 556 virtual memory machine 535f. Welch window 554 Winograd algorithms 509 see also Discrete Fourier transform (DFT); Fourier transform; Spectral density Faure sequence 310 Fax (facsimile) Group 3 standard 909 fcomplex (data type) 24, 948 Feasible vector 431 FFT see Fast Fourier transform (FFT) Field, in data record 338 Figure-of-merit function 656 Filon’s method 590 Filter 558ff. acausal 559 bilinear transformation method 561 causal 559, 650 characteristic polynomial 561 data smoothing 650 digital 558ff. DISPO 650 by fast Fourier transform (FFT) 530, 558ff. finite impulse response (FIR) 538, 559f. homogeneous modes of 561 infinite impulse response (IIR) 559, 573 Kalman 705 linear 559ff. low-pass for smoothing 650 nonrecursive 559f. optimal (Wiener) 542, 547ff., 565f., 650 quadrature mirror 592, 600 realizable 559, 561 recursive 559, 573 Remes exchange algorithm 560 Savitzky-Golay 189, 650ff. stability of 561 in the time domain 558ff. Fine-to-coarse operator 873