NAG Library Function Document nag_fft_hermitian (c06ebc)

Transcription

1 c06 Fourier Trasforms NAG Library Fuctio Documet ag_fft_hermitia () 1 Purpose ag_fft_hermitia () calculates the discrete Fourier trasform of a Hermitia sequece of complex data values. (No extra workspace required.) 2 Specificatio #iclude <ag.h> #iclude <agc06.h> void ag_fft_hermitia (double x[], Iteger, NagError *fail) 3 Descriptio Give a Hermitia sequece of complex data values z j (i.e., a sequece such that z 0 is real ad z j is the complex cojugate of z j,forj ¼ 1; 2;...; 1), ag_fft_hermitia () calculates their discrete Fourier trasform defied by ^x k ¼ 1 pffiffiffi X 1 z j exp i 2jk ; k ¼ 0; 1;...; 1: j¼0 (Note the scale factor of p1ffiffi i this defiitio.) The trasformed values ^x k are purely real (see also the c06 Chapter Itroductio). To compute the iverse discrete Fourier trasform defied by p X 1 ^y k ¼ 1 ffiffiffi j¼0 z j exp þi 2jk ; this fuctio should be preceded by a call of ag_cojugate_hermitia (c06gbc) to form the complex cojugates of the z j. ag_fft_hermitia () uses the fast Fourier trasform (FFT) algorithm (see Brigham (1974)). There are some restrictios o the value of (see Sectio 5). 4 Refereces Brigham E O (1974) The Fast Fourier Trasform Pretice Hall 5 Argumets 1: x½š double Iput/Output O etry: the sequece to be trasformed stored i Hermitia form. If the data values z j are writte as x j þ iy j, ad if x is declared with bouds ð0 : 1Þ i the fuctio from which ag_fft_hermitia () is called, the for 0 j =2, x j is cotaied i x½j 1Š, adfor 1 j ð 1Þ=2, y j is cotaied i x½ jš. (See also Sectio i the c06 Chapter Itroductio ad Sectio 10.) O exit: the compoets of the discrete Fourier trasform ^x k. If x is declared with bouds ð0 : 1Þ i the fuctio from which ag_fft_hermitia () is called, the ^x k is stored i x½kš, fork ¼ 0; 1;...; 1..1

2 NAG Library Maual 2: Iteger Iput O etry:, the umber of data values. The largest prime factor of must ot exceed 19, ad the total umber of prime factors of, coutig repetitios, must ot exceed 20. Costrait: > 1. 3: fail NagError* Iput/Output The NAG error argumet (see Sectio 3.6 i the Essetial Itroductio). 6 Error Idicators ad Warigs NE_ALLOC_FAIL Dyamic memory allocatio failed. See Sectio i the Essetial Itroductio for further iformatio. NE_BAD_PARAM O etry, argumet NE_INT O etry, ¼ hvaluei. Costrait: > 1. hvaluei had a illegal value. NE_INTERNAL_ERROR A iteral error has occurred i this fuctio. Check the fuctio call ad ay array sizes. If the call is correct the please cotact NAG for assistace. A uexpected error has bee triggered by this fuctio. Please cotact NAG. See Sectio i the Essetial Itroductio for further iformatio. NE_NO_LICENCE Your licece key may have expired or may ot have bee istalled correctly. See Sectio i the Essetial Itroductio for further iformatio. NE_PRIME_FACTOR At least oe of the prime factors of is greater tha 19. ¼ hvaluei. NE_TOO_MANY_FACTORS has more tha 20 prime factors. ¼ hvaluei. 7 Accuracy Some idicatio of accuracy ca be obtaied by performig a subsequet iverse trasform ad comparig the results with the origial sequece (i exact arithmetic they would be idetical). 8 Parallelism ad Performace Not applicable. 9 Further Commets The time take is approximately proportioal to log ðþ, but also depeds o the factorizatio of. ag_fft_hermitia () is faster if the oly prime factors of are 2, 3 or 5; ad fastest of all if is a power of 2..2

3 c06 Fourier Trasforms O the other had, ag_fft_hermitia () is particularly slow if has several upaired prime factors, i.e., if the square-free part of has several factors. 10 Example This example reads i a sequece of real data values which is assumed to be a Hermitia sequece of complex data values stored i Hermitia form. The iput sequece is expaded ito a full complex sequece ad prited alogside the origial sequece. The discrete Fourier trasform (as computed by ag_fft_hermitia ()) is prited out. It the performs a iverse trasform usig ag_fft_real (c06eac) ad ag_cojugate_hermitia (c06gbc), ad prits the sequece so obtaied alogside the origial data values Program Text /* ag_fft_hermitia () Example Program. * * Copyright 2014 Numerical Algorithms Group. * * Mark 1, * Mark 8 revised, #iclude <ag.h> #iclude <stdio.h> #iclude <ag_stdlib.h> #iclude <agc06.h> it mai(void) Iteger exit_status = 0, j,, 2, j; NagError fail; double *u = 0, *v = 0, *x = 0, *xx = 0; INIT_FAIL(fail); pritf("ag_fft_hermitia () Example Program Results\"); /* Skip headig i data file scaf_s("%*[^\]"); scaf("%*[^\]"); #edif while (scaf_s("%"nag_ifmt"", &)!= EOF) while (scaf("%"nag_ifmt"", &)!= EOF) #edif if ( > 1) if (!(u = NAG_ALLOC(, double))!(v = NAG_ALLOC(, double))!(x = NAG_ALLOC(, double))!(xx = NAG_ALLOC(, double))) pritf("allocatio failure\"); exit_status = -1; else pritf("ivalid.\"); retur exit_status; for (j = 0; j < ; j++).3

4 NAG Library Maual scaf_s("%lf", &x[j]); scaf("%lf", &x[j]); #edif xx[j] = x[j]; /* Calculate full complex form of Hermitia sequece u[0] = x[0]; v[0] = 0.0; 2 = (-1)/2; for (j = 1; j <= 2; j++) j = - j; u[j] = x[j]; u[j] = x[j]; v[j] = x[j]; v[j] = -x[j]; if ( % 2 == 0) u[2+1] = x[2+1]; v[2+1] = 0.0; pritf("\origial ad correspodig complex sequece\"); pritf("\ Data Real Imag \\"); for (j = 0; j < ; j++) pritf("%3"nag_ifmt" %10.5f %10.5f %10.5f\", j, x[j], u[j], v[j]); /* Calculate trasform /* ag_fft_hermitia (). * Sigle oe-dimesioal Hermitia discrete Fourier * trasform ag_fft_hermitia(, x, &fail); pritf("error from ag_fft_hermitia ().\%s\", pritf("\compoets of discrete Fourier trasform\\"); for (j = 0; j < ; j++) pritf("%3"nag_ifmt" %10.5f\", j, x[j]); /* Calculate iverse trasform /* ag_fft_real (c06eac). * Sigle oe-dimesioal real discrete Fourier trasform ag_fft_real(, x, &fail); pritf("error from ag_fft_real (c06eac).\%s\", /* ag_cojugate_hermitia (c06gbc). * Complex cojugate of Hermitia sequece ag_cojugate_hermitia(, x, &fail); pritf("error from ag_cojugate_hermitia (c06gbc).\%s\", pritf("\origial sequece as restored by iverse trasform\"); pritf("\ Origial Restored\\"); for (j = 0; j < ; j++).4

5 c06 Fourier Trasforms pritf("%3"nag_ifmt" %10.5f %10.5f\", j, xx[j], x[j]); END: NAG_FREE(u); NAG_FREE(v); NAG_FREE(x); NAG_FREE(xx); retur exit_status; 10.2 Program Data ag_fft_hermitia () Example Program Data Program Results ag_fft_hermitia () Example Program Results Origial ad correspodig complex sequece Data Real Imag Compoets of discrete Fourier trasform Origial sequece as restored by iverse trasform Origial Restored (last)