WebSVN - shark - Blame - Rev 1618 - /shark/branches/xen/ports/fftw/doc/fftw

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		3	<HEAD>
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		5	from fftw.texi on 18 May 1999 -->
		6
		7	<TITLE>FFTW - Tutorial</TITLE>
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		9	<BODY TEXT="#000000" BGCOLOR="#FFFFFF">
		10	Go to the <A HREF="fftw_1.html">first</A>, <A HREF="fftw_1.html">previous</A>, <A HREF="fftw_3.html">next</A>, <A HREF="fftw_10.html">last</A> section, <A HREF="fftw_toc.html">table of contents</A>.
		11	<P><HR><P>
		12
		13
		14	<H1><A NAME="SEC2">Tutorial</A></H1>
		15	<P>
		16	<A NAME="IDX14"></A>
		17	This chapter describes the basic usage of FFTW, i.e., how to compute the
		18	Fourier transform of a single array. This chapter tells the truth, but
		19	not the <EM>whole</EM> truth. Specifically, FFTW implements additional
		20	routines and flags, providing extra functionality, that are not
		21	documented here. See Section <A HREF="fftw_3.html#SEC16">FFTW Reference</A>, for more complete information.
		22	(Note that you need to compile and install FFTW before you can use it in
		23	a program. See Section <A HREF="fftw_6.html#SEC66">Installation and Customization</A>, for the details of
		24	the installation.)
		25
		26
		27	<P>
		28	This tutorial chapter is structured as follows. Section <A HREF="fftw_2.html#SEC3">Complex One-dimensional Transforms Tutorial</A> describes the basic usage of the
		29	one-dimensional transform of complex data. Section <A HREF="fftw_2.html#SEC4">Complex Multi-dimensional Transforms Tutorial</A> describes the basic usage of the
		30	multi-dimensional transform of complex data. Section <A HREF="fftw_2.html#SEC5">Real One-dimensional Transforms Tutorial</A> describes the one-dimensional transform of real
		31	data and its inverse. Finally, Section <A HREF="fftw_2.html#SEC6">Real Multi-dimensional Transforms Tutorial</A> describes the multi-dimensional transform of real data and its
		32	inverse. We recommend that you read these sections in the order that
		33	they are presented. We then discuss two topics in detail. In
		34	Section <A HREF="fftw_2.html#SEC7">Multi-dimensional Array Format</A>, we discuss the various
		35	alternatives for storing multi-dimensional arrays in memory. Section <A HREF="fftw_2.html#SEC13">Words of Wisdom</A> shows how you can save FFTW's plans for future use.
		36
		37
		38
		39
		40	<H2><A NAME="SEC3">Complex One-dimensional Transforms Tutorial</A></H2>
		41	<P>
		42	<A NAME="IDX15"></A>
		43	<A NAME="IDX16"></A>
		44
		45
		46	<P>
		47	The basic usage of FFTW is simple. A typical call to FFTW looks like:
		48
		49
		50
		51	<PRE>
		52	#include <fftw.h>
		53	...
		54	{
		55	fftw_complex in[N], out[N];
		56	fftw_plan p;
		57	...
		58	p = fftw_create_plan(N, FFTW_FORWARD, FFTW_ESTIMATE);
		59	...
		60	fftw_one(p, in, out);
		61	...
		62	fftw_destroy_plan(p);
		63	}
		64	</PRE>
		65
		66	<P>
		67	The first thing we do is to create a <EM>plan</EM>, which is an object
		68	<A NAME="IDX17"></A>
		69	that contains all the data that FFTW needs to compute the FFT, using the
		70	following function:
		71
		72
		73
		74	<PRE>
		75	fftw_plan fftw_create_plan(int n, fftw_direction dir, int flags);
		76	</PRE>
		77
		78	<P>
		79	<A NAME="IDX18"></A>
		80	<A NAME="IDX19"></A>
		81	<A NAME="IDX20"></A>
		82
		83
		84	<P>
		85	The first argument, <CODE>n</CODE>, is the size of the transform you are
		86	trying to compute. The size <CODE>n</CODE> can be any positive integer, but
		87	sizes that are products of small factors are transformed most
		88	efficiently. The second argument, <CODE>dir</CODE>, can be either
		89	<CODE>FFTW_FORWARD</CODE> or <CODE>FFTW_BACKWARD</CODE>, and indicates the direction
		90	of the transform you
		91	<A NAME="IDX21"></A>
		92	<A NAME="IDX22"></A>
		93	are interested in. Alternatively, you can use the sign of the exponent
		94	in the transform, -1 or +1, which corresponds to
		95	<CODE>FFTW_FORWARD</CODE> or <CODE>FFTW_BACKWARD</CODE> respectively. The
		96	<CODE>flags</CODE> argument is either <CODE>FFTW_MEASURE</CODE> or
		97	<A NAME="IDX23"></A>
		98	<CODE>FFTW_ESTIMATE</CODE>. <CODE>FFTW_MEASURE</CODE> means that FFTW actually runs
		99	<A NAME="IDX24"></A>
		100	and measures the execution time of several FFTs in order to find the
		101	best way to compute the transform of size <CODE>n</CODE>. This may take some
		102	time, depending on your installation and on the precision of the timer
		103	in your machine. <CODE>FFTW_ESTIMATE</CODE>, on the contrary, does not run
		104	any computation, and just builds a
		105	<A NAME="IDX25"></A>
		106	reasonable plan, which may be sub-optimal. In other words, if your
		107	program performs many transforms of the same size and initialization
		108	time is not important, use <CODE>FFTW_MEASURE</CODE>; otherwise use the
		109	estimate. (A compromise between these two extremes exists. See Section <A HREF="fftw_2.html#SEC13">Words of Wisdom</A>.)
		110
		111
		112	<P>
		113	Once the plan has been created, you can use it as many times as you like
		114	for transforms on arrays of the same size. When you are done with the
		115	plan, you deallocate it by calling <CODE>fftw_destroy_plan(plan)</CODE>.
		116	<A NAME="IDX26"></A>
		117
		118
		119	<P>
		120	The transform itself is computed by passing the plan along with the
		121	input and output arrays to <CODE>fftw_one</CODE>:
		122
		123
		124
		125	<PRE>
		126	void fftw_one(fftw_plan plan, fftw_complex in, fftw_complex out);
		127	</PRE>
		128
		129	<P>
		130	<A NAME="IDX27"></A>
		131
		132
		133	<P>
		134	Note that the transform is out of place: <CODE>in</CODE> and <CODE>out</CODE> must
		135	point to distinct arrays. It operates on data of type
		136	<CODE>fftw_complex</CODE>, a data structure with real (<CODE>in[i].re</CODE>) and
		137	imaginary (<CODE>in[i].im</CODE>) floating-point components. The <CODE>in</CODE>
		138	and <CODE>out</CODE> arrays should have the length specified when the plan was
		139	created. An alternative function, <CODE>fftw</CODE>, allows you to
		140	efficiently perform multiple and/or strided transforms (see Section <A HREF="fftw_3.html#SEC16">FFTW Reference</A>).
		141	<A NAME="IDX28"></A>
		142
		143
		144	<P>
		145	The DFT results are stored in-order in the array <CODE>out</CODE>, with the
		146	zero-frequency (DC) component in <CODE>out[0]</CODE>.
		147	<A NAME="IDX29"></A>
		148	The array <CODE>in</CODE> is not modified. Users should note that FFTW
		149	computes an unnormalized DFT, the sign of whose exponent is given by the
		150	<CODE>dir</CODE> parameter of <CODE>fftw_create_plan</CODE>. Thus, computing a
		151	forward followed by a backward transform (or vice versa) results in the
		152	original array scaled by <CODE>n</CODE>. See Section <A HREF="fftw_3.html#SEC23">What FFTW Really Computes</A>,
		153	for the definition of DFT.
		154	<A NAME="IDX30"></A>
		155
		156
		157	<P>
		158	A program using FFTW should be linked with <CODE>-lfftw -lm</CODE> on Unix
		159	systems, or with the FFTW and standard math libraries in general.
		160	<A NAME="IDX31"></A>
		161
		162
		163
		164
		165	<H2><A NAME="SEC4">Complex Multi-dimensional Transforms Tutorial</A></H2>
		166	<P>
		167	<A NAME="IDX32"></A>
		168	<A NAME="IDX33"></A>
		169
		170
		171	<P>
		172	FFTW can also compute transforms of any number of dimensions
		173	(<EM>rank</EM>). The syntax is similar to that for the one-dimensional
		174	<A NAME="IDX34"></A>
		175	transforms, with <SAMP>`fftw_'</SAMP> replaced by <SAMP>`fftwnd_'</SAMP> (which stands
		176	for "<CODE>fftw</CODE> in <CODE>N</CODE> dimensions").
		177
		178
		179	<P>
		180	As before, we <CODE>#include <fftw.h></CODE> and create a plan for the
		181	transforms, this time of type <CODE>fftwnd_plan</CODE>:
		182
		183
		184
		185	<PRE>
		186	fftwnd_plan fftwnd_create_plan(int rank, const int *n,
		187	fftw_direction dir, int flags);
		188	</PRE>
		189
		190	<P>
		191	<A NAME="IDX35"></A>
		192	<A NAME="IDX36"></A>
		193	<A NAME="IDX37"></A>
		194
		195
		196	<P>
		197	<CODE>rank</CODE> is the dimensionality of the array, and can be any
		198	non-negative integer. The next argument, <CODE>n</CODE>, is a pointer to an
		199	integer array of length <CODE>rank</CODE> containing the (positive) sizes of
		200	each dimension of the array. (Note that the array will be stored in
		201	row-major order. See Section <A HREF="fftw_2.html#SEC7">Multi-dimensional Array Format</A>, for information
		202	on row-major order.) The last two parameters are the same as in
		203	<CODE>fftw_create_plan</CODE>. We now, however, have an additional possible
		204	flag, <CODE>FFTW_IN_PLACE</CODE>, since <CODE>fftwnd</CODE> supports true in-place
		205	<A NAME="IDX38"></A>
		206	<A NAME="IDX39"></A>
		207	<A NAME="IDX40"></A>
		208	transforms. Multiple flags are combined using a bitwise <EM>or</EM>
		209	(<SAMP>`\|'</SAMP>). (An <EM>in-place</EM> transform is one in which the output
		210	data overwrite the input data. It thus requires half as much memory
		211	as--and is often faster than--its opposite, an <EM>out-of-place</EM>
		212	transform.)
		213	<A NAME="IDX41"></A>
		214	<A NAME="IDX42"></A>
		215
		216
		217	<P>
		218	For two- and three-dimensional transforms, FFTWND provides alternative
		219	routines that accept the sizes of each dimension directly, rather than
		220	indirectly through a rank and an array of sizes. These are otherwise
		221	identical to <CODE>fftwnd_create_plan</CODE>, and are sometimes more
		222	convenient:
		223
		224
		225
		226	<PRE>
		227	fftwnd_plan fftw2d_create_plan(int nx, int ny,
		228	fftw_direction dir, int flags);
		229	fftwnd_plan fftw3d_create_plan(int nx, int ny, int nz,
		230	fftw_direction dir, int flags);
		231	</PRE>
		232
		233	<P>
		234	<A NAME="IDX43"></A>
		235	<A NAME="IDX44"></A>
		236
		237
		238	<P>
		239	Once the plan has been created, you can use it any number of times for
		240	transforms of the same size. When you do not need a plan anymore, you
		241	can deallocate the plan by calling <CODE>fftwnd_destroy_plan(plan)</CODE>.
		242	<A NAME="IDX45"></A>
		243
		244
		245	<P>
		246	Given a plan, you can compute the transform of an array of data by
		247	calling:
		248
		249
		250
		251	<PRE>
		252	void fftwnd_one(fftwnd_plan plan, fftw_complex in, fftw_complex out);
		253	</PRE>
		254
		255	<P>
		256	<A NAME="IDX46"></A>
		257
		258
		259	<P>
		260	Here, <CODE>in</CODE> and <CODE>out</CODE> point to multi-dimensional arrays in
		261	row-major order, of the size specified when the plan was created. In
		262	the case of an in-place transform, the <CODE>out</CODE> parameter is ignored
		263	and the output data are stored in the <CODE>in</CODE> array. The results are
		264	stored in-order, unnormalized, with the zero-frequency component in
		265	<CODE>out[0]</CODE>.
		266	<A NAME="IDX47"></A>
		267	A forward followed by a backward transform (or vice-versa) yields the
		268	original data multiplied by the size of the array (i.e. the product of
		269	the dimensions). See Section <A HREF="fftw_3.html#SEC28">What FFTWND Really Computes</A>, for a discussion
		270	of what FFTWND computes.
		271	<A NAME="IDX48"></A>
		272
		273
		274	<P>
		275	For example, code to perform an in-place FFT of a three-dimensional
		276	array might look like:
		277
		278
		279
		280	<PRE>
		281	#include <fftw.h>
		282	...
		283	{
		284	fftw_complex in[L][M][N];
		285	fftwnd_plan p;
		286	...
		287	p = fftw3d_create_plan(L, M, N, FFTW_FORWARD,
		288	FFTW_MEASURE \| FFTW_IN_PLACE);
		289	...
		290	fftwnd_one(p, &in[0][0][0], NULL);
		291	...
		292	fftwnd_destroy_plan(p);
		293	}
		294	</PRE>
		295
		296	<P>
		297	Note that <CODE>in</CODE> is a statically-declared array, which is
		298	automatically in row-major order, but we must take the address of the
		299	first element in order to fit the type expected by <CODE>fftwnd_one</CODE>.
		300	(See Section <A HREF="fftw_2.html#SEC7">Multi-dimensional Array Format</A>.)
		301
		302
		303
		304
		305	<H2><A NAME="SEC5">Real One-dimensional Transforms Tutorial</A></H2>
		306	<P>
		307	<A NAME="IDX49"></A>
		308	<A NAME="IDX50"></A>
		309	<A NAME="IDX51"></A>
		310
		311
		312	<P>
		313	If the input data are purely real, you can save roughly a factor of two
		314	in both time and storage by using the <EM>rfftw</EM> transforms, which are
		315	FFTs specialized for real data. The output of a such a transform is a
		316	<EM>halfcomplex</EM> array, which consists of only half of the complex DFT
		317	amplitudes (since the negative-frequency amplitudes for real data are
		318	the complex conjugate of the positive-frequency amplitudes).
		319	<A NAME="IDX52"></A>
		320
		321
		322	<P>
		323	In exchange for these speed and space advantages, the user sacrifices
		324	some of the simplicity of FFTW's complex transforms. First of all, to
		325	allow maximum performance, the output format of the one-dimensional real
		326	transforms is different from that used by the multi-dimensional
		327	transforms. Second, the inverse transform (halfcomplex to real) has the
		328	side-effect of destroying its input array. Neither of these
		329	inconveniences should pose a serious problem for users, but it is
		330	important to be aware of them. (Both the inconvenient output format
		331	and the side-effect of the inverse transform can be ameliorated for
		332	one-dimensional transforms, at the expense of some performance, by using
		333	instead the multi-dimensional transform routines with a rank of one.)
		334
		335
		336	<P>
		337	The computation of the plan is similar to that for the complex
		338	transforms. First, you <CODE>#include <rfftw.h></CODE>. Then, you create a
		339	plan (of type <CODE>rfftw_plan</CODE>) by calling:
		340
		341
		342
		343	<PRE>
		344	rfftw_plan rfftw_create_plan(int n, fftw_direction dir, int flags);
		345	</PRE>
		346
		347	<P>
		348	<A NAME="IDX53"></A>
		349	<A NAME="IDX54"></A>
		350	<A NAME="IDX55"></A>
		351
		352
		353	<P>
		354	<CODE>n</CODE> is the length of the <EM>real</EM> array in the transform (even
		355	for halfcomplex-to-real transforms), and can be any positive integer
		356	(although sizes with small factors are transformed more efficiently).
		357	<CODE>dir</CODE> is either <CODE>FFTW_REAL_TO_COMPLEX</CODE> or
		358	<CODE>FFTW_COMPLEX_TO_REAL</CODE>.
		359	<A NAME="IDX56"></A>
		360	<A NAME="IDX57"></A>
		361	The <CODE>flags</CODE> parameter is the same as in <CODE>fftw_create_plan</CODE>.
		362
		363
		364	<P>
		365	Once created, a plan can be used for any number of transforms, and is
		366	deallocated when you are done with it by calling
		367	<CODE>rfftw_destroy_plan(plan)</CODE>.
		368	<A NAME="IDX58"></A>
		369
		370
		371	<P>
		372	Given a plan, a real-to-complex or complex-to-real transform is computed
		373	by calling:
		374
		375
		376
		377	<PRE>
		378	void rfftw_one(rfftw_plan plan, fftw_real in, fftw_real out);
		379	</PRE>
		380
		381	<P>
		382	<A NAME="IDX59"></A>
		383
		384
		385	<P>
		386	(Note that <CODE>fftw_real</CODE> is an alias for the floating-point type for
		387	which FFTW was compiled.) Depending upon the direction of the plan,
		388	either the input or the output array is halfcomplex, and is stored in
		389	the following format:
		390	<A NAME="IDX60"></A>
		391
		392
		393	<p align=center>
		394	r<sub>0</sub>, r<sub>1</sub>, r<sub>2</sub>, ..., r<sub>n/2</sub>, i<sub>(n+1)/2-1</sub>, ..., i<sub>2</sub>, i<sub>1</sub>
		395	</p>
		396
		397	<P>
		398	Here,
		399	r<sub>k</sub>
		400	is the real part of the kth output, and
		401	i<sub>k</sub>
		402	is the imaginary part. (We follow here the C convention that integer
		403	division is rounded down, e.g. 7 / 2 = 3.) For a halfcomplex
		404	array <CODE>hc[]</CODE>, the kth component has its real part in
		405	<CODE>hc[k]</CODE> and its imaginary part in <CODE>hc[n-k]</CODE>, with the
		406	exception of <CODE>k</CODE> <CODE>==</CODE> <CODE>0</CODE> or <CODE>n/2</CODE> (the latter only
		407	if n is even)---in these two cases, the imaginary part is zero due to
		408	symmetries of the real-complex transform, and is not stored. Thus, the
		409	transform of <CODE>n</CODE> real values is a halfcomplex array of length
		410	<CODE>n</CODE>, and vice versa. <A NAME="DOCF1" HREF="fftw_foot.html#FOOT1">(1)</A> This is actually only half of the DFT
		411	spectrum of the data. Although the other half can be obtained by
		412	complex conjugation, it is not required by many applications such as
		413	convolution and filtering.
		414
		415
		416	<P>
		417	Like the complex transforms, the RFFTW transforms are unnormalized, so a
		418	forward followed by a backward transform (or vice-versa) yields the
		419	original data scaled by the length of the array, <CODE>n</CODE>.
		420	<A NAME="IDX61"></A>
		421
		422
		423	<P>
		424	Let us reiterate here our warning that an <CODE>FFTW_COMPLEX_TO_REAL</CODE>
		425	transform has the side-effect of destroying its (halfcomplex) input.
		426	The <CODE>FFTW_REAL_TO_COMPLEX</CODE> transform, however, leaves its (real)
		427	input untouched, just as you would hope.
		428
		429
		430	<P>
		431	As an example, here is an outline of how you might use RFFTW to compute
		432	the power spectrum of a real array (i.e. the squares of the absolute
		433	values of the DFT amplitudes):
		434	<A NAME="IDX62"></A>
		435
		436
		437
		438	<PRE>
		439	#include <rfftw.h>
		440	...
		441	{
		442	fftw_real in[N], out[N], power_spectrum[N/2+1];
		443	rfftw_plan p;
		444	int k;
		445	...
		446	p = rfftw_create_plan(N, FFTW_REAL_TO_COMPLEX, FFTW_ESTIMATE);
		447	...
		448	rfftw_one(p, in, out);
		449	power_spectrum[0] = out[0]out[0]; / DC component */
		450	for (k = 1; k < (N+1)/2; ++k) /* (k < N/2 rounded up) */
		451	power_spectrum[k] = out[k]out[k] + out[N-k]out[N-k];
		452	if (N % 2 == 0) /* N is even */
		453	power_spectrum[N/2] = out[N/2]out[N/2]; / Nyquist freq. */
		454	...
		455	rfftw_destroy_plan(p);
		456	}
		457	</PRE>
		458
		459	<P>
		460	Programs using RFFTW should link with <CODE>-lrfftw -lfftw -lm</CODE> on Unix,
		461	or with the FFTW, RFFTW, and math libraries in general.
		462	<A NAME="IDX63"></A>
		463
		464
		465
		466
		467	<H2><A NAME="SEC6">Real Multi-dimensional Transforms Tutorial</A></H2>
		468	<P>
		469	<A NAME="IDX64"></A>
		470
		471
		472	<P>
		473	FFTW includes multi-dimensional transforms for real data of any rank.
		474	As with the one-dimensional real transforms, they save roughly a factor
		475	of two in time and storage over complex transforms of the same size.
		476	Also as in one dimension, these gains come at the expense of some
		477	increase in complexity--the output format is different from the
		478	one-dimensional RFFTW (and is more similar to that of the complex FFTW)
		479	and the inverse (complex to real) transforms have the side-effect of
		480	overwriting their input data.
		481
		482
		483	<P>
		484	To use the real multi-dimensional transforms, you first <CODE>#include
		485	<rfftw.h></CODE> and then create a plan for the size and direction of
		486	transform that you are interested in:
		487
		488
		489
		490	<PRE>
		491	rfftwnd_plan rfftwnd_create_plan(int rank, const int *n,
		492	fftw_direction dir, int flags);
		493	</PRE>
		494
		495	<P>
		496	<A NAME="IDX65"></A>
		497	<A NAME="IDX66"></A>
		498
		499
		500	<P>
		501	The first two parameters describe the size of the real data (not the
		502	halfcomplex data, which will have different dimensions). The last two
		503	parameters are the same as those for <CODE>rfftw_create_plan</CODE>. Just as
		504	for fftwnd, there are two alternate versions of this routine,
		505	<CODE>rfftw2d_create_plan</CODE> and <CODE>rfftw3d_create_plan</CODE>, that are
		506	sometimes more convenient for two- and three-dimensional transforms.
		507	<A NAME="IDX67"></A>
		508	<A NAME="IDX68"></A>
		509	Also as in fftwnd, rfftwnd supports true in-place transforms, specified
		510	by including <CODE>FFTW_IN_PLACE</CODE> in the flags.
		511
		512
		513	<P>
		514	Once created, a plan can be used for any number of transforms, and is
		515	deallocated by calling <CODE>rfftwnd_destroy_plan(plan)</CODE>.
		516
		517
		518	<P>
		519	Given a plan, the transform is computed by calling one of the following
		520	two routines:
		521
		522
		523
		524	<PRE>
		525	void rfftwnd_one_real_to_complex(rfftwnd_plan plan,
		526	fftw_real in, fftw_complex out);
		527	void rfftwnd_one_complex_to_real(rfftwnd_plan plan,
		528	fftw_complex in, fftw_real out);
		529	</PRE>
		530
		531	<P>
		532	<A NAME="IDX69"></A>
		533	<A NAME="IDX70"></A>
		534
		535
		536	<P>
		537	As is clear from their names and parameter types, the former function is
		538	for <CODE>FFTW_REAL_TO_COMPLEX</CODE> transforms and the latter is for
		539	<CODE>FFTW_COMPLEX_TO_REAL</CODE> transforms. (We could have used only a
		540	single routine, since the direction of the transform is encoded in the
		541	plan, but we wanted to correctly express the datatypes of the
		542	parameters.) The latter routine, as we discuss elsewhere, has the
		543	side-effect of overwriting its input (except when the rank of the array
		544	is one). In both cases, the <CODE>out</CODE> parameter is ignored for
		545	in-place transforms.
		546
		547
		548	<P>
		549	The format of the complex arrays deserves careful attention.
		550	<A NAME="IDX71"></A>
		551	Suppose that the real data has dimensions
		552	n<sub>1</sub> x n<sub>2</sub> x ... x n<sub>d</sub>
		553	(in row-major order). Then, after a real-to-complex transform, the
		554	output is an
		555	n<sub>1</sub> x n<sub>2</sub> x ... x (n<sub>d</sub>/2+1)
		556	array of <CODE>fftw_complex</CODE> values in row-major order, corresponding to
		557	slightly over half of the output of the corresponding complex transform.
		558	(Note that the division is rounded down.) The ordering of the data is
		559	otherwise exactly the same as in the complex case. (In principle, the
		560	output could be exactly half the size of the complex transform output,
		561	but in more than one dimension this requires too complicated a format to
		562	be practical.) Note that, unlike the one-dimensional RFFTW, the real
		563	and imaginary parts of the DFT amplitudes are here stored together in
		564	the natural way.
		565
		566
		567	<P>
		568	Since the complex data is slightly larger than the real data, some
		569	complications arise for in-place transforms. In this case, the final
		570	dimension of the real data must be padded with extra values to
		571	accommodate the size of the complex data--two extra if the last
		572	dimension is even and one if it is odd.
		573	<A NAME="IDX72"></A>
		574	That is, the last dimension of the real data must physically contain
		575	2 * (n<sub>d</sub>/2+1)
		576	<CODE>fftw_real</CODE> values (exactly enough to hold the complex data).
		577	This physical array size does not, however, change the <EM>logical</EM>
		578	array size--only
		579	n<sub>d</sub>
		580	values are actually stored in the last dimension, and
		581	n<sub>d</sub>
		582	is the last dimension passed to <CODE>rfftwnd_create_plan</CODE>.
		583
		584
		585	<P>
		586	For example, consider the transform of a two-dimensional real array of
		587	size <CODE>nx</CODE> by <CODE>ny</CODE>. The output of the <CODE>rfftwnd</CODE> transform
		588	is a two-dimensional real array of size <CODE>nx</CODE> by <CODE>ny/2+1</CODE>,
		589	where the <CODE>y</CODE> dimension has been cut nearly in half because of
		590	redundancies in the output. Because <CODE>fftw_complex</CODE> is twice the
		591	size of <CODE>fftw_real</CODE>, the output array is slightly bigger than the
		592	input array. Thus, if we want to compute the transform in place, we
		593	must <EM>pad</EM> the input array so that it is of size <CODE>nx</CODE> by
		594	<CODE>2*(ny/2+1)</CODE>. If <CODE>ny</CODE> is even, then there are two padding
		595	elements at the end of each row (which need not be initialized, as they
		596	are only used for output).
		597	The following illustration depicts the input and output arrays just
		598	described, for both the out-of-place and in-place transforms (with the
		599	arrows indicating consecutive memory locations):
		600
		601	<p align=center><img src="rfftwnd.gif" width=389 height=583>
		602
		603
		604	<P>
		605	Figure 1 depicts the input and output arrays just
		606	described, for both the out-of-place and in-place transforms (with the
		607	arrows indicating consecutive memory locations).
		608
		609
		610	<P>
		611	The RFFTWND transforms are unnormalized, so a forward followed by a
		612	backward transform will result in the original data scaled by the number
		613	of real data elements--that is, the product of the (logical) dimensions
		614	of the real data.
		615	<A NAME="IDX73"></A>
		616
		617
		618	<P>
		619	Below, we illustrate the use of RFFTWND by showing how you might use it
		620	to compute the (cyclic) convolution of two-dimensional real arrays
		621	<CODE>a</CODE> and <CODE>b</CODE> (using the identity that a convolution corresponds
		622	to a pointwise product of the Fourier transforms). For variety,
		623	in-place transforms are used for the forward FFTs and an out-of-place
		624	transform is used for the inverse transform.
		625	<A NAME="IDX74"></A>
		626	<A NAME="IDX75"></A>
		627
		628
		629
		630	<PRE>
		631	#include <rfftw.h>
		632	...
		633	{
		634	fftw_real a[M][2(N/2+1)], b[M][2(N/2+1)], c[M][N];
		635	fftw_complex A, B, C[M][N/2+1];
		636	rfftwnd_plan p, pinv;
		637	fftw_real scale = 1.0 / (M * N);
		638	int i, j;
		639	...
		640	p = rfftw2d_create_plan(M, N, FFTW_REAL_TO_COMPLEX,
		641	FFTW_ESTIMATE \| FFTW_IN_PLACE);
		642	pinv = rfftw2d_create_plan(M, N, FFTW_COMPLEX_TO_REAL,
		643	FFTW_ESTIMATE);
		644
		645	/* aliases for accessing complex transform outputs: */
		646	A = (fftw_complex*) &a[0][0];
		647	B = (fftw_complex*) &b[0][0];
		648	...
		649	for (i = 0; i < M; ++i)
		650	for (j = 0; j < N; ++j) {
		651	a[i][j] = ... ;
		652	b[i][j] = ... ;
		653	}
		654	...
		655	rfftwnd_one_real_to_complex(p, &a[0][0], NULL);
		656	rfftwnd_one_real_to_complex(p, &b[0][0], NULL);
		657
		658	for (i = 0; i < M; ++i)
		659	for (j = 0; j < N/2+1; ++j) {
		660	int ij = i*(N/2+1) + j;
		661	C[i][j].re = (A[ij].re * B[ij].re
		662	- A[ij].im * B[ij].im) * scale;
		663	C[i][j].im = (A[ij].re * B[ij].im
		664	+ A[ij].im * B[ij].re) * scale;
		665	}
		666
		667	/* inverse transform to get c, the convolution of a and b;
		668	this has the side effect of overwriting C */
		669	rfftwnd_one_complex_to_real(pinv, &C[0][0], &c[0][0]);
		670	...
		671	rfftwnd_destroy_plan(p);
		672	rfftwnd_destroy_plan(pinv);
		673	}
		674	</PRE>
		675
		676	<P>
		677	We access the complex outputs of the in-place transforms by casting
		678	each real array to a <CODE>fftw_complex</CODE> pointer. Because this is a
		679	"flat" pointer, we have to compute the row-major index <CODE>ij</CODE>
		680	explicitly in the convolution product loop.
		681	<A NAME="IDX76"></A>
		682	In order to normalize the convolution, we must multiply by a scale
		683	factor--we can do so either before or after the inverse transform, and
		684	choose the former because it obviates the necessity of an additional
		685	loop.
		686	<A NAME="IDX77"></A>
		687	Notice the limits of the loops and the dimensions of the various arrays.
		688
		689
		690	<P>
		691	As with the one-dimensional RFFTW, an out-of-place
		692	<CODE>FFTW_COMPLEX_TO_REAL</CODE> transform has the side-effect of overwriting
		693	its input array. (The real-to-complex transform, on the other hand,
		694	leaves its input array untouched.) If you use RFFTWND for a rank-one
		695	transform, however, this side-effect does not occur. Because of this
		696	fact (and the simpler output format), users may find the RFFTWND
		697	interface more convenient than RFFTW for one-dimensional transforms.
		698	However, RFFTWND in one dimension is slightly slower than RFFTW because
		699	RFFTWND uses an extra buffer array internally.
		700
		701
		702
		703
		704	<H2><A NAME="SEC7">Multi-dimensional Array Format</A></H2>
		705
		706	<P>
		707	This section describes the format in which multi-dimensional arrays are
		708	stored. We felt that a detailed discussion of this topic was necessary,
		709	since it is often a source of confusion among users and several
		710	different formats are common. Although the comments below refer to
		711	<CODE>fftwnd</CODE>, they are also applicable to the <CODE>rfftwnd</CODE> routines.
		712
		713
		714
		715
		716	<H3><A NAME="SEC8">Row-major Format</A></H3>
		717	<P>
		718	<A NAME="IDX78"></A>
		719
		720
		721	<P>
		722	The multi-dimensional arrays passed to <CODE>fftwnd</CODE> are expected to be
		723	stored as a single contiguous block in <EM>row-major</EM> order (sometimes
		724	called "C order"). Basically, this means that as you step through
		725	adjacent memory locations, the first dimension's index varies most
		726	slowly and the last dimension's index varies most quickly.
		727
		728
		729	<P>
		730	To be more explicit, let us consider an array of rank d whose
		731	dimensions are
		732	n<sub>1</sub> x n<sub>2</sub> x n<sub>3</sub> x ... x n<sub>d</sub>.
		733	Now, we specify a location in the array by a sequence of (zero-based) indices,
		734	one for each dimension:
		735	(i<sub>1</sub>, i<sub>2</sub>, i<sub>3</sub>,..., i<sub>d</sub>).
		736	If the array is stored in row-major
		737	order, then this element is located at the position
		738	i<sub>d</sub> + n<sub>d</sub> * (i<sub>d-1</sub> + n<sub>d-1</sub> * (... + n<sub>2</sub> * i<sub>1</sub>)).
		739
		740
		741	<P>
		742	Note that each element of the array must be of type <CODE>fftw_complex</CODE>;
		743	i.e. a (real, imaginary) pair of (double-precision) numbers. Note also
		744	that, in <CODE>fftwnd</CODE>, the expression above is multiplied by the stride
		745	to get the actual array index--this is useful in situations where each
		746	element of the multi-dimensional array is actually a data structure or
		747	another array, and you just want to transform a single field. In most
		748	cases, however, you use a stride of 1.
		749	<A NAME="IDX79"></A>
		750
		751
		752
		753
		754	<H3><A NAME="SEC9">Column-major Format</A></H3>
		755	<P>
		756	<A NAME="IDX80"></A>
		757
		758
		759	<P>
		760	Readers from the Fortran world are used to arrays stored in
		761	<EM>column-major</EM> order (sometimes called "Fortran order"). This is
		762	essentially the exact opposite of row-major order in that, here, the
		763	<EM>first</EM> dimension's index varies most quickly.
		764
		765
		766	<P>
		767	If you have an array stored in column-major order and wish to transform
		768	it using <CODE>fftwnd</CODE>, it is quite easy to do. When creating the plan,
		769	simply pass the dimensions of the array to <CODE>fftwnd_create_plan</CODE> in
		770	<EM>reverse order</EM>. For example, if your array is a rank three
		771	<CODE>N x M x L</CODE> matrix in column-major order, you should pass the
		772	dimensions of the array as if it were an <CODE>L x M x N</CODE> matrix (which
		773	it is, from perspective of <CODE>fftwnd</CODE>). This is done for you
		774	automatically by the FFTW Fortran wrapper routines (see Section <A HREF="fftw_5.html#SEC62">Calling FFTW from Fortran</A>).
		775	<A NAME="IDX81"></A>
		776
		777
		778
		779
		780	<H3><A NAME="SEC10">Static Arrays in C</A></H3>
		781	<P>
		782	<A NAME="IDX82"></A>
		783
		784
		785	<P>
		786	Multi-dimensional arrays declared statically (that is, at compile time,
		787	not necessarily with the <CODE>static</CODE> keyword) in C are <EM>already</EM>
		788	in row-major order. You don't have to do anything special to transform
		789	them. (See Section <A HREF="fftw_2.html#SEC4">Complex Multi-dimensional Transforms Tutorial</A>, for an
		790	example of this sort of code.)
		791
		792
		793
		794
		795	<H3><A NAME="SEC11">Dynamic Arrays in C</A></H3>
		796
		797	<P>
		798	Often, especially for large arrays, it is desirable to allocate the
		799	arrays dynamically, at runtime. This isn't too hard to do, although it
		800	is not as straightforward for multi-dimensional arrays as it is for
		801	one-dimensional arrays.
		802
		803
		804	<P>
		805	Creating the array is simple: using a dynamic-allocation routine like
		806	<CODE>malloc</CODE>, allocate an array big enough to store N <CODE>fftw_complex</CODE>
		807	values, where N is the product of the sizes of the array dimensions
		808	(i.e. the total number of complex values in the array). For example,
		809	here is code to allocate a 5x12x27 rank 3 array:
		810	<A NAME="IDX83"></A>
		811
		812
		813
		814	<PRE>
		815	fftw_complex *an_array;
		816
		817	an_array = (fftw_complex ) malloc(5 12 * 27 * sizeof(fftw_complex));
		818	</PRE>
		819
		820	<P>
		821	Accessing the array elements, however, is more tricky--you can't simply
		822	use multiple applications of the <SAMP>`[]'</SAMP> operator like you could for
		823	static arrays. Instead, you have to explicitly compute the offset into
		824	the array using the formula given earlier for row-major arrays. For
		825	example, to reference the (i,j,k)-th element of the array
		826	allocated above, you would use the expression <CODE>an_array[k + 27 * (j
		827	+ 12 * i)]</CODE>.
		828
		829
		830	<P>
		831	This pain can be alleviated somewhat by defining appropriate macros, or,
		832	in C++, creating a class and overloading the <SAMP>`()'</SAMP> operator.
		833
		834
		835
		836
		837	<H3><A NAME="SEC12">Dynamic Arrays in C--The Wrong Way</A></H3>
		838
		839	<P>
		840	A different method for allocating multi-dimensional arrays in C is often
		841	suggested that is incompatible with <CODE>fftwnd</CODE>: <EM>using it will
		842	cause FFTW to die a painful death</EM>. We discuss the technique here,
		843	however, because it is so commonly known and used. This method is to
		844	create arrays of pointers of arrays of pointers of ...etcetera. For
		845	example, the analogue in this method to the example above is:
		846
		847
		848
		849	<PRE>
		850	int i,j;
		851	fftw_complex **a_bad_array; / another way to make a 5x12x27 array */
		852
		853	a_bad_array = (fftw_complex **) malloc(5 sizeof(fftw_complex **));
		854	for (i = 0; i < 5; ++i) {
		855	a_bad_array[i] =
		856	(fftw_complex *) malloc(12 sizeof(fftw_complex *));
		857	for (j = 0; j < 12; ++j)
		858	a_bad_array[i][j] =
		859	(fftw_complex ) malloc(27 sizeof(fftw_complex));
		860	}
		861	</PRE>
		862
		863	<P>
		864	As you can see, this sort of array is inconvenient to allocate (and
		865	deallocate). On the other hand, it has the advantage that the
		866	(i,j,k)-th element can be referenced simply by
		867	<CODE>a_bad_array[i][j][k]</CODE>.
		868
		869
		870	<P>
		871	If you like this technique and want to maximize convenience in accessing
		872	the array, but still want to pass the array to FFTW, you can use a
		873	hybrid method. Allocate the array as one contiguous block, but also
		874	declare an array of arrays of pointers that point to appropriate places
		875	in the block. That sort of trick is beyond the scope of this
		876	documentation; for more information on multi-dimensional arrays in C,
		877	see the <CODE>comp.lang.c</CODE>
		878	<A HREF="http://www.eskimo.com/~scs/C-faq/s6.html">FAQ</A>.
		879
		880
		881
		882
		883	<H2><A NAME="SEC13">Words of Wisdom</A></H2>
		884	<P>
		885	<A NAME="IDX84"></A>
		886	<A NAME="IDX85"></A>
		887
		888
		889	<P>
		890	FFTW implements a method for saving plans to disk and restoring them.
		891	In fact, what FFTW does is more general than just saving and loading
		892	plans. The mechanism is called <EM><CODE>wisdom</CODE></EM>. Here, we describe
		893	this feature at a high level. See Section <A HREF="fftw_3.html#SEC16">FFTW Reference</A>, for a less casual
		894	(but more complete) discussion of how to use <CODE>wisdom</CODE> in FFTW.
		895
		896
		897	<P>
		898	Plans created with the <CODE>FFTW_MEASURE</CODE> option produce near-optimal
		899	FFT performance, but it can take a long time to compute a plan because
		900	FFTW must actually measure the runtime of many possible plans and select
		901	the best one. This is designed for the situations where so many
		902	transforms of the same size must be computed that the start-up time is
		903	irrelevant. For short initialization times but slightly slower
		904	transforms, we have provided <CODE>FFTW_ESTIMATE</CODE>. The <CODE>wisdom</CODE>
		905	mechanism is a way to get the best of both worlds. There are, however,
		906	certain caveats that the user must be aware of in using <CODE>wisdom</CODE>.
		907	For this reason, <CODE>wisdom</CODE> is an optional feature which is not
		908	enabled by default.
		909
		910
		911	<P>
		912	At its simplest, <CODE>wisdom</CODE> provides a way of saving plans to disk so
		913	that they can be reused in other program runs. You create a plan with
		914	the flags <CODE>FFTW_MEASURE</CODE> and <CODE>FFTW_USE_WISDOM</CODE>, and then save
		915	the <CODE>wisdom</CODE> using <CODE>fftw_export_wisdom</CODE>:
		916	<A NAME="IDX86"></A>
		917
		918
		919
		920	<PRE>
		921	plan = fftw_create_plan(..., ... \| FFTW_MEASURE \| FFTW_USE_WISDOM);
		922	fftw_export_wisdom(...);
		923	</PRE>
		924
		925	<P>
		926	<A NAME="IDX87"></A>
		927
		928
		929	<P>
		930	The next time you run the program, you can restore the <CODE>wisdom</CODE>
		931	with <CODE>fftw_import_wisdom</CODE>, and then recreate the plan using the
		932	same flags as before. This time, however, the same optimal plan will be
		933	created very quickly without measurements. (FFTW still needs some time
		934	to compute trigonometric tables, however.) The basic outline is:
		935
		936
		937
		938	<PRE>
		939	fftw_import_wisdom(...);
		940	plan = fftw_create_plan(..., ... \| FFTW_USE_WISDOM);
		941	</PRE>
		942
		943	<P>
		944	<A NAME="IDX88"></A>
		945
		946
		947	<P>
		948	Wisdom is more than mere rote memorization, however. FFTW's
		949	<CODE>wisdom</CODE> encompasses all of the knowledge and measurements that
		950	were used to create the plan for a given size. Therefore, existing
		951	<CODE>wisdom</CODE> is also applied to the creation of other plans of
		952	different sizes.
		953
		954
		955	<P>
		956	Whenever a plan is created with the <CODE>FFTW_MEASURE</CODE> and
		957	<CODE>FFTW_USE_WISDOM</CODE> flags, <CODE>wisdom</CODE> is generated. Thereafter,
		958	plans for any transform with a similar factorization will be computed
		959	more quickly, so long as they use the <CODE>FFTW_USE_WISDOM</CODE> flag. In
		960	fact, for transforms with the same factors and of equal or lesser size,
		961	no measurements at all need to be made and an optimal plan can be
		962	created with negligible delay!
		963
		964
		965	<P>
		966	For example, suppose that you create a plan for
		967	N = 2<sup>16</sup>.
		968	Then, for any equal or smaller power of two, FFTW can create a
		969	plan (with the same direction and flags) quickly, using the
		970	precomputed <CODE>wisdom</CODE>. Even for larger powers of two, or sizes that
		971	are a power of two times some other prime factors, plans will be
		972	computed more quickly than they would otherwise (although some
		973	measurements still have to be made).
		974
		975
		976	<P>
		977	The <CODE>wisdom</CODE> is cumulative, and is stored in a global, private data
		978	structure managed internally by FFTW. The storage space required is
		979	minimal, proportional to the logarithm of the sizes the <CODE>wisdom</CODE> was
		980	generated from. The <CODE>wisdom</CODE> can be forgotten (and its associated
		981	memory freed) by a call to <CODE>fftw_forget_wisdom()</CODE>; otherwise, it is
		982	remembered until the program terminates. It can also be exported to a
		983	file, a string, or any other medium using <CODE>fftw_export_wisdom</CODE> and
		984	restored during a subsequent execution of the program (or a different
		985	program) using <CODE>fftw_import_wisdom</CODE> (these functions are described
		986	below).
		987
		988
		989	<P>
		990	Because <CODE>wisdom</CODE> is incorporated into FFTW at a very low level, the
		991	same <CODE>wisdom</CODE> can be used for one-dimensional transforms,
		992	multi-dimensional transforms, and even the parallel extensions to FFTW.
		993	Just include <CODE>FFTW_USE_WISDOM</CODE> in the flags for whatever plans you
		994	create (i.e., always plan wisely).
		995
		996
		997	<P>
		998	Plans created with the <CODE>FFTW_ESTIMATE</CODE> plan can use <CODE>wisdom</CODE>,
		999	but cannot generate it; only <CODE>FFTW_MEASURE</CODE> plans actually produce
		1000	<CODE>wisdom</CODE>. Also, plans can only use <CODE>wisdom</CODE> generated from
		1001	plans created with the same direction and flags. For example, a size
		1002	<CODE>42</CODE> <CODE>FFTW_BACKWARD</CODE> transform will not use <CODE>wisdom</CODE>
		1003	produced by a size <CODE>42</CODE> <CODE>FFTW_FORWARD</CODE> transform. The only
		1004	exception to this rule is that <CODE>FFTW_ESTIMATE</CODE> plans can use
		1005	<CODE>wisdom</CODE> from <CODE>FFTW_MEASURE</CODE> plans.
		1006
		1007
		1008
		1009
		1010	<H3><A NAME="SEC14">Caveats in Using Wisdom</A></H3>
		1011	<P>
		1012	<A NAME="IDX89"></A>
		1013
		1014
		1015
		1016	<BLOCKQUOTE>
		1017	<i>
		1018	<P>
		1019	For in much wisdom is much grief, and he that increaseth knowledge
		1020	increaseth sorrow.
		1021	</i>
		1022	[Ecclesiastes 1:18]
		1023	<A NAME="IDX90"></A>
		1024	</BLOCKQUOTE>
		1025
		1026	<P>
		1027	There are pitfalls to using <CODE>wisdom</CODE>, in that it can negate FFTW's
		1028	ability to adapt to changing hardware and other conditions. For example,
		1029	it would be perfectly possible to export <CODE>wisdom</CODE> from a program
		1030	running on one processor and import it into a program running on another
		1031	processor. Doing so, however, would mean that the second program would
		1032	use plans optimized for the first processor, instead of the one it is
		1033	running on.
		1034
		1035
		1036	<P>
		1037	It should be safe to reuse <CODE>wisdom</CODE> as long as the hardware and
		1038	program binaries remain unchanged. (Actually, the optimal plan may
		1039	change even between runs of the same binary on identical hardware, due
		1040	to differences in the virtual memory environment, etcetera. Users
		1041	seriously interested in performance should worry about this problem,
		1042	too.) It is likely that, if the same <CODE>wisdom</CODE> is used for two
		1043	different program binaries, even running on the same machine, the plans
		1044	may be sub-optimal because of differing code alignments. It is
		1045	therefore wise to recreate <CODE>wisdom</CODE> every time an application is
		1046	recompiled. The more the underlying hardware and software changes
		1047	between the creation of <CODE>wisdom</CODE> and its use, the greater grows the
		1048	risk of sub-optimal plans.
		1049
		1050
		1051
		1052
		1053	<H3><A NAME="SEC15">Importing and Exporting Wisdom</A></H3>
		1054	<P>
		1055	<A NAME="IDX91"></A>
		1056
		1057
		1058
		1059	<PRE>
		1060	void fftw_export_wisdom_to_file(FILE *output_file);
		1061	fftw_status fftw_import_wisdom_from_file(FILE *input_file);
		1062	</PRE>
		1063
		1064	<P>
		1065	<A NAME="IDX92"></A>
		1066	<A NAME="IDX93"></A>
		1067
		1068
		1069	<P>
		1070	<CODE>fftw_export_wisdom_to_file</CODE> writes the <CODE>wisdom</CODE> to
		1071	<CODE>output_file</CODE>, which must be a file open for
		1072	writing. <CODE>fftw_import_wisdom_from_file</CODE> reads the <CODE>wisdom</CODE>
		1073	from <CODE>input_file</CODE>, which must be a file open for reading, and
		1074	returns <CODE>FFTW_SUCCESS</CODE> if successful and <CODE>FFTW_FAILURE</CODE>
		1075	otherwise. In both cases, the file is left open and must be closed by
		1076	the caller. It is perfectly fine if other data lie before or after the
		1077	<CODE>wisdom</CODE> in the file, as long as the file is positioned at the
		1078	beginning of the <CODE>wisdom</CODE> data before import.
		1079
		1080
		1081
		1082	<PRE>
		1083	char *fftw_export_wisdom_to_string(void);
		1084	fftw_status fftw_import_wisdom_from_string(const char *input_string)
		1085	</PRE>
		1086
		1087	<P>
		1088	<A NAME="IDX94"></A>
		1089	<A NAME="IDX95"></A>
		1090
		1091
		1092	<P>
		1093	<CODE>fftw_export_wisdom_to_string</CODE> allocates a string, exports the
		1094	<CODE>wisdom</CODE> to it in <CODE>NULL</CODE>-terminated format, and returns a
		1095	pointer to the string. If there is an error in allocating or writing
		1096	the data, it returns <CODE>NULL</CODE>. The caller is responsible for
		1097	deallocating the string (with <CODE>fftw_free</CODE>) when she is done with
		1098	it. <CODE>fftw_import_wisdom_from_string</CODE> imports the <CODE>wisdom</CODE> from
		1099	<CODE>input_string</CODE>, returning <CODE>FFTW_SUCCESS</CODE> if successful and
		1100	<CODE>FFTW_FAILURE</CODE> otherwise.
		1101
		1102
		1103	<P>
		1104	Exporting <CODE>wisdom</CODE> does not affect the store of <CODE>wisdom</CODE>. Imported
		1105	<CODE>wisdom</CODE> supplements the current store rather than replacing it
		1106	(except when there is conflicting <CODE>wisdom</CODE>, in which case the older
		1107	<CODE>wisdom</CODE> is discarded). The format of the exported <CODE>wisdom</CODE> is
		1108	"nerd-readable" LISP-like ASCII text; we will not document it here
		1109	except to note that it is insensitive to white space (interested users
		1110	can contact us for more details).
		1111	<A NAME="IDX96"></A>
		1112	<A NAME="IDX97"></A>
		1113
		1114
		1115	<P>
		1116	See Section <A HREF="fftw_3.html#SEC16">FFTW Reference</A>, for more information, and for a description of
		1117	how you can implement <CODE>wisdom</CODE> import/export for other media
		1118	besides files and strings.
		1119
		1120
		1121	<P>
		1122	The following is a brief example in which the <CODE>wisdom</CODE> is read from
		1123	a file, a plan is created (possibly generating more <CODE>wisdom</CODE>), and
		1124	then the <CODE>wisdom</CODE> is exported to a string and printed to
		1125	<CODE>stdout</CODE>.
		1126
		1127
		1128
		1129	<PRE>
		1130	{
		1131	fftw_plan plan;
		1132	char *wisdom_string;
		1133	FILE *input_file;
		1134
		1135	/* open file to read wisdom from */
		1136	input_file = fopen("sample.wisdom", "r");
		1137	if (FFTW_FAILURE == fftw_import_wisdom_from_file(input_file))
		1138	printf("Error reading wisdom!\n");
		1139	fclose(input_file); /* be sure to close the file! */
		1140
		1141	/* create a plan for N=64, possibly creating and/or using wisdom */
		1142	plan = fftw_create_plan(64,FFTW_FORWARD,
		1143	FFTW_MEASURE \| FFTW_USE_WISDOM);
		1144
		1145	/* ... do some computations with the plan ... */
		1146
		1147	/* always destroy plans when you are done */
		1148	fftw_destroy_plan(plan);
		1149
		1150	/* write the wisdom to a string */
		1151	wisdom_string = fftw_export_wisdom_to_string();
		1152	if (wisdom_string != NULL) {
		1153	printf("Accumulated wisdom: %s\n",wisdom_string);
		1154
		1155	/* Just for fun, destroy and restore the wisdom */
		1156	fftw_forget_wisdom(); /* all gone! */
		1157	fftw_import_wisdom_from_string(wisdom_string);
		1158	/* wisdom is back! */
		1159
		1160	fftw_free(wisdom_string); /* deallocate it since we're done */
		1161	}
		1162	}
		1163	</PRE>
		1164
		1165	<P><HR><P>
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