SCFFT(3S) SCFFT(3S) NAME SCFFT, DZFFT, CSFFT, ZDFFT - Computes a real-to-complex or complex-to- real Fast Fourier Transform (FFT) SYNOPSIS Single precision -> Single precision complex Fortran: CALL SCFFT (isign, n, scale, x, y, table, work, isys) C/C++: #include <scsl_fft.h> int scfft (int isign, int n, float scale, float *x, scsl_complex *y, float *table, float *work, int *isys); C++ STL: #include <complex.h> #include <scsl_fft.h> int scfft (int isign, int n, float scale, float *x, complex<float> *y, float*table, float *work, int *isys); Double precision -> Double precision complex Fortran: CALL DZFFT (isign, n, scale, x, y, table, work, isys) C/C++: #include <scsl_fft.h> int dzfft (int isign, int n, double scale, float *x, scsl_zomplex *y, double *table, double *work, int *isys); C++ STL: #include <complex.h> #include <scsl_fft.h> int dzfft (int isign, int n, double scale, double *x, complex<double> *y, double *table, double *work, int *isys); Single precision complex -> Single precision Fortran: CALL CSFFT (isign, n, scale, x, y, table, work, isys) C/C++: #include <scsl_fft.h> int csfft (int isign, int n, float scale, scsl_complex *x, float *y, float *table, float *work, int *isys); C++ STL: #include <complex.h> #include <scsl_fft.h> int csfft (int isign, int n, float scale, complex<float> *x, float *y, float *table, float *work, int *isys); Double precision complex -> Double precision Fortran: CALL ZDFFT (isign, n, scale, x, y, table, work, isys) C/C++: #include <scsl_fft.h> int zdfft (int isign, int n, double scale, scsl_zomplex *x, double *y, double *table, double *work, int *isys); C++ STL: #include <complex.h> #include <scsl_fft.h> int zdfft (int isign, int n, double scale, complex<double> *x, double *y, double *table, double *work, int *isys); IMPLEMENTATION These routines are part of the SCSL Scientific Library and can be loaded using either the -lscs or the -lscs_mp option. The -lscs_mp option directs the linker to use the multi-processor version of the library. When linking to SCSL with -lscs or -lscs_mp, the default integer size is 4 bytes (32 bits). Another version of SCSL is available in which integers are 8 bytes (64 bits). This version allows the user access to larger memory sizes and helps when porting legacy Cray codes. It can be loaded by using the -lscs_i8 option or the -lscs_i8_mp option. A program may use only one of the two versions; 4-byte integer and 8-byte integer library calls cannot be mixed. The C and C++ prototypes shown above are appropriate for the 4-byte integer version of SCSL. When using the 8-byte integer version, the variables of type int become long long and the <scsl_fft_i8.h> header file should be included. DESCRIPTION SCFFT/DZFFT computes the FFT of the real array X, and it stores the results in the complex array Y. CSFFT/ZDFFT computes the corresponding inverse complex-to-real transform. It is customary in FFT applications to use zero-based subscripts; the formulas are simpler that way. For these routines, suppose that the arrays are declared as follows: Fortran: REAL X(0:n-1) COMPLEX Y(0:n/2) C/C++: float x[n]; scsl_complex y[n/2+1]; C++ STL: float x[n]; complex<float> y[n/2+1]; Then the output array is the FFT of the input array, using the following formula for the FFT: n-1 Y(k) = scale * Sum [ X(j)*w**(isign*j*k) ] for k = 0, ..., n/2 j=0 where: w = exp(2*pi*i/n), i = + sqrt(-1), pi = 3.14159..., isign = +1 or -1. Different authors use different conventions for which of the transforms, isign = +1 or isign = -1, is the forward or inverse transform, and what the scale factor should be in either case. You can make these routines compute any of the various possible definitions, however, by choosing the appropriate values for isign and scale. The relevant fact from FFT theory is this: If you call SCFFT with any particular values of isign and scale, the mathematical inverse function is computed by calling CSFFT with -isign and 1/(n*scale). In particular, if you use isign = +1 and scale = 1.0 in SCFFT for the forward FFT, you can compute the inverse FFT by using CSFFT with isign = -1 and scale = 1.0/n. See the NOTES section of this man page for information about the interpretation of the data types described in the following arguments. These routines have the following arguments: isign Integer. (input) Specifies whether to initialize the table array or to do the forward or inverse Fourier transform, as follows: If isign = 0, the routine initializes the table array and returns. In this case, the only arguments used or checked are isign, n, and table. If isign = +1 or -1, the value of isign is the sign of the exponent used in the FFT formula. n Integer. (input) Size of transform. If n is not positive, the routine returns without calculating the transform. scale Scale factor. (input) SCFFT: Single precision. DZFFT: Double precision. CSFFT: Single precision. ZDFFT: Double precision. Each element of the output array is multiplied by scale after taking the Fourier transform, as defined in the preceding formula. x Input array of values to be transformed. (input) SCFFT: Single precision array of dimension n. DZFFT: Double precision array of dimension n. CSFFT: Single precision complex array of dimension n/2 + 1. (input) ZDFFT: Double precision complex array of dimension n/2 + 1. (input) y Output array of transformed values. (output) SCFFT: Single precision complex array of dimension n/2 + 1. (input) DZFFT: Double precision complex array of dimension n/2 + 1. (input) CSFFT: Single precision array of dimension n. ZDFFT: Double precision array of dimension n. The output array, y, is the FFT of the the input array, x, computed according to the preceding formula. The output array may be equivalenced to the input array in the calling program. Be careful when dimensioning the arrays, in this case, to allow for the fact that the complex array contains two (real) words more than the real array. table Array of dimension (n + NFR) (input or output) SCFFT, CSFFT: Single precision array. DZFFT, ZDFFT: Double precision array. Table of factors and roots of unity. See the description of the isys argument for the value of NFR. If isign = 0, the routine initializes table (table is output only). If isign = +1 or -1, the values in table are assumed to be initialized already by a prior call with isign = 0 (table is input only). work Array of dimension n + 2 SCFFT, CSFFT: Single precision array. DZFFT, ZDFFT: Double precision array. Work array used for intermediate calculations. Its address space must be different from that of the input and output arrays. isys Integer array dimensioned 0..isys(0). An array that gives implementation-specific information. All features and functions of the FFT routines specific to any particular implementation are confined to this isys array. In the Origin series implementation, isys(0)=0 and isys(0)=1 are supported. In SCSL versions prior to 1.3, only isys(0)=0 was allowed. For isys(0)=0, NFR=15, and for isys(0)=1, NFR=256. The NFR words of storage in the table array contain a factorization of the length of the transform. The smaller value of NFR for isys(0)=0 is historical. It is too small to store all the required factors for the highest performing FFT, so when isys(0)=0, extra space is allocated when the table array is initialized. To avoid memory leaks, this extra space must be deallocated when the table array is no longer needed. The SCFFTF routine is used to release this memory. Due to the potential for memory leaks, the use of isys(0)=0 should be avoided. For isys(0)=1, the value of NFR is large enough so that no extra memory needs to be allocated, and there is no need to call SCFFTF to release memory. If called, it does nothing. NOTE: isys(0)=1 means that isys is an integer array with two elements. The second element, isys(1), will not be accessed. NOTES The following data types are described in this documentation: Term Used Data type Fortran: Array dimensioned 0..n-1 x(0:n-1) Array of dimensions (m,n) x(m,n) Array of dimensions (m,n,p) x(m,n,p) Integer INTEGER (INTEGER*8 for -lscs_i8[_mp]) Single precision REAL Double precision DOUBLE PRECISION Single precision complex COMPLEX Double precision complex DOUBLE COMPLEX C/C++: Array dimensioned 0..n-1 x[n] Array of dimensions (m,n) x[m*n] or x[n][m] Array of dimensions (m,n,p) x[m*n*p] or x[p][n][m] Integer int (long long for -lscs_i8[_mp]) Single precision float Double precision double Single precision complex scsl_complex Double precision complex scsl_zomplex C++ STL: Array dimensioned 0..n-1 x[n] Array of dimensions (m,n) x[m*n] or x[n][m] Array of dimensions (m,n,p) x[m*n*p] or x[p][n][m] Integer int (long long for -lscs_i8[_mp]) Single precision float Double precision double Single precision complex complex<float> Double precision complex complex<double> CAUTIONS Transform sizes with a prime factor exceeding 232-1 are not supported for the 8-byte integer version of the library. In addition to the work array, the FFT routines also dynamically allocate scratch space from the stack. The amount of space allocated can be slightly bigger than the size of the largest processor cache. For single processor runs, the default stack size is large enough that these allocations generally cause no problems. But for parallel runs, you need to ensure that the stack size of slave threads is big enough to hold this scratch space. Failure to reserve sufficient stack space will cause programs to dump core due to stack overflows. The stack size of MP library slave threads is controlled via the MP_SLAVE_STACKSIZE environment variable or the mp_set_slave_stacksize() library routine. See the mp(3C), mp(3F) and pe_environ(5) reference pages for more information on controlling the slave stack size. For pthreads applications, the thread's stack size is specified as one of many creation attributes provided in the pthread_attr_t argument to pthread_create(3P). The stacksize attribute should be set explicitly to a non-default value using the pthread_attr_setstacksize(3P) call, described in the pthread_attr_init(3P) man page. Care must be exercised if copies of the table array are used: even though a copy exists, the original must persist. As an example, the following code will not work: #include <scsl_fft.h> float x[1024]; scsl_complex y[513]; float table[1024+256]; float work[1024+2]; int isys[2]; isys[0] = 1; { float table_orig[1024+256]; scfft(0, 1024, 1.0f, x, y, table_orig, work, isys) bcopy(table_orig, table, (1024+256)*sizeof(float)); } scfft(1, 1024, 1.0f, x, y, table, work, isys) In this example, because table_orig is a stack variable that does not persist outside of the code block delimited by the braces, the data in the copy, table, are not guaranteed to be valid. However, the following code will work because table_orig is persistent: #include <scsl_fft.h> float x[1024]; scsl_complex y[513]; float table_orig[1024+256]; float table[1024+256]; float work[1024+2]; int isys[2]; isys[0] = 1; scfft(0, 1024, 1.0f, x, y, table_orig, work, isys) bcopy(table_orig, table, (1024+256)*sizeof(float)); scfft(1, 1024, 1.0f, x, y, table, work, isys) EXAMPLES These examples use the table and workspace sizes appropriate to Origin series. Example 1: Initialize the TABLE array in preparation for doing an FFT of size 1024. In this case only the arguments isign, n, and table are used. You can use dummy arguments or zeros for the other arguments in the subroutine call. Fortran: REAL TABLE(1024+256) INTEGER ISYS(0:1) ISYS(0) = 1 CALL SCFFT(0, 1024, 0.0, DUMMY, DUMMY, TABLE, DUMMY, ISYS) C/C++: #include <scsl_fft.h> float table[1024+256]; int isys[2]; isys[0] = 1; scfft(0, 1024, 0.0f, NULL, NULL, table, NULL, isys); C++ STL: #include <complex.h> #include <scsl_fft.h> float table[1024+256]; int isys[2]; isys[0] = 1; scfft(0, 1024, 0.0f, NULL, NULL, table, NULL, isys); Example 2: X is a real array dimensioned (0...1023), and Y is a complex array dimensioned (0...:512). Take the FFT of X and store the results in Y. Before taking the FFT, initialize the TABLE array, as in example 1. Fortran: REAL X(0:1023) COMPLEX Y(0:512) REAL TABLE(1024+256) REAL WORK(1024+2) INTEGER ISYS(0:1) ISYS(0) = 1 CALL SCFFT(0, 1024, 1.0, X, Y, TABLE, WORK, ISYS) CALL SCFFT(1, 1024, 1.0, X, Y, TABLE, WORK, ISYS) C/C++: #include <scsl_fft.h> float x[1024]; scsl_complex y[513]; float table[1024+256]; float work[1024+2]; int isys[2]; isys[0] = 1; scfft(0, 1024, 1.0f, x, y, table, work, isys) scfft(1, 1024, 1.0f, x, y, table, work, isys) C++ STL: #include <complex.h> #include <scsl_fft.h> float x[1024]; complex<float> y[513]; float table[1024+256]; float work[1024+2]; int isys[2]; isys[0] = 1; scfft(0, 1024, 1.0f, x, y, table, work, isys) scfft(1, 1024, 1.0f, x, y, table, work, isys) Example 3: With X and Y as in example 2, take the inverse FFT of Y and store it back in X. The scale factor 1/1024 is used. Assume that the TABLE array is initialized already. Fortran: CALL CSFFT(-1, 1024, 1.0/1024.0, Y, X, TABLE, WORK, ISYS) C/C++ and C++ STL: csfft(-1, 1024, 1.0f/1024.0f, y, x, table, work, isys) Example 4: Perform the same computation as in example 2, but equivalence the input and output arrays to save storage space. Use the 8-byte integer version of SCSL. Fortran: REAL X(0:1023) COMPLEX Y(0:512) EQUIVALENCE ( X(1), Y(1) ) REAL TABLE(1024+256) REAL WORK(1024+2) INTEGER*8 ISYS(0:1) ISYS(0) = 1_8 CALL SCFFT(0_8, 1024_8, 1.0, X, Y, TABLE, WORK, ISYS) CALL SCFFT(1_8, 1024_8, 1.0, X, Y, TABLE, WORK, ISYS) C/C++: #include <scsl_fft_i8.h> float *x; scsl_complex y[513]; float table[1024+256]; float work[1024+2]; long long isys[2]; isys[0] = 1LL; x = (float *) &y[0]; scfft(0LL, 1024LL, 1.0f, x, y, table, work, isys) scfft(1LL, 1024LL, 1.0f, x, y, table, work, isys) C++ STL: #include <complex.h> #include <scsl_fft_i8.h> float *x; complex<float> y[513]; float table[1024+256]; float work[1024+2]; long long isys[2]; isys[0] = 1LL; x = (float *) &y[0]; scfft(0LL, 1024LL, 1.0f, x, y, table, work, isys) scfft(1LL, 1024LL, 1.0f, x, y, table, work, isys) Example 5: Perform the same computation as in example 2, but assume that the lower bound of each Fortran array is 1, rather than 0. The subroutine calls are not changed. Fortran: REAL X(1024) COMPLEX Y(513) CALL SCFFT(0, 1024, 1.0, X, Y, TABLE, WORK, ISYS) CALL SCFFT(1, 1024, 1.0, X, Y, TABLE, WORK, ISYS) SEE ALSO INTRO_FFT(3S), INTRO_SCSL(3S), CCFFT(3S), CCFFTM(3S), SCFFTM(3S) Page 11