IRIX Base Documentation 2002 November/SGI IRIX Base Documentation 2002 November.iso/usr/share/catman/p_man/cat3/SCSL/ccfft3d.z/ccfft3d

Text File | 2002-10-03 | 33.5 KB | 727 lines

CCCCCCCCFFFFFFFFTTTT3333DDDD((((3333SSSS))))                                                        CCCCCCCCFFFFFFFFTTTT3333DDDD((((3333SSSS))))



NNNNAAAAMMMMEEEE
     CCCCCCCCFFFFFFFFTTTT3333DDDD, ZZZZZZZZFFFFFFFFTTTT3333DDDD - Applies a three-dimensional complex-to-complex Fast
     Fourier Transform (FFT)

SSSSYYYYNNNNOOOOPPPPSSSSIIIISSSS
     Single precision complex -> Single precision complex

          Fortran:
               CCCCAAAALLLLLLLL CCCCCCCCFFFFFFFFTTTT3333DDDD ((((_i_s_i_g_n,,,, _n_1,,,, _n_2,,,, _n_3,,,, _s_c_a_l_e,,,, _x,,,, _l_d_x,,,, _l_d_x_2,,,, _y,,,, _l_d_y,,,,
               _l_d_y_2,,,, _t_a_b_l_e,,,, _w_o_r_k,,,, _i_s_y_s))))

          C/C++:
               ####iiiinnnncccclllluuuuddddeeee <<<<ssssccccssssllll____fffffffftttt....hhhh>>>>
               iiiinnnntttt ccccccccfffffffftttt3333dddd ((((iiiinnnntttt _i_s_i_g_n,,,, iiiinnnntttt _n_1,,,, iiiinnnntttt _n_2,,,, iiiinnnntttt _n_3,,,, ffffllllooooaaaatttt _s_c_a_l_e,,,,
               ssssccccssssllll____ccccoooommmmpppplllleeeexxxx *_x,,,, iiiinnnntttt _l_d_x,,,, iiiinnnntttt _l_d_x_2,,,, ssssccccssssllll____ccccoooommmmpppplllleeeexxxx *_y,,,, iiiinnnntttt _l_d_y,,,,
               iiiinnnntttt _l_d_y_2,,,, ffffllllooooaaaatttt *_t_a_b_l_e,,,, ffffllllooooaaaatttt *_w_o_r_k,,,, iiiinnnntttt *_i_s_y_s))));;;;

          C++ STL:
               ####iiiinnnncccclllluuuuddddeeee <<<<ccccoooommmmpppplllleeeexxxx....hhhh>>>>
               ####iiiinnnncccclllluuuuddddeeee <<<<ssssccccssssllll____fffffffftttt....hhhh>>>>
               iiiinnnntttt ccccccccfffffffftttt3333dddd ((((iiiinnnntttt _i_s_i_g_n,,,, iiiinnnntttt _n_1,,,, iiiinnnntttt _n_2,,,, iiiinnnntttt _n_3,,,, ffffllllooooaaaatttt _s_c_a_l_e,,,,
               ccccoooommmmpppplllleeeexxxx<<<<ffffllllooooaaaatttt>>>> *_x,,,, iiiinnnntttt _l_d_x,,,, iiiinnnntttt _l_d_x_2,,,, ccccoooommmmpppplllleeeexxxx<<<<ffffllllooooaaaatttt>>>> *_y,,,, iiiinnnntttt
               _l_d_y,,,, iiiinnnntttt _l_d_y_2,,,, ffffllllooooaaaatttt *_t_a_b_l_e,,,, ffffllllooooaaaatttt *_w_o_r_k,,,, iiiinnnntttt *_i_s_y_s))));;;;

     Double precision complex -> Double precision complex

          Fortran:
               CCCCAAAALLLLLLLL ZZZZZZZZFFFFFFFFTTTT3333DDDD ((((_i_s_i_g_n,,,, _n_1,,,, _n_2,,,, _n_3,,,, _s_c_a_l_e,,,, _x,,,, _l_d_x,,,, _l_d_x_2,,,, _y,,,, _l_d_y,,,,
               _l_d_y_2,,,, _t_a_b_l_e,,,, _w_o_r_k,,,, _i_s_y_s))))

          C/C++:
               ####iiiinnnncccclllluuuuddddeeee <<<<ssssccccssssllll____fffffffftttt....hhhh>>>>
               iiiinnnntttt zzzzzzzzfffffffftttt3333dddd ((((iiiinnnntttt _i_s_i_g_n,,,, iiiinnnntttt _n_1,,,, iiiinnnntttt _n_2,,,, iiiinnnntttt _n_3,,,, ddddoooouuuubbbblllleeee _s_c_a_l_e,,,,
               ssssccccssssllll____zzzzoooommmmpppplllleeeexxxx *_x,,,, iiiinnnntttt _l_d_x,,,, iiiinnnntttt _l_d_x_2,,,, ssssccccssssllll____zzzzoooommmmpppplllleeeexxxx *_y,,,, iiiinnnntttt _l_d_y,,,,
               iiiinnnntttt _l_d_y_2,,,, ddddoooouuuubbbblllleeee *_t_a_b_l_e,,,, ddddoooouuuubbbblllleeee *_w_o_r_k,,,, iiiinnnntttt *_i_s_y_s))));;;;

          C++ STL:
               ####iiiinnnncccclllluuuuddddeeee <<<<ccccoooommmmpppplllleeeexxxx....hhhh>>>>
               ####iiiinnnncccclllluuuuddddeeee <<<<ssssccccssssllll____fffffffftttt....hhhh>>>>
               iiiinnnntttt zzzzzzzzfffffffftttt3333dddd ((((iiiinnnntttt _i_s_i_g_n,,,, iiiinnnntttt _n_1,,,, iiiinnnntttt _n_2,,,, iiiinnnntttt _n_3,,,, ddddoooouuuubbbblllleeee _s_c_a_l_e,,,,
               ccccoooommmmpppplllleeeexxxx<<<<ddddoooouuuubbbblllleeee>>>> *_x,,,, iiiinnnntttt _l_d_x,,,, iiiinnnntttt _l_d_x_2,,,, ccccoooommmmpppplllleeeexxxx<<<<ddddoooouuuubbbblllleeee>>>> *_y,,,, iiiinnnntttt
               _l_d_y,,,, iiiinnnntttt _l_d_y_2,,,, ddddoooouuuubbbblllleeee *_t_a_b_l_e,,,, ddddoooouuuubbbblllleeee *_w_o_r_k,,,, iiiinnnntttt *_i_s_y_s))));;;;

IIIIMMMMPPPPLLLLEEEEMMMMEEEENNNNTTTTAAAATTTTIIIIOOOONNNN
     These routines are part of the SCSL Scientific Library and can be loaded
     using either the ----llllssssccccssss or the ----llllssssccccssss____mmmmpppp option.  The ----llllssssccccssss____mmmmpppp option
     directs the linker to use the multi-processor version of the library.

     When linking to SCSL with ----llllssssccccssss or ----llllssssccccssss____mmmmpppp, 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



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CCCCCCCCFFFFFFFFTTTT3333DDDD((((3333SSSS))))                                                        CCCCCCCCFFFFFFFFTTTT3333DDDD((((3333SSSS))))



     by using the ----llllssssccccssss____iiii8888 option or the ----llllssssccccssss____iiii8888____mmmmpppp 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 iiiinnnntttt become lllloooonnnngggg lllloooonnnngggg and the <<<<ssssccccssssllll____fffffffftttt____iiii8888....hhhh>>>> header
     file should be included.

DDDDEEEESSSSCCCCRRRRIIIIPPPPTTTTIIIIOOOONNNN
     These routines compute the three-dimensional complex FFT of the complex
     matrix _X, and store the results in the complex matrix _Y.

     In FFT applications, it is customary to use zero-based subscripts; the
     formulas are simpler that way.  So suppose the arrays are declared as
     follows:

          Fortran:

               COMPLEX X(0:ldx-1, 0:ldx2-1, 0:n3-1)
               COMPLEX Y(0:ldy-1, 0:ldy2-1, 0:n3-1)


          C/C++:

               scsl_complex x[n3][ldx2][ldx], y[n3][ldy2][ldy];


          C++ STL:

               complex<float> x[n3][ldx2][ldx], y[n3][ldy2][ldy];


     These routines compute the formula:

     Y(k1,k2,k3) =

           n1-1 n2-1 n3-1
     scale*Sum  Sum  Sum  [X(j1,j2,j3)*w1**(j1*k1)*w2**(j2*k2)*w3**(j3*k3)]
           j1=0 j2=0 j3=0

     for _k_1 = 0, ..., _n_1 - 1,
         _k_2 = 0, ..., _n_2 - 1,
         _k_3 = 0, ..., _n_3 - 1,


     where:

     _w_1        = exp(_i_s_i_g_n*2*_p_i*_i/_n_1),






                                                                        PPPPaaaaggggeeee 2222






CCCCCCCCFFFFFFFFTTTT3333DDDD((((3333SSSS))))                                                        CCCCCCCCFFFFFFFFTTTT3333DDDD((((3333SSSS))))



     _w_2        = exp(_i_s_i_g_n*2*_p_i*_i/_n_2),

     _w_3        = exp(_i_s_i_g_n*2*_p_i*_i/_n_3),

     _i         = + sqrt(-1)

     _p_i        = 3.14159...

     _i_s_i_g_n     = +1 or -1

     Different authors use different conventions for which of the transforms,
     _i_s_i_g_n = +1 or _i_s_i_g_n = -1, is the forward or inverse transform, and what
     the _s_c_a_l_e factor should be in either case.  You can make this routine
     compute any of the various possible definitions, however, by choosing the
     appropriate values for _i_s_i_g_n and _s_c_a_l_e.

     The relevant fact from FFT theory is this:  If you take the FFT with any
     particular values of _i_s_i_g_n and _s_c_a_l_e, the mathematical inverse function
     is computed by taking the FFT with -_i_s_i_g_n and 1/(_n_1 * _n_2 * _n_3 * _s_c_a_l_e).
     In particular, if you use _i_s_i_g_n = +1 and _s_c_a_l_e = 1.0 for the forward FFT,
     you can compute the inverse FFT by using _i_s_i_g_n = -1 and _s_c_a_l_e = 1/(_n_1 *
     _n_2 * _n_3).

     See the NOTES section of this man page for information about the
     interpretation of the data types described in the following arguments.

     This routine has the following arguments:

     _i_s_i_g_n     Integer.  (input)
               Specifies whether to initialize the _t_a_b_l_e array or to do the
               forward or inverse Fourier transform, as follows:

               If _i_s_i_g_n = 0, the routine initializes the _t_a_b_l_e array and
               returns.  In this case, the only arguments used or checked are
               _i_s_i_g_n, _n_1, _n_2, _n_3, and _t_a_b_l_e.

               If _i_s_i_g_n = +1 or -1, the value of _i_s_i_g_n is the sign of the
               exponent used in the FFT formula.

     _n_1        Integer.  (input)
               Transform size in the first dimension.  If _n_1 is not positive,
               the routine returns without computing a transform.

     _n_2        Integer.  (input)
               Transform size in the second dimension.  If _n_2 is not positive,
               the routine returns without computing a transform.

     _n_3        Integer.  (input)
               Transform size in the third dimension.  If _n_3 is not positive,
               the routine returns without computing a transform.





                                                                        PPPPaaaaggggeeee 3333






CCCCCCCCFFFFFFFFTTTT3333DDDD((((3333SSSS))))                                                        CCCCCCCCFFFFFFFFTTTT3333DDDD((((3333SSSS))))



     _s_c_a_l_e     Scale factor. (input)
               CCCCCCCCFFFFFFFFTTTT3333DDDD: Single precision.
               ZZZZZZZZFFFFFFFFTTTT3333DDDD: Double precision.
               Each element of the output array is multiplied by _s_c_a_l_e after
               taking the Fourier transform, as defined previously.

     _x         Array of dimensions (_l_d_x, _l_d_x_2, _n_3).  (input)
               CCCCCCCCFFFFFFFFTTTT3333DDDD: Single precision complex array.
               ZZZZZZZZFFFFFFFFTTTT3333DDDD: Double complex array.
               Input array of values to be transformed.

     _l_d_x       Integer.  (input)
               The first dimension of _x, as it was declared in the calling
               program (the leading dimension of _x).  _l_d_x >= MMMMAAAAXXXX(_n_1, 1).

     _l_d_x_2      Integer.  (input)
               The second dimension of _x, as it was declared in the calling
               program.  _l_d_x_2 >= MMMMAAAAXXXX(_n_2, 1).

     _y         Array of dimensions (_l_d_y, _l_d_y_2, _n_3).  (output)
               CCCCCCCCFFFFFFFFTTTT3333DDDD: Single precision complex array.
               ZZZZZZZZFFFFFFFFTTTT3333DDDD: Double complex array.
               Output array of transformed values.  The output array may be
               the same as the input array, in which case, the transform is
               done in place; that is, the input array is overwritten with the
               transformed values.  In this case, it is necessary that
               _l_d_x = _l_d_y, and _l_d_x_2 = _l_d_y_2.

     _l_d_y       Integer.  (input)
               The first dimension of _y, as it was declared in the calling
               program (the leading dimension of _y).  _l_d_y >= MMMMAAAAXXXX(_n_1, 1).

     _l_d_y_2      Integer.  (input)
               The second dimension of _y, as it was declared in the calling
               program.  _l_d_y_2 >= MMMMAAAAXXXX(_n_2, 1).

     _t_a_b_l_e     Array of dimension (2* _n_1 + _N_F) + (2 * _n_2 + _N_F) + (2 * _n_3 + _N_F)
               (input or output)
               CCCCCCCCFFFFFFFFTTTT3333DDDD: Single precision array.
               ZZZZZZZZFFFFFFFFTTTT3333DDDD: Double precision array.

               Table of factors and root of unity.  See the description of the
               _i_s_y_s argument for the value of _N_F.

               If _i_s_i_g_n = 0, the routine initializes _t_a_b_l_e (_t_a_b_l_e is output
               only).

               If _i_s_i_g_n = +1 or -1, the values in _t_a_b_l_e are assumed to be
               initialized already by a prior call with _i_s_i_g_n = 0 (_t_a_b_l_e is
               input only).





                                                                        PPPPaaaaggggeeee 4444






CCCCCCCCFFFFFFFFTTTT3333DDDD((((3333SSSS))))                                                        CCCCCCCCFFFFFFFFTTTT3333DDDD((((3333SSSS))))



     _w_o_r_k      Array of dimension 2 * MMMMAAAAXXXX((((_n_1,,,, _n_2,,,, _n_3))))
               CCCCCCCCFFFFFFFFTTTT3333DDDD: Single precision array.
               ZZZZZZZZFFFFFFFFTTTT3333DDDD: Double precision array.

               Work array.  This is a scratch array used for intermediate
               calculations.  Its address space must be different from that of
               the input and output arrays.

     _i_s_y_s      Integer array dimensioned 0000........_i_s_y_s((((0000)))).
               An array that gives implementation-specific information.  All
               features and functions of the FFT routines specific to any
               particular implementation are confined to this _i_s_y_s array.

               In the Origin series implementation, _i_s_y_s((((0000))))====0000 and _i_s_y_s((((0000))))====1111
               are supported.  In SCSL versions prior to 1.3, only _i_s_y_s((((0000))))====0000
               was allowed. For _i_s_y_s((((0000))))====0000, _N_F====33330000, and for _i_s_y_s((((0000))))====1111, _N_F====222255556666.
               The _N_F words of storage in the _t_a_b_l_e array contain a
               factorization of the length of the transform.

               The smaller value of _N_F for _i_s_y_s((((0000))))====0000 is historical. It is too
               small to store all the required factors for the highest
               performing FFT, so when _i_s_y_s((((0000))))====0000, extra space is allocated
               when the _t_a_b_l_e array is initialized. To avoid memory leaks,
               this extra space must be deallocated when the _t_a_b_l_e array is no
               longer needed. The CCCCCCCCFFFFFFFFTTTT3333DDDDFFFF routine is used to release this
               memory. Due to the potential for memory leaks, the use of
               _i_s_y_s((((0000))))====0000 should be avoided.

               For _i_s_y_s((((0000))))====1111, the value of _N_F is large enough so that no extra
               memory needs to be allocated, and there is no need to call
               CCCCCCCCFFFFFFFFTTTT3333DDDDFFFF to release memory. If called, it does nothing.

               NOTE: _i_s_y_s((((0000))))====1111 means that _i_s_y_s is an integer array with two
               elements. The second element, _i_s_y_s((((1111)))), will not be accessed.

NNNNOOOOTTTTEEEESSSS
     The following data types are described in this documentation:

          TTTTeeeerrrrmmmm UUUUsssseeeedddd                     DDDDaaaattttaaaa ttttyyyyppppeeee

     Fortran:

          Array dimensioned 0000........_n----1111      xxxx((((0000::::nnnn----1111))))

          Array of dimensions (_m,_n)     xxxx((((mmmm,,,,nnnn))))

          Array of dimensions (_m,_n,_p)   xxxx((((mmmm,,,,nnnn,,,,pppp))))

          Integer                       IIIINNNNTTTTEEEEGGGGEEEERRRR (IIIINNNNTTTTEEEEGGGGEEEERRRR****8888 for ----llllssssccccssss____iiii8888[[[[____mmmmpppp]]]])






                                                                        PPPPaaaaggggeeee 5555






CCCCCCCCFFFFFFFFTTTT3333DDDD((((3333SSSS))))                                                        CCCCCCCCFFFFFFFFTTTT3333DDDD((((3333SSSS))))



          Single precision              RRRREEEEAAAALLLL

          Double precision              DDDDOOOOUUUUBBBBLLLLEEEE PPPPRRRREEEECCCCIIIISSSSIIIIOOOONNNN

          Single precision complex      CCCCOOOOMMMMPPPPLLLLEEEEXXXX

          Double precision complex      DDDDOOOOUUUUBBBBLLLLEEEE CCCCOOOOMMMMPPPPLLLLEEEEXXXX

     C/C++:

          Array dimensioned 0000........_n----1111      xxxx[[[[_n]]]]

          Array of dimensions (_m,_n)     xxxx[[[[mmmm****nnnn]]]] oooorrrr xxxx[[[[nnnn]]]][[[[mmmm]]]]

          Array of dimensions (_m,_n,_p)   xxxx[[[[mmmm****nnnn****pppp]]]] oooorrrr xxxx[[[[pppp]]]][[[[nnnn]]]][[[[mmmm]]]]

          Integer                       iiiinnnntttt (lllloooonnnngggg lllloooonnnngggg for ----llllssssccccssss____iiii8888[[[[____mmmmpppp]]]])

          Single precision              ffffllllooooaaaatttt

          Double precision              ddddoooouuuubbbblllleeee

          Single precision complex      ssssccccssssllll____ccccoooommmmpppplllleeeexxxx

          Double precision complex      ssssccccssssllll____zzzzoooommmmpppplllleeeexxxx

     C++ STL:

          Array dimensioned 0000........_n----1111      xxxx[[[[_n]]]]

          Array of dimensions (_m,_n)     xxxx[[[[mmmm****nnnn]]]] oooorrrr xxxx[[[[nnnn]]]][[[[mmmm]]]]

          Array of dimensions (_m,_n,_p)   xxxx[[[[mmmm****nnnn****pppp]]]] oooorrrr xxxx[[[[pppp]]]][[[[nnnn]]]][[[[mmmm]]]]

          Integer                       iiiinnnntttt (lllloooonnnngggg lllloooonnnngggg for ----llllssssccccssss____iiii8888[[[[____mmmmpppp]]]])

          Single precision              ffffllllooooaaaatttt

          Double precision              ddddoooouuuubbbblllleeee

          Single precision complex      ccccoooommmmpppplllleeeexxxx<<<<ffffllllooooaaaatttt>>>>

          Double precision complex      ccccoooommmmpppplllleeeexxxx<<<<ddddoooouuuubbbblllleeee>>>>

CCCCAAAAUUUUTTTTIIIIOOOONNNNSSSS
     Transform sizes with a prime factor exceeding 22223332222----1111 are not supported for
     the 8-byte integer version of the library.

     In addition to the _w_o_r_k 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



                                                                        PPPPaaaaggggeeee 6666






CCCCCCCCFFFFFFFFTTTT3333DDDD((((3333SSSS))))                                                        CCCCCCCCFFFFFFFFTTTT3333DDDD((((3333SSSS))))



     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 MMMMPPPP____SSSSLLLLAAAAVVVVEEEE____SSSSTTTTAAAACCCCKKKKSSSSIIIIZZZZEEEE
     environment variable or the mmmmpppp____sssseeeetttt____ssssllllaaaavvvveeee____ssssttttaaaacccckkkkssssiiiizzzzeeee(((()))) library routine. See
     the mmmmpppp(3C), mmmmpppp(3F) and ppppeeee____eeeennnnvvvviiiirrrroooonnnn(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 pppptttthhhhrrrreeeeaaaadddd____ccccrrrreeeeaaaatttteeee(3P).  The
     stacksize attribute should be set explicitly to a non-default value using
     the pppptttthhhhrrrreeeeaaaadddd____aaaattttttttrrrr____sssseeeettttssssttttaaaacccckkkkssssiiiizzzzeeee(3P) call, described in the
     pppptttthhhhrrrreeeeaaaadddd____aaaattttttttrrrr____iiiinnnniiiitttt(3P) man page.

     Care must be exercised if copies of the _t_a_b_l_e array are used: even though
     a copy exists, the original must persist. As an example, the following
     code will nnnnooootttt work:

     #include <scsl_fft.h>
     scsl_complex x[129][129][129], y[129][129][129];
     float table[(2*128 + 256) + (2*128 + 256) + (2*128 + 256)];
     float work[2*128];
     int isys[2];
     isys[0] = 1;
     {
       float table_orig[(2*128 + 256) + (2*128 + 256) + (2*128 + 256)];

       ccfft3d(0, 128, 128, 128, 1.0f, (scsl_complex *) x, 129,
              129, (scsl_complex *) y, 129, 129, table_orig,
              work, isys);
       bcopy(table_orig, table,
             ((2*128+256)+(2*256+256)+(2*256+256))*sizeof(float));
     }
     ccfft3d(1, 128, 128, 128, 1.0f, (scsl_complex *) x, 129,
            129, (scsl_complex *) y, 129, 129, table, work, isys);


     In this example, because _t_a_b_l_e__o_r_i_g is a stack variable that does not
     persist outside of the code block delimited by the braces, the data in
     the copy, _t_a_b_l_e, are not guaranteed to be valid. However, the following
     code will work because _t_a_b_l_e__o_r_i_g is persistent:

     #include <scsl_fft.h>
     scsl_complex x[129][129][129], y[129][129][129];
     float table_orig[(2*128 + 256) + (2*128 + 256) + (2*128 + 256)];
     float table[(2*128 + 256) + (2*128 + 256) + (2*128 + 256)];
     float work[2*128];
     int isys[2];
     isys[0] = 1;
     ccfft3d(0, 128, 128, 128, 1.0f, (scsl_complex *) x, 129,
             129, (scsl_complex *) y, 129, 129, table_orig,
             work, isys);



                                                                        PPPPaaaaggggeeee 7777






CCCCCCCCFFFFFFFFTTTT3333DDDD((((3333SSSS))))                                                        CCCCCCCCFFFFFFFFTTTT3333DDDD((((3333SSSS))))



     bcopy(table_orig, table,
             ((2*128+256)+(2*256+256)+(2*256+256))*sizeof(float));
     ccfft3d(1, 128, 128, 128, 1.0f, (scsl_complex *) x, 129,
             129, (scsl_complex *) y, 129, 129, table,
             work, isys);


EEEEXXXXAAAAMMMMPPPPLLLLEEEESSSS
     The following examples are for Origin series only.

     Example 1:  Initialize the TTTTAAAABBBBLLLLEEEE array in preparation for doing a three-
     dimensional FFT of size 128 by 128 by 128.  In this case, only the _i_s_i_g_n,
     _n_1, _n_2, _n_3, and _t_a_b_l_e arguments are used; you can use dummy arguments or
     zeros for other arguments.

     Fortran:

      REAL TABLE ((2*128 + 256) + (2*128 + 256) + (2*128 + 256))
      INTEGER ISYS(0:1)
      ISYS(0) = 1
      CALL CCFFT3D (0, 128, 128, 128, 0.0, DUMMY, 1, 1, DUMMY, 1, 1,
     &              TABLE, DUMMY, ISYS)


     C/C++:

          #include <scsl_fft.h>
          float table[(2*128 + 256) + (2*128 + 256) + (2*128 + 256)];
          int isys[2];
          isys[0] = 1;
          ccfft3d(0, 128, 128, 128, 0.0f, NULL, 1, 1, NULL, 1, 1,
                  table, NULL, isys);


     C++ STL:

          #include <complex.h>
          #include <scsl_fft.h>
          float table[(2*128 + 256) + (2*128 + 256) + (2*128 + 256)];
          int isys[2];
          isys[0] = 1;
          ccfft3d(0, 128, 128, 128, 0.0f, NULL, 1, 1, NULL, 1, 1,
                  table, NULL, isys);


     Example 2:  XXXX and YYYY are complex arrays of dimension (0:128, 0:128,
     0:128).  The first 128 elements of each dimension contain data; for
     performance reasons, the extra element forces the leading dimensions to
     be odd numbers.  Take the three-dimensional FFT of XXXX and store it in YYYY.
     Initialize the TTTTAAAABBBBLLLLEEEE array, as in example 1.





                                                                        PPPPaaaaggggeeee 8888






CCCCCCCCFFFFFFFFTTTT3333DDDD((((3333SSSS))))                                                        CCCCCCCCFFFFFFFFTTTT3333DDDD((((3333SSSS))))



     Fortran:

      COMPLEX X(0:128, 0:128, 0:128)
      COMPLEX Y(0:128, 0:128, 0:128)
      REAL TABLE((2*128 + 256) + (2*128 + 256) + (2*128 + 256))
      REAL WORK (2*128)
      INTEGER ISYS(0:1)
      ISYS(0) = 1
      CALL CCFFT3D(0, 128, 128, 128, 1.0, X, 129, 129, Y, 129, 129,
     &             TABLE, WORK, ISYS)
      CALL CCFFT3D(1, 128, 128, 128, 1.0, X, 129, 129, Y, 129, 129,
     &             TABLE, WORK, ISYS)


     C/C++:

     #include <scsl_fft.h>
     scsl_complex x[129][129][129], y[129][129][129];
     float table[(2*128 + 256) + (2*128 + 256) + (2*128 + 256)];
     float work[2*128];
     int isys[2];
     isys[0] = 1;
     ccfft3d(0, 128, 128, 128, 1.0f, (scsl_complex *) x, 129, 129,
            (scsl_complex *) y, 129, 129, table, work, isys);
     ccfft3d(1, 128, 128, 128, 1.0f, (scsl_complex *) x, 129, 129,
            (scsl_complex *) y, 129, 129, table, work, isys);


     C++ STL:

     #include <complex.h>
     #include <scsl_fft.h>
     complex<float> x[129][129][129], y[129][129][129];
     float table[(2*128 + 256) + (2*128 + 256) + (2*128 + 256)];
     float work[2*128];
     int isys[2];
     isys[0] = 1;
     ccfft3d(0, 128, 128, 128, 1.0f, (complex<float> *) x, 129, 129,
            (complex<float> *) y, 129, 129, table, work, isys);
     ccfft3d(1, 128, 128, 128, 1.0f, (complex<float> *) x, 129, 129,
            (complex<float> *) y, 129, 129, table, work, isys);


     Example 3:  With XXXX and YYYY as in example 2, take the inverse FFT of YYYY and
     store it back in XXXX.  The _s_c_a_l_e factor 1.0/(128.0*128.0*128.0) is used.
     Assume that the TTTTAAAABBBBLLLLEEEE array is already initialized.

     Fortran:

      CALL CCFFT3D(-1, 128, 128, 128, 1.0/(128.0**3), Y, 129, 129,
     &             X, 129, 129, TABLE, WORK, ISYS)




                                                                        PPPPaaaaggggeeee 9999






CCCCCCCCFFFFFFFFTTTT3333DDDD((((3333SSSS))))                                                        CCCCCCCCFFFFFFFFTTTT3333DDDD((((3333SSSS))))



     C/C++:

     ccfft3d(-1, 128, 128, 128, 1.0f/(128.0f*128.0f*128.0f),
            (scsl_complex *) y, 129, 129, (scsl_complex *) x, 129,
            129, table, work, isys);


     C++ STL:

     ccfft3d(-1, 128, 128, 128, 1.0f/(128.0f*128.0f*128.0f),
            (complex<float> *) y, 129, 129, (complex<float> *) x,
            129, 129, table, work, isys);


     Example 4:  Perform the same computation as in example 2, but put the
     output back in the array _X to save storage space. Use the 8-byte integer
     version of SCSL.

     Fortran:

      COMPLEX X(0:128, 0:128, 0:128)
      REAL TABLE((2*128 + 256) + (2*128 + 256) + (2*128 + 256))
      REAL WORK (2*128)
      INTEGER ISYS(0:1)
      ISYS(0) = 1_8
      CALL CCFFT3D(0_8, 128_8, 128_8, 128_8, 1.0, X, 129_8, 129_8,
     &             X, 129_8, 129_8, TABLE, WORK, ISYS)
      CALL CCFFT3D(1_8, 128_8, 128_8, 128_8, 1.0, X, 129_8, 129_8,
     &             X, 129_8, 129_8, TABLE, WORK, ISYS)


     C/C++:

     #include <scsl_fft_i8.h>
     scsl_complex x[129][129][129];
     float table[(2*128 + 256) + (2*128 + 256) + (2*128 + 256)];
     float work[2*128];
     long long isys[2];
     isys[0] = 1LL;
     ccfft3d(0LL, 128LL, 128LL, 128LL, 1.0f, (scsl_complex *) x, 129LL,
            129LL, (scsl_complex *) x, 129LL, 129LL, table,
            work, isys);
     ccfft3d(1LL, 128LL, 128LL, 128LL, 1.0f, (scsl_complex *) x, 129LL,
            129LL, (scsl_complex *) x, 129LL, 129LL, table,
            work, isys);


     C++ STL:

     #include <complex.h>
     #include <scsl_fft_i8.h>
     complex<float> x[129][129][129];



                                                                       PPPPaaaaggggeeee 11110000






CCCCCCCCFFFFFFFFTTTT3333DDDD((((3333SSSS))))                                                        CCCCCCCCFFFFFFFFTTTT3333DDDD((((3333SSSS))))



     float table[(2*128 + 256) + (2*128 + 256) + (2*128 + 256)];
     float work[2*128];
     int isys[2];
     isys[0] = 1;
     ccfft3d(0LL, 128LL, 128LL, 128LL, 1.0f, (complex<float> *) x, 129LL,
             129LL, (complex<float> *) x, 129LL, 129LL, table,
             work, isys);
     ccfft3d(1LL, 128LL, 128LL, 128LL, 1.0f, (complex<float> *) x, 129LL,
             129LL, (complex<float> *) x, 129LL, 129LL, 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 do not change.

     Fortran:

           COMPLEX X(129, 129, 129)
           COMPLEX Y(129, 129, 129)
           ...
           CALL CCFFT3D(0, 128, 128, 128, 1.0, X, 129, 129,
          &             Y, 129, 129, TABLE, WORK, ISYS)
           CALL CCFFT3D(1, 128, 128, 128, 1.0, X, 129, 129,
          &             Y, 129, 129, TABLE, WORK, ISYS)


SSSSEEEEEEEE AAAALLLLSSSSOOOO
     IIIINNNNTTTTRRRROOOO____FFFFFFFFTTTT(3S), IIIINNNNTTTTRRRROOOO____SSSSCCCCSSSSLLLL(3S), CCCCCCCCFFFFFFFFTTTT(3S), CCCCCCCCFFFFFFFFTTTT2222DDDD(3S), CCCCCCCCFFFFFFFFTTTTMMMM(3S),
     SSSSCCCCFFFFFFFFTTTT(3S), SSSSCCCCFFFFFFFFTTTT2222DDDD(3S), SSSSCCCCFFFFFFFFTTTT3333DDDD(3S), SSSSCCCCFFFFFFFFTTTTMMMM(3S)






















                                                                       PPPPaaaaggggeeee 11111111