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CCFFTMR(3S)							   CCFFTMR(3S)

NAME
     CCFFTMR, ZZFFTMR - Applies multiple complex-to-complex Fast Fourier
     Transforms (FFTs) to the rows of a two-dimensional (2D) array

SYNOPSIS
     Single precision complex -> Single precision complex

	  Fortran:
	       CALL CCFFTMR (isign, n, lot, scale, x, ldx, y, ldy, table,
	       work, isys)

	  C/C++:
	       #include <scsl_fft.h>
	       int ccfftmr (int isign, int n, int lot, float scale,
	       scsl_complex *x, int ldx, scsl_complex *y, int ldy, float
	       *table, float *work, int *isys);

	  C++ STL:
	       #include <complex.h>
	       #include <scsl_fft.h>
	       int ccfftmr (int *isign, int *n, int *lot, float *scale,
	       complex<float> *x, int *ldx, complex<float> *y, int *ldy, float
	       *table, float *work, int *isys);

     Double precision complex -> Double precision complex

	  Fortran:
	       CALL ZZFFTMR (isign, n, lot, scale, x, ldx, y, ldy, table,
	       work, isys)

	  C/C++:
	       #include <scsl_fft.h>
	       int zzfftmr (int isign, int n, int lot, double scale,
	       scsl_zomplex *x, int ldx, scsl_zomplex *y, int ldy, double
	       *table, double *work, int *isys);

	  C++ STL:
	       #include <complex.h>
	       #include <scsl_fft.h>
	       int zzfftmr (int *isign, int *n, int *lot, double *scale,
	       complex<double> *x, int *ldx, complex<double> *y, int *ldy,
	       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

									Page 1

CCFFTMR(3S)							   CCFFTMR(3S)

     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
     CCFFTMR/ZZFFTMR computes the FFT of each row of the complex matrix X, and
     stores the results in the rows of complex matrix Y.

     Suppose the arrays are declared as follows:

	  Fortran:

	       COMPLEX X(0:ldx-1, 0:n-1)
	       COMPLEX Y(0:ldy-1, 0:n-1)

	  C/C++:

	       scsl_complex x[n][ldx], y[n][ldy];

	  C++ STL:

	       complex<float> x[n][ldx], y[n][ldy];

     where ldx >= lot, ldy >= lot.

     Then row L of the output array is the FFT of row L of the input array,
     using the following formula for the FFT:

		     n-1	isign * j * k
     Y	   = scale * Sum [X  * w	      ]
      k,L	     j=0    j

     for k = 0, ..., n-1
	 L = 0, ..., lot-1

     where:

     w	       = exp(2*pi*i/n),

     i	       = + sqrt(-1),

									Page 2

CCFFTMR(3S)							   CCFFTMR(3S)

     pi	       = 3.14159...,

     isign     = +1 or -1

     lot       = the number of rows to transform

     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 this routine
     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 take the FFT with any
     particular values of isign and scale, the mathematical inverse function
     is computed by taking the FFT with -isign and 1/(n * scale).  In
     particular, if you use isign = +1 and scale = 1.0 for the forward FFT,
     you can compute the inverse FFT by using the following:  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.

     This routine has 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 table 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 each transform (the number of elements in each row of
	       the input and output matrix to be transformed). n >= 0; if n =
	       0, the routine returns.

     lot       Integer.	  (input)
	       The number of transforms to be computed (lot size).  This is
	       the number of elements in each column of the input and output
	       matrix.	lot >= 0.  If lot = 0, the routine returns.

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

									Page 3

CCFFTMR(3S)							   CCFFTMR(3S)

     x	       Array of dimensions (ldx, n).   (input)
	       CCFFTMR: Single precision complex array.
	       ZZFFTMR: Double precision complex array.
	       Input array of values to be transformed.

     ldx       Integer.	  (input)
	       The number of rows in x, as it was declared in the calling
	       program (the leading dimension of X).  ldx >= MAX(lot, 1).

     y	       Array of dimensions (ldy, n).   (output)
	       CCFFTMR: Single precision complex array.
	       ZZFFTMR: Double precision complex array.
	       Output array of transformed values.  Each row of the output
	       array, y, is the FFT of the corresponding row of the input
	       array, x, computed according to the preceding formula.

	       The output array may be the same as the input array. In that
	       case, the transform is done in place.  The input array is
	       overwritten with the transformed values.	 In this case, it is
	       necessary that ldx = ldy.

     ldy       Integer.	  (input)
	       The number of rows in the Y array, as it was declared in the
	       calling program (the leading dimension of Y).  ldy >= MAX(lot,
	       1).

     table     Array of dimension (2 * n +  NF) (input or output)
	       CCFFTMR: Single precision array.
	       ZZFFTMR: Double precision array.
	       Table of factors and roots of unity.  See the description of
	       the isys argument for the value of NF.

	       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 (2 * n).
	       CCFFTM: Single precision array.
	       ZZFFTM: Double precision array.
	       Work array.  This is a scratch array used for intermediate
	       calculations.  Its address space must be different from 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.

									Page 4

CCFFTMR(3S)							   CCFFTMR(3S)

	       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, NF=30, and for isys(0)=1, NF=256.
	       The NF words of storage in the table array contain a
	       factorization of the length of the transform.

	       The smaller value of NF 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 CCFFTMRF 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 NF is large enough so that no extra
	       memory needs to be allocated, and there is no need to call
	       CCFFTMRF 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]

									Page 5

CCFFTMR(3S)							   CCFFTMR(3S)

	  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.

									Page 6

CCFFTMR(3S)							   CCFFTMR(3S)

     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>
	  scsl_complex x[131][65], y[131][65];
	  float table[2*128 + 256];
	  float work[2*128];
	  int isys[2];
	  isys[0] = 1;
	  {
	     float table_orig[2*128 + 256];

	     ccfftmr(0, 128, 64, 1.0f, (scsl_complex *) x, 65,
		    (scsl_complex *) y, 65, table_orig, work, isys);
	     bcopy(table_orig, table, (2*128+256)*sizeof(float));
	  }
	  ccfftmr(1, 128, 64, 1.0f, (scsl_complex *) x, 65,
		    (scsl_complex *) y, 65, 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>
	  scsl_complex x[131][65], y[131][65];
	  float table_orig[2*128 + 256];
	  float table[2*128 + 256];
	  float work[2*128];
	  int isys[2];
	  isys[0] = 1;
	  ccfftmr(0, 128, 64, 1.0f, (scsl_complex *) x, 65,
		 (scsl_complex *) y, 65, table_orig, work, isys);
	  bcopy(table_orig, table, (2*128+256)*sizeof(float));
	  ccfftmr(1, 128, 64, 1.0f, (scsl_complex *) x, 65,
		 (scsl_complex *) y, 65, table, work, isys);

EXAMPLES
     The following examples are for Origin series only.

     Example 1:	 Initialize the table array in preparation for doing FFTs of
     size 128.	Only the isign, n, and table arguments are used in this case.
     You can use dummy arguments or zeros for the other arguments in the
     subroutine call.

     Fortran:

	   REAL TABLE(2*128 + 256)
	   CALL CCFFTMR(0, 128, 0, 0.0, DUMMY, 1, DUMMY, 1,

									Page 7

CCFFTMR(3S)							   CCFFTMR(3S)

	  &		TABLE, DUMMY, 0)

     C/C++:

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

     C++ STL:

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

     Example 2:	 X and Y are complex arrays  dimensioned (0...64) by
     (0...130).	 The first 64 elements of each column contain data.  For
     performance reasons, the extra element forces the leading dimension to be
     an odd number.  Take the FFT of the first 64 rows of X and store the
     results in the first 64 rows of Y.	 Before taking the FFT, initialize the
     TABLE array, as in example 1.

     Fortran:

     COMPLEX X(0:64, 0:130)
     COMPLEX Y(0:64, 0:130)
     REAL TABLE(2*128 + 256)
     REAL WORK(2*128)
     INTEGER ISYS(0:1)
     ISYS(0) = 1
     CALL CCFFTMR(0, 128, 64, 1.0, X, 65, Y, 65, TABLE, WORK, ISYS)
     CALL CCFFTMR(1, 128, 64, 1.0, X, 65, Y, 65, TABLE, WORK, ISYS)

     C/C++:

	  #include <scsl_fft.h>
	  scsl_complex x[131][65], y[131][65];
	  float table[2*128 + 256];
	  float work[2*128];
	  int isys[2];
	  isys[0] = 1;
	  ccfftmr(0, 128, 64, 1.0f, (scsl_complex *) x, 65,

									Page 8

CCFFTMR(3S)							   CCFFTMR(3S)

		  (scsl_complex *) y, 65, table, work, isys);
	  ccfftmr(1, 128, 64, 1.0f, (scsl_complex *) x, 65,
		  (scsl_complex *) y, 65, table, work, isys);

     C++ STL:

	  #include <complex.h>
	  #include <scsl_fft.h>
	  complex<float> x[131][65], y[131][65];
	  float table[2*128 + 128];
	  float work[2*128];
	  int isys[2];
	  isys[0] = 1;
	  ccfftmr(0, 128, 64, 1.0f, (complex<float> *) x, 65,
		  (complex<float> *) y, 65, table, work, isys);
	  ccfftmr(1, 128, 64, 1.0f, (complex<float> *) x, 65,
		  (complex<float> *) y, 65, 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 1/128 is used.  Assume that the
     TABLE array is already initialized.

     Fortran:

	   CALL CCFFTMR(-1, 128, 64, 1.0/128.0, Y, 65, X, 65,
	  &		TABLE,WORK,0)

     C/C++:

	  ccfftmr(-1, 128, 64, 1.0f/128.0f, (scsl_complex *) y, 65,
		 (scsl_complex *) x, 65, table, work, isys);

     C++ STL:

     ccfftmr(-1, 128, 64, 1.0f/128.0f, (complex<float> *) y, 65,
	    (complex<float> *) x, 65, table, work, isys);

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

     Fortran:

	   COMPLEX X(0:64, 0:130)
	   REAL TABLE(2*128 + 256)
	   REAL WORK(2*128)
	   INTEGER*8 ISYS(0:1)

									Page 9

CCFFTMR(3S)							   CCFFTMR(3S)

	   ISYS(0) = 1_8
	   CALL CCFFTMR(0_8, 128_8, 64_8, 1.0, X, 65_8, X, 65_8,
	  &		TABLE, WORK, ISYS)
	   CALL CCFFTMR(1_8, 128_8, 64_8, 1.0, X, 65_8, X, 65_8,
	  &		TABLE, WORK, ISYS)

     C/C++:

	  #include <scsl_fft_i8.h>
	  scsl_complex x[131][65], y[131][65];
	  float table[2*128 + 256];
	  float work[2*128];
	  long long isys[2];
	  isys[0] = 1LL;
	  ccfftmr(0LL, 128LL, 64LL, 1.0f, (scsl_complex *) x, 65LL,
		 (scsl_complex *) x, 65LL, table, work, isys);
	  ccfftmr(1LL, 128LL, 64LL, 1.0f, (scsl_complex *) x, 65LL,
		 (scsl_complex *) x, 65LL, table, work, isys);

     C++ STL:

     #include <complex.h>
     #include <scsl_fft_i8.h>
     complex<float> x[131][65], y[131][65];
     float table[2*128 + 256];
     float work[2*128];
     long long isys[2];
     isys[0] = 1LL;
     ccfftmr(0LL, 128LL, 64LL, 1.0f, (complex<float> *) x, 65LL,
	    (complex<float> *) x, 65LL, table, work, isys);
     ccfftmr(1LL, 128LL, 64LL, 1.0f, (complex<float> *) x, 65LL,
	    (complex<float> *) x, 65LL, 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:

     COMPLEX X(65, 131)
     COMPLEX Y(65, 131)
     CALL CCFFTMR(0, 128, 64, 1.0, X, 65, Y, 65, TABLE, WORK, ISYS)
     CALL CCFFTMR(1, 128, 64, 1.0, X, 65, Y, 65, TABLE, WORK, ISYS)

SEE ALSO
     INTRO_FFT(3S), INTRO_SCSL(3S), CCFFT(3S), CCFFTM(3S), SCFFT(3S),
     SCFFTM(3S)

								       Page 10

CCFFTMR(3S)							   CCFFTMR(3S)

								       Page 11

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