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That occurs when computing the two-dimensional FFT. However, the size of the original array is modified to contain one or two additional rows, which are needed to store the values identically equal to zero. The values of the arguments used with the real two-dimensional FFT routines depend upon whether an in-place or out-of place transform is performed, and whether the results are stored in a full or partial result matrix, as shown in TABLE 16. TABLE 16 Relationships Between Values of Arguments for Real Two-Dimensional FFT Routines Full Result Matrix Partial Result Matrix In-Place Transform B unused B unused LDB unused LDB unused LDA must be even LDA must be even LDA 2*M LDA M+2 if M is evenLDA M+1 if M is odd A(1:2*M, 1:N) A(1:M+2, 1:N) if M is evenA(1:M+1, 1:N) if M is odd Out-of-Place Transform A unchanged A unchanged LDA M LDA M 2*LDB M 2*LDB M+2 if M is even2*LDB M+1 if M is odd B(1:2*M, 1:N) B(1:M+2, 1:N) if M is evenB(1:M+1, 1:N) if M is odd When computing the real two-dimensional FFT of an input sequence of M rows and N columns, the computed Fourier coefficients will be stored in a result matrix with 2*M rows and N columns when using the Full storage option. When using the Partial storage option, the Fourier coefficients will be stored in a result matrix with M+2 rows and N columns when M is even, or in a result matrix with M+1 rows and N columns when

M CALL RFFT2F ('O', IS_FULL, M, N, AT, LDA, B, LDB, WT, LWORK) PRINT *, 'Transformed Out-of-Place, Full' DO I = 1, LDB_ACTUAL, N PRINT '(100('' ('', F8.3, '','', F8.3, '')'' :))', $ (B(I,J), B(I+1,J), J = 1, N) END DO * B(M+3:LDB,1:N) = 0 * PRINT *, 'Transformed, last half clear:' * DO I = 1, LDB, N * PRINT '(100('' ('', F8.3, '','', F8.3, '')'' :))', * $ (B(I,J), B(I+1,J), J = 1, N) * END DO CALL RFFT2B ('O', M, N, AT, LDA, B, LDB, WT, LWORK) PRINT *, 'Inverse: Scaled Output, Out-of-Place, Full' DO I = 1, M PRINT '(100(F8.3))', (AT(I,J) / (M * N), J = 1, N) END DO PRINT * * * Example 2 * in-place, full * LDA must be at least 2*M * AT = INPUT IS_FULL = 'F' CALL RFFT2F ('I', IS_FULL, M, N, AT, LDA, 0, 0, WT, LWORK) PRINT *, 'Transformed In-Place, Full' DO I = 1, LDA, 2 PRINT '(100('' ('', F8.3, '','', F8.3, '')'' :))', $ (AT(I,J), AT(I+1,J), J = 1, N) END DO CALL RFFT2B ('I', M, N, AT, LDA, 0, 0, WT, LWORK) PRINT *, 'Inverse: Scaled Output, In-Place, Full' DO I = 1, M PRINT '(100(F8.3))', (AT(I,J) / (M * N), J = 1, N) END DO PRINT * DEALLOCATE(AT,WT,B) END SUBROUTINE my_system% f95 -dalign fft_ex16.f -xlic_lib=sunperf my_system% a.out Original Sequence 0.968 0.654 0.067 0.021 0.478 0.512 0.910 0.202 0.352 0.940 0.933 0.204 Transformed Out-of-Place, Full ( 6.241, 0.000) ( 1.173, 0.000) (

Where one row contains the real part of the complex coefficient and the next row contains the imaginary part of the complex coefficient. In CODE EXAMPLE 15, to better display the complex conjugate symmetry, the real and imaginary parts of each complex coefficient are displayed on one line. For example, the following output: Transformed Out-of-Place, Full ( 6.241, 0.000) ( 1.173, 0.000) ( -0.018, 1.169) ( 0.304, 0.111) represents the following values for the Fourier coefficients. Column 1 Column 2 Re(X0) Im(X0) Re(X0) Im(X0) Re(X1) Im(X1) Re(X1) Im(X1) The inverse transform is unnormalized and can be normalized by dividing each value by M*N. CODE EXAMPLE 16 RFFT2F and RFFT2B Example Showing In-Place and Out-of-Place Storage my_system% cat fft_ex16.f PROGRAM TESTFFT INTEGER M, N PARAMETER(M = 6, N = 2) CALL FFT(M,N) END SUBROUTINE FFT(M, N) CHARACTER*1 IS_FULL INTEGER I, J, M, N, ISTAT, LWORK, LDA, LDB, LDB_ACTUAL REAL RNUM, RAND EXTERNAL RFFT2F, RFFT2B, RFFT2I, RAND REAL, DIMENSION(:,:), ALLOCATABLE :: AT, B, INPUT REAL, DIMENSION(:), ALLOCATABLE :: WT LDA = 2*M LDB = 2*M LWORK = M+2*N+MAX(M,2*N)+30 ALLOCATE(AT(LDA,N), INPUT(LDA,N), WT(LWORK), B(LDB_ACTUAL,N)) CALL RFFT2I (M, N, WT) DO I = 1, N DO J = 1, M INPUT(J,I) = RAND(0) END DO END DO AT = INPUT * PRINT *, 'Original Sequence' DO I = 1, M PRINT '(100(F8.3))', (AT(I,J), J = 1, N) END DO PRINT * * * Example 1 * Out-of-place, full * leading dimension of B (2*LDB) must be at least 2*M * IS_FULL = 'F' LDB =

Computing the three-dimensional FFT. However, the size of the original array is modified to contain one or two additional rows, which are needed to store the values identically equal to zero. The values of the arguments used with the real three-dimensional FFT routines depend upon whether an in-place or out-of place transform is performed, and whether the results are stored in a full or partial result matrix, as shown in TABLE 19. TABLE 19 Relationship Between Values of Arguments for Real Three-Dimensional FFT Routines Full Result Array Partial Result Array In-Place Transform B unused B unused LDB unused LDB unused LDA must be even LDA must be even LDA 2*M LDA M+2 if M is evenLDA M+1 if M is odd A(1:2*M, 1:N) A(1:M+2, 1:N) if M is evenA(1:M+1, 1:N) if M is odd Out-of-Place Transform A unchanged A unchanged LDA M LDA M LDB 2*M LDB M/2+1 if M is evenLDB (M-1)/2+1 if M is odd B(1:2*M, 1:N, 1:K) B(1:M+2, 1:N, 1:K) if M is evenB(1:M+1, 1:N, 1:K) if M is odd When computing the real 3D FFT of an input sequence of M rows, N columns, and K planes, the computed Fourier coefficients will be stored in a result matrix with 2*M rows, N columns for each value of K when using the Full storage option. When using the Partial storage option, the Fourier coefficients will be stored in a result matrix with M+2 rows and N columns for each value of K when M is even, or