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589 lines
15 KiB
Go
589 lines
15 KiB
Go
// Copyright ©2018 The Gonum Authors. All rights reserved.
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// Use of this source code is governed by a BSD-style
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// license that can be found in the LICENSE file.
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// This is a translation of the FFTPACK cfft functions by
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// Paul N Swarztrauber, placed in the public domain at
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// http://www.netlib.org/fftpack/.
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package fftpack
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import (
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"math"
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"math/cmplx"
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)
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// Cffti initializes the array work which is used in both Cfftf
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// and Cfftb. the prime factorization of n together with a
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// tabulation of the trigonometric functions are computed and
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// stored in work.
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//
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// Input parameter:
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//
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// n The length of the sequence to be transformed.
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//
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// Output parameters:
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//
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// work A work array which must be dimensioned at least 4*n.
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// the same work array can be used for both Cfftf and Cfftb
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// as long as n remains unchanged. Different work arrays
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// are required for different values of n. The contents of
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// work must not be changed between calls of Cfftf or Cfftb.
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//
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// ifac A work array containing the factors of n. ifac must have
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// length 15.
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func Cffti(n int, work []float64, ifac []int) {
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if len(work) < 4*n {
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panic("fourier: short work")
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}
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if len(ifac) < 15 {
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panic("fourier: short ifac")
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}
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if n == 1 {
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return
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}
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cffti1(n, work[2*n:4*n], ifac[:15])
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}
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func cffti1(n int, wa []float64, ifac []int) {
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ntryh := [4]int{3, 4, 2, 5}
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nl := n
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nf := 0
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outer:
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for j, ntry := 0, 0; ; j++ {
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if j < 4 {
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ntry = ntryh[j]
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} else {
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ntry += 2
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}
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for {
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if nl%ntry != 0 {
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continue outer
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}
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ifac[nf+2] = ntry
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nl /= ntry
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nf++
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if ntry == 2 && nf != 1 {
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for i := 1; i < nf; i++ {
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ib := nf - i + 1
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ifac[ib+1] = ifac[ib]
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}
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ifac[2] = 2
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}
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if nl == 1 {
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break outer
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}
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}
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}
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ifac[0] = n
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ifac[1] = nf
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argh := 2 * math.Pi / float64(n)
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i := 1
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l1 := 1
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for k1 := 0; k1 < nf; k1++ {
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ip := ifac[k1+2]
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ld := 0
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l2 := l1 * ip
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ido := n / l2
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idot := 2*ido + 2
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for j := 0; j < ip-1; j++ {
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i1 := i
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wa[i-1] = 1
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wa[i] = 0
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ld += l1
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var fi float64
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argld := float64(ld) * argh
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for ii := 3; ii < idot; ii += 2 {
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i += 2
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fi++
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arg := fi * argld
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wa[i-1] = math.Cos(arg)
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wa[i] = math.Sin(arg)
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}
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if ip > 5 {
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wa[i1-1] = wa[i-1]
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wa[i1] = wa[i]
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}
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}
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l1 = l2
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}
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}
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// Cfftf computes the forward complex Discrete Fourier transform
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// (the Fourier analysis). Equivalently, Cfftf computes the
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// Fourier coefficients of a complex periodic sequence. The
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// transform is defined below at output parameter c.
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//
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// Input parameters:
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//
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// n The length of the array c to be transformed. The method
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// is most efficient when n is a product of small primes.
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// n may change so long as different work arrays are provided.
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//
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// c A complex array of length n which contains the sequence
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// to be transformed.
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//
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// work A real work array which must be dimensioned at least 4*n.
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// in the program that calls Cfftf. The work array must be
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// initialized by calling subroutine Cffti(n,work,ifac) and a
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// different work array must be used for each different
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// value of n. This initialization does not have to be
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// repeated so long as n remains unchanged thus subsequent
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// transforms can be obtained faster than the first.
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// the same work array can be used by Cfftf and Cfftb.
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//
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// ifac A work array containing the factors of n. ifac must have
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// length of at least 15.
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//
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// Output parameters:
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//
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// c for j=0, ..., n-1
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// c[j]=the sum from k=0, ..., n-1 of
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// c[k]*exp(-i*j*k*2*pi/n)
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//
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// where i=sqrt(-1)
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//
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// This transform is unnormalized since a call of Cfftf
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// followed by a call of Cfftb will multiply the input
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// sequence by n.
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//
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// The n elements of c are represented in n pairs of real
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// values in r where c[j] = r[j*2]+r[j*2+1]i.
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//
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// work Contains results which must not be destroyed between
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// calls of Cfftf or Cfftb.
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// ifac Contains results which must not be destroyed between
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// calls of Cfftf or Cfftb.
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func Cfftf(n int, r, work []float64, ifac []int) {
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if len(r) < 2*n {
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panic("fourier: short sequence")
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}
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if len(work) < 4*n {
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panic("fourier: short work")
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}
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if len(ifac) < 15 {
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panic("fourier: short ifac")
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}
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if n == 1 {
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return
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}
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cfft1(n, r[:2*n], work[:2*n], work[2*n:4*n], ifac[:15], -1)
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}
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// Cfftb computes the backward complex Discrete Fourier Transform
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// (the Fourier synthesis). Equivalently, Cfftf computes the computes
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// a complex periodic sequence from its Fourier coefficients. The
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// transform is defined below at output parameter c.
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//
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// Input parameters:
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//
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// n The length of the array c to be transformed. The method
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// is most efficient when n is a product of small primes.
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// n may change so long as different work arrays are provided.
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//
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// c A complex array of length n which contains the sequence
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// to be transformed.
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//
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// work A real work array which must be dimensioned at least 4*n.
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// in the program that calls Cfftb. The work array must be
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// initialized by calling subroutine Cffti(n,work,ifac) and a
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// different work array must be used for each different
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// value of n. This initialization does not have to be
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// repeated so long as n remains unchanged thus subsequent
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// transforms can be obtained faster than the first.
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// The same work array can be used by Cfftf and Cfftb.
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//
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// ifac A work array containing the factors of n. ifac must have
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// length of at least 15.
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//
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// Output parameters:
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//
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// c for j=0, ..., n-1
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// c[j]=the sum from k=0, ..., n-1 of
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// c[k]*exp(i*j*k*2*pi/n)
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//
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// where i=sqrt(-1)
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//
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// This transform is unnormalized since a call of Cfftf
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// followed by a call of Cfftb will multiply the input
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// sequence by n.
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//
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// The n elements of c are represented in n pairs of real
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// values in r where c[j] = r[j*2]+r[j*2+1]i.
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//
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// work Contains results which must not be destroyed between
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// calls of Cfftf or Cfftb.
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// ifac Contains results which must not be destroyed between
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// calls of Cfftf or Cfftb.
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func Cfftb(n int, c, work []float64, ifac []int) {
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if len(c) < 2*n {
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panic("fourier: short sequence")
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}
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if len(work) < 4*n {
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panic("fourier: short work")
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}
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if len(ifac) < 15 {
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panic("fourier: short ifac")
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}
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if n == 1 {
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return
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}
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cfft1(n, c[:2*n], work[:2*n], work[2*n:4*n], ifac[:15], 1)
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}
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// cfft1 implements cfftf1 and cfftb1 depending on sign.
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func cfft1(n int, c, ch, wa []float64, ifac []int, sign float64) {
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nf := ifac[1]
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na := false
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l1 := 1
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iw := 0
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for k1 := 1; k1 <= nf; k1++ {
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ip := ifac[k1+1]
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l2 := ip * l1
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ido := n / l2
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idot := 2 * ido
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idl1 := idot * l1
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switch ip {
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case 4:
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ix2 := iw + idot
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ix3 := ix2 + idot
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if na {
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pass4(idot, l1, ch, c, wa[iw:], wa[ix2:], wa[ix3:], sign)
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} else {
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pass4(idot, l1, c, ch, wa[iw:], wa[ix2:], wa[ix3:], sign)
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}
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na = !na
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case 2:
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if na {
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pass2(idot, l1, ch, c, wa[iw:], sign)
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} else {
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pass2(idot, l1, c, ch, wa[iw:], sign)
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}
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na = !na
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case 3:
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ix2 := iw + idot
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if na {
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pass3(idot, l1, ch, c, wa[iw:], wa[ix2:], sign)
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} else {
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pass3(idot, l1, c, ch, wa[iw:], wa[ix2:], sign)
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}
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na = !na
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case 5:
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ix2 := iw + idot
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ix3 := ix2 + idot
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ix4 := ix3 + idot
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if na {
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pass5(idot, l1, ch, c, wa[iw:], wa[ix2:], wa[ix3:], wa[ix4:], sign)
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} else {
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pass5(idot, l1, c, ch, wa[iw:], wa[ix2:], wa[ix3:], wa[ix4:], sign)
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}
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na = !na
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default:
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var nac bool
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if na {
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nac = pass(idot, ip, l1, idl1, ch, ch, ch, c, c, wa[iw:], sign)
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} else {
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nac = pass(idot, ip, l1, idl1, c, c, c, ch, ch, wa[iw:], sign)
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}
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if nac {
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na = !na
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}
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}
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l1 = l2
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iw += (ip - 1) * idot
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}
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if na {
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for i := 0; i < 2*n; i++ {
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c[i] = ch[i]
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}
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}
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}
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// pass2 implements passf2 and passb2 depending on sign.
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func pass2(ido, l1 int, cc, ch, wa1 []float64, sign float64) {
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cc3 := newThreeArray(ido, 2, l1, cc)
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ch3 := newThreeArray(ido, l1, 2, ch)
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if ido <= 2 {
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for k := 0; k < l1; k++ {
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ch3.setCmplx(0, k, 0, cc3.atCmplx(0, 0, k)+cc3.atCmplx(0, 1, k))
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ch3.setCmplx(0, k, 1, cc3.atCmplx(0, 0, k)-cc3.atCmplx(0, 1, k))
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}
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return
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}
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for k := 0; k < l1; k++ {
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for i := 1; i < ido; i += 2 {
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ch3.setCmplx(i-1, k, 0, cc3.atCmplx(i-1, 0, k)+cc3.atCmplx(i-1, 1, k))
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t2 := cc3.atCmplx(i-1, 0, k) - cc3.atCmplx(i-1, 1, k)
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ch3.setCmplx(i-1, k, 1, complex(wa1[i-1], sign*wa1[i])*t2)
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}
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}
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}
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// pass3 implements passf3 and passb3 depending on sign.
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func pass3(ido, l1 int, cc, ch, wa1, wa2 []float64, sign float64) {
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const (
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taur = -0.5
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taui = 0.866025403784439 // sqrt(3)/2
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)
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cc3 := newThreeArray(ido, 3, l1, cc)
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ch3 := newThreeArray(ido, l1, 3, ch)
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if ido == 2 {
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for k := 0; k < l1; k++ {
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t2 := cc3.atCmplx(0, 1, k) + cc3.atCmplx(0, 2, k)
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ch3.setCmplx(0, k, 0, cc3.atCmplx(0, 0, k)+t2)
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c2 := cc3.atCmplx(0, 0, k) + scale(taur, t2)
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c3 := cmplx.Conj(swap(scale(sign*taui, cc3.atCmplx(0, 1, k)-cc3.atCmplx(0, 2, k))))
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ch3.setCmplx(0, k, 1, c2-c3)
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ch3.setCmplx(0, k, 2, c2+c3)
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}
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return
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}
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for k := 0; k < l1; k++ {
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for i := 1; i < ido; i += 2 {
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t2 := cc3.atCmplx(i-1, 1, k) + cc3.atCmplx(i-1, 2, k)
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ch3.setCmplx(i-1, k, 0, cc3.atCmplx(i-1, 0, k)+t2)
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c2 := cc3.atCmplx(i-1, 0, k) + scale(taur, t2)
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c3 := cmplx.Conj(swap(scale(sign*taui, cc3.atCmplx(i-1, 1, k)-cc3.atCmplx(i-1, 2, k))))
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d2 := c2 - c3
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d3 := c2 + c3
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ch3.setCmplx(i-1, k, 1, complex(wa1[i-1], sign*wa1[i])*d2)
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ch3.setCmplx(i-1, k, 2, complex(wa2[i-1], sign*wa2[i])*d3)
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}
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}
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}
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// pass4 implements passf4 and passb4 depending on sign.
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func pass4(ido, l1 int, cc, ch, wa1, wa2, wa3 []float64, sign float64) {
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cc3 := newThreeArray(ido, 4, l1, cc)
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ch3 := newThreeArray(ido, l1, 4, ch)
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if ido == 2 {
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for k := 0; k < l1; k++ {
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t1 := cc3.atCmplx(0, 0, k) - cc3.atCmplx(0, 2, k)
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t2 := cc3.atCmplx(0, 0, k) + cc3.atCmplx(0, 2, k)
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t3 := cc3.atCmplx(0, 1, k) + cc3.atCmplx(0, 3, k)
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t4 := cmplx.Conj(swap(scale(sign, cc3.atCmplx(0, 3, k)-cc3.atCmplx(0, 1, k))))
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ch3.setCmplx(0, k, 0, t2+t3)
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ch3.setCmplx(0, k, 1, t1+t4)
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ch3.setCmplx(0, k, 2, t2-t3)
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ch3.setCmplx(0, k, 3, t1-t4)
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}
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return
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}
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for k := 0; k < l1; k++ {
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for i := 1; i < ido; i += 2 {
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t1 := cc3.atCmplx(i-1, 0, k) - cc3.atCmplx(i-1, 2, k)
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t2 := cc3.atCmplx(i-1, 0, k) + cc3.atCmplx(i-1, 2, k)
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t3 := cc3.atCmplx(i-1, 1, k) + cc3.atCmplx(i-1, 3, k)
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t4 := cmplx.Conj(swap(scale(sign, cc3.atCmplx(i-1, 3, k)-cc3.atCmplx(i-1, 1, k))))
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ch3.setCmplx(i-1, k, 0, t2+t3)
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c2 := t1 + t4
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c3 := t2 - t3
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c4 := t1 - t4
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ch3.setCmplx(i-1, k, 1, complex(wa1[i-1], sign*wa1[i])*c2)
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ch3.setCmplx(i-1, k, 2, complex(wa2[i-1], sign*wa2[i])*c3)
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ch3.setCmplx(i-1, k, 3, complex(wa3[i-1], sign*wa3[i])*c4)
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}
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}
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}
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// pass5 implements passf5 and passb5 depending on sign.
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func pass5(ido, l1 int, cc, ch, wa1, wa2, wa3, wa4 []float64, sign float64) {
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const (
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tr11 = 0.309016994374947
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ti11 = 0.951056516295154
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tr12 = -0.809016994374947
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ti12 = 0.587785252292473
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)
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cc3 := newThreeArray(ido, 5, l1, cc)
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ch3 := newThreeArray(ido, l1, 5, ch)
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if ido == 2 {
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for k := 0; k < l1; k++ {
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t2 := cc3.atCmplx(0, 1, k) + cc3.atCmplx(0, 4, k)
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t3 := cc3.atCmplx(0, 2, k) + cc3.atCmplx(0, 3, k)
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t4 := cc3.atCmplx(0, 2, k) - cc3.atCmplx(0, 3, k)
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t5 := cc3.atCmplx(0, 1, k) - cc3.atCmplx(0, 4, k)
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ch3.setCmplx(0, k, 0, cc3.atCmplx(0, 0, k)+t2+t3)
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c2 := cc3.atCmplx(0, 0, k) + scale(tr11, t2) + scale(tr12, t3)
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c3 := cc3.atCmplx(0, 0, k) + scale(tr12, t2) + scale(tr11, t3)
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c4 := cmplx.Conj(swap(scale(sign, scale(ti12, t5)-scale(ti11, t4))))
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c5 := cmplx.Conj(swap(scale(sign, scale(ti11, t5)+scale(ti12, t4))))
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ch3.setCmplx(0, k, 1, c2-c5)
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ch3.setCmplx(0, k, 2, c3-c4)
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ch3.setCmplx(0, k, 3, c3+c4)
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ch3.setCmplx(0, k, 4, c2+c5)
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}
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return
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}
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for k := 0; k < l1; k++ {
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for i := 1; i < ido; i += 2 {
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t2 := cc3.atCmplx(i-1, 1, k) + cc3.atCmplx(i-1, 4, k)
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t3 := cc3.atCmplx(i-1, 2, k) + cc3.atCmplx(i-1, 3, k)
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t4 := cc3.atCmplx(i-1, 2, k) - cc3.atCmplx(i-1, 3, k)
|
|
t5 := cc3.atCmplx(i-1, 1, k) - cc3.atCmplx(i-1, 4, k)
|
|
ch3.setCmplx(i-1, k, 0, cc3.atCmplx(i-1, 0, k)+t2+t3)
|
|
|
|
c2 := cc3.atCmplx(i-1, 0, k) + scale(tr11, t2) + scale(tr12, t3)
|
|
c3 := cc3.atCmplx(i-1, 0, k) + scale(tr12, t2) + scale(tr11, t3)
|
|
c4 := cmplx.Conj(swap(scale(sign, scale(ti12, t5)-scale(ti11, t4))))
|
|
c5 := cmplx.Conj(swap(scale(sign, scale(ti11, t5)+scale(ti12, t4))))
|
|
d2 := c2 - c5
|
|
d3 := c3 - c4
|
|
d4 := c3 + c4
|
|
d5 := c2 + c5
|
|
ch3.setCmplx(i-1, k, 1, complex(wa1[i-1], sign*wa1[i])*d2)
|
|
ch3.setCmplx(i-1, k, 2, complex(wa2[i-1], sign*wa2[i])*d3)
|
|
ch3.setCmplx(i-1, k, 3, complex(wa3[i-1], sign*wa3[i])*d4)
|
|
ch3.setCmplx(i-1, k, 4, complex(wa4[i-1], sign*wa4[i])*d5)
|
|
}
|
|
}
|
|
}
|
|
|
|
// pass implements passf and passb depending on sign.
|
|
func pass(ido, ip, l1, idl1 int, cc, c1, c2, ch, ch2, wa []float64, sign float64) (nac bool) {
|
|
cc3 := newThreeArray(ido, ip, l1, cc)
|
|
c13 := newThreeArray(ido, l1, ip, c1)
|
|
ch3 := newThreeArray(ido, l1, ip, ch)
|
|
c2m := newTwoArray(idl1, ip, c2)
|
|
ch2m := newTwoArray(idl1, ip, ch2)
|
|
|
|
idot := ido / 2
|
|
ipph := (ip + 1) / 2
|
|
idp := ip * ido
|
|
|
|
if ido < l1 {
|
|
for j := 1; j < ipph; j++ {
|
|
jc := ip - j
|
|
for i := 0; i < ido; i++ {
|
|
for k := 0; k < l1; k++ {
|
|
ch3.set(i, k, j, cc3.at(i, j, k)+cc3.at(i, jc, k))
|
|
ch3.set(i, k, jc, cc3.at(i, j, k)-cc3.at(i, jc, k))
|
|
}
|
|
}
|
|
}
|
|
for i := 0; i < ido; i++ {
|
|
for k := 0; k < l1; k++ {
|
|
ch3.set(i, k, 0, cc3.at(i, 0, k))
|
|
}
|
|
}
|
|
} else {
|
|
for j := 1; j < ipph; j++ {
|
|
jc := ip - j
|
|
for k := 0; k < l1; k++ {
|
|
for i := 0; i < ido; i++ {
|
|
ch3.set(i, k, j, cc3.at(i, j, k)+cc3.at(i, jc, k))
|
|
ch3.set(i, k, jc, cc3.at(i, j, k)-cc3.at(i, jc, k))
|
|
}
|
|
}
|
|
}
|
|
for k := 0; k < l1; k++ {
|
|
for i := 0; i < ido; i++ {
|
|
ch3.set(i, k, 0, cc3.at(i, 0, k))
|
|
}
|
|
}
|
|
}
|
|
|
|
idl := 1 - ido
|
|
inc := 0
|
|
for l := 1; l < ipph; l++ {
|
|
lc := ip - l
|
|
idl += ido
|
|
for ik := 0; ik < idl1; ik++ {
|
|
c2m.set(ik, l, ch2m.at(ik, 0)+wa[idl-1]*ch2m.at(ik, 1))
|
|
c2m.set(ik, lc, sign*wa[idl]*ch2m.at(ik, ip-1))
|
|
}
|
|
idlj := idl
|
|
inc += ido
|
|
for j := 2; j < ipph; j++ {
|
|
jc := ip - j
|
|
idlj += inc
|
|
if idlj > idp {
|
|
idlj -= idp
|
|
}
|
|
war := wa[idlj-1]
|
|
wai := wa[idlj]
|
|
for ik := 0; ik < idl1; ik++ {
|
|
c2m.add(ik, l, war*ch2m.at(ik, j))
|
|
c2m.add(ik, lc, sign*wai*ch2m.at(ik, jc))
|
|
}
|
|
}
|
|
}
|
|
|
|
for j := 1; j < ipph; j++ {
|
|
for ik := 0; ik < idl1; ik++ {
|
|
ch2m.add(ik, 0, ch2m.at(ik, j))
|
|
}
|
|
}
|
|
|
|
for j := 1; j < ipph; j++ {
|
|
jc := ip - j
|
|
for ik := 1; ik < idl1; ik += 2 {
|
|
ch2m.setCmplx(ik-1, j, c2m.atCmplx(ik-1, j)-cmplx.Conj(swap(c2m.atCmplx(ik-1, jc))))
|
|
ch2m.setCmplx(ik-1, jc, c2m.atCmplx(ik-1, j)+cmplx.Conj(swap(c2m.atCmplx(ik-1, jc))))
|
|
}
|
|
}
|
|
|
|
if ido == 2 {
|
|
return true
|
|
}
|
|
|
|
for ik := 0; ik < idl1; ik++ {
|
|
c2m.set(ik, 0, ch2m.at(ik, 0))
|
|
}
|
|
|
|
for j := 1; j < ip; j++ {
|
|
for k := 0; k < l1; k++ {
|
|
c13.setCmplx(0, k, j, ch3.atCmplx(0, k, j))
|
|
}
|
|
}
|
|
|
|
if idot > l1 {
|
|
idj := 1 - ido
|
|
for j := 1; j < ip; j++ {
|
|
idj += ido
|
|
for k := 0; k < l1; k++ {
|
|
idij := idj
|
|
for i := 3; i < ido; i += 2 {
|
|
idij += 2
|
|
c13.setCmplx(i-1, k, j, complex(wa[idij-1], sign*wa[idij])*ch3.atCmplx(i-1, k, j))
|
|
}
|
|
}
|
|
}
|
|
|
|
return false
|
|
}
|
|
|
|
idij := -1
|
|
for j := 1; j < ip; j++ {
|
|
idij += 2
|
|
for i := 3; i < ido; i += 2 {
|
|
idij += 2
|
|
for k := 0; k < l1; k++ {
|
|
c13.setCmplx(i-1, k, j, complex(wa[idij-1], sign*wa[idij])*ch3.atCmplx(i-1, k, j))
|
|
}
|
|
}
|
|
}
|
|
return false
|
|
}
|