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653 lines
18 KiB
Go
653 lines
18 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 "math"
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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, 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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// 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.set(0, k, 0, cc3.at(0, 0, k)+cc3.at(0, 1, k))
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ch3.set(0, k, 1, cc3.at(0, 0, k)-cc3.at(0, 1, k))
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ch3.set(1, k, 0, cc3.at(1, 0, k)+cc3.at(1, 1, k))
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ch3.set(1, k, 1, cc3.at(1, 0, k)-cc3.at(1, 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.set(i-1, k, 0, cc3.at(i-1, 0, k)+cc3.at(i-1, 1, k))
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tr2 := cc3.at(i-1, 0, k) - cc3.at(i-1, 1, k)
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ch3.set(i, k, 0, cc3.at(i, 0, k)+cc3.at(i, 1, k))
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ti2 := cc3.at(i, 0, k) - cc3.at(i, 1, k)
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ch3.set(i, k, 1, wa1[i-1]*ti2+sign*wa1[i]*tr2)
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ch3.set(i-1, k, 1, wa1[i-1]*tr2-sign*wa1[i]*ti2)
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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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tr2 := cc3.at(0, 1, k) + cc3.at(0, 2, k)
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cr2 := cc3.at(0, 0, k) + taur*tr2
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ch3.set(0, k, 0, cc3.at(0, 0, k)+tr2)
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ti2 := cc3.at(1, 1, k) + cc3.at(1, 2, k)
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ci2 := cc3.at(1, 0, k) + taur*ti2
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ch3.set(1, k, 0, cc3.at(1, 0, k)+ti2)
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cr3 := sign * taui * (cc3.at(0, 1, k) - cc3.at(0, 2, k))
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ci3 := sign * taui * (cc3.at(1, 1, k) - cc3.at(1, 2, k))
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ch3.set(0, k, 1, cr2-ci3)
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ch3.set(0, k, 2, cr2+ci3)
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ch3.set(1, k, 1, ci2+cr3)
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ch3.set(1, k, 2, ci2-cr3)
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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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tr2 := cc3.at(i-1, 1, k) + cc3.at(i-1, 2, k)
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cr2 := cc3.at(i-1, 0, k) + taur*tr2
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ch3.set(i-1, k, 0, cc3.at(i-1, 0, k)+tr2)
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ti2 := cc3.at(i, 1, k) + cc3.at(i, 2, k)
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ci2 := cc3.at(i, 0, k) + taur*ti2
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ch3.set(i, k, 0, cc3.at(i, 0, k)+ti2)
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cr3 := sign * taui * (cc3.at(i-1, 1, k) - cc3.at(i-1, 2, k))
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ci3 := sign * taui * (cc3.at(i, 1, k) - cc3.at(i, 2, k))
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dr2 := cr2 - ci3
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dr3 := cr2 + ci3
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di2 := ci2 + cr3
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di3 := ci2 - cr3
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ch3.set(i, k, 1, wa1[i-1]*di2+sign*wa1[i]*dr2)
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ch3.set(i-1, k, 1, wa1[i-1]*dr2-sign*wa1[i]*di2)
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ch3.set(i, k, 2, wa2[i-1]*di3+sign*wa2[i]*dr3)
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ch3.set(i-1, k, 2, wa2[i-1]*dr3-sign*wa2[i]*di3)
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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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ti1 := cc3.at(1, 0, k) - cc3.at(1, 2, k)
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ti2 := cc3.at(1, 0, k) + cc3.at(1, 2, k)
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tr4 := sign * (cc3.at(1, 3, k) - cc3.at(1, 1, k))
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ti3 := cc3.at(1, 1, k) + cc3.at(1, 3, k)
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tr1 := cc3.at(0, 0, k) - cc3.at(0, 2, k)
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tr2 := cc3.at(0, 0, k) + cc3.at(0, 2, k)
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ti4 := sign * (cc3.at(0, 1, k) - cc3.at(0, 3, k))
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tr3 := cc3.at(0, 1, k) + cc3.at(0, 3, k)
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ch3.set(0, k, 0, tr2+tr3)
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ch3.set(0, k, 2, tr2-tr3)
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ch3.set(1, k, 0, ti2+ti3)
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ch3.set(1, k, 2, ti2-ti3)
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ch3.set(0, k, 1, tr1+tr4)
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ch3.set(0, k, 3, tr1-tr4)
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ch3.set(1, k, 1, ti1+ti4)
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ch3.set(1, k, 3, ti1-ti4)
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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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ti1 := cc3.at(i, 0, k) - cc3.at(i, 2, k)
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ti2 := cc3.at(i, 0, k) + cc3.at(i, 2, k)
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ti3 := cc3.at(i, 1, k) + cc3.at(i, 3, k)
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tr4 := sign * (cc3.at(i, 3, k) - cc3.at(i, 1, k))
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tr1 := cc3.at(i-1, 0, k) - cc3.at(i-1, 2, k)
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tr2 := cc3.at(i-1, 0, k) + cc3.at(i-1, 2, k)
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ti4 := sign * (cc3.at(i-1, 1, k) - cc3.at(i-1, 3, k))
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tr3 := cc3.at(i-1, 1, k) + cc3.at(i-1, 3, k)
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ch3.set(i-1, k, 0, tr2+tr3)
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cr3 := tr2 - tr3
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ch3.set(i, k, 0, ti2+ti3)
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ci3 := ti2 - ti3
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cr2 := tr1 + tr4
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cr4 := tr1 - tr4
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ci2 := ti1 + ti4
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ci4 := ti1 - ti4
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ch3.set(i-1, k, 1, wa1[i-1]*cr2-sign*wa1[i]*ci2)
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ch3.set(i, k, 1, wa1[i-1]*ci2+sign*wa1[i]*cr2)
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ch3.set(i-1, k, 2, wa2[i-1]*cr3-sign*wa2[i]*ci3)
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ch3.set(i, k, 2, wa2[i-1]*ci3+sign*wa2[i]*cr3)
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ch3.set(i-1, k, 3, wa3[i-1]*cr4-sign*wa3[i]*ci4)
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ch3.set(i, k, 3, wa3[i-1]*ci4+sign*wa3[i]*cr4)
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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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ti5 := cc3.at(1, 1, k) - cc3.at(1, 4, k)
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ti2 := cc3.at(1, 1, k) + cc3.at(1, 4, k)
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ti4 := cc3.at(1, 2, k) - cc3.at(1, 3, k)
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ti3 := cc3.at(1, 2, k) + cc3.at(1, 3, k)
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tr5 := cc3.at(0, 1, k) - cc3.at(0, 4, k)
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tr2 := cc3.at(0, 1, k) + cc3.at(0, 4, k)
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tr4 := cc3.at(0, 2, k) - cc3.at(0, 3, k)
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tr3 := cc3.at(0, 2, k) + cc3.at(0, 3, k)
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ch3.set(0, k, 0, cc3.at(0, 0, k)+tr2+tr3)
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ch3.set(1, k, 0, cc3.at(1, 0, k)+ti2+ti3)
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cr2 := cc3.at(0, 0, k) + tr11*tr2 + tr12*tr3
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ci2 := cc3.at(1, 0, k) + tr11*ti2 + tr12*ti3
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cr3 := cc3.at(0, 0, k) + tr12*tr2 + tr11*tr3
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ci3 := cc3.at(1, 0, k) + tr12*ti2 + tr11*ti3
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cr5 := sign * (ti11*tr5 + ti12*tr4)
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ci5 := sign * (ti11*ti5 + ti12*ti4)
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cr4 := sign * (ti12*tr5 - ti11*tr4)
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ci4 := sign * (ti12*ti5 - ti11*ti4)
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ch3.set(0, k, 1, cr2-ci5)
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ch3.set(0, k, 4, cr2+ci5)
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ch3.set(1, k, 1, ci2+cr5)
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ch3.set(1, k, 2, ci3+cr4)
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ch3.set(0, k, 2, cr3-ci4)
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ch3.set(0, k, 3, cr3+ci4)
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ch3.set(1, k, 3, ci3-cr4)
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ch3.set(1, k, 4, ci2-cr5)
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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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ti5 := cc3.at(i, 1, k) - cc3.at(i, 4, k)
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ti2 := cc3.at(i, 1, k) + cc3.at(i, 4, k)
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ti4 := cc3.at(i, 2, k) - cc3.at(i, 3, k)
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ti3 := cc3.at(i, 2, k) + cc3.at(i, 3, k)
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tr5 := cc3.at(i-1, 1, k) - cc3.at(i-1, 4, k)
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tr2 := cc3.at(i-1, 1, k) + cc3.at(i-1, 4, k)
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tr4 := cc3.at(i-1, 2, k) - cc3.at(i-1, 3, k)
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tr3 := cc3.at(i-1, 2, k) + cc3.at(i-1, 3, k)
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ch3.set(i-1, k, 0, cc3.at(i-1, 0, k)+tr2+tr3)
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ch3.set(i, k, 0, cc3.at(i, 0, k)+ti2+ti3)
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cr2 := cc3.at(i-1, 0, k) + tr11*tr2 + tr12*tr3
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ci2 := cc3.at(i, 0, k) + tr11*ti2 + tr12*ti3
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cr3 := cc3.at(i-1, 0, k) + tr12*tr2 + tr11*tr3
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ci3 := cc3.at(i, 0, k) + tr12*ti2 + tr11*ti3
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cr5 := sign * (ti11*tr5 + ti12*tr4)
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ci5 := sign * (ti11*ti5 + ti12*ti4)
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cr4 := sign * (ti12*tr5 - ti11*tr4)
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ci4 := sign * (ti12*ti5 - ti11*ti4)
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dr3 := cr3 - ci4
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dr4 := cr3 + ci4
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di3 := ci3 + cr4
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di4 := ci3 - cr4
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dr5 := cr2 + ci5
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dr2 := cr2 - ci5
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di5 := ci2 - cr5
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di2 := ci2 + cr5
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ch3.set(i-1, k, 1, wa1[i-1]*dr2-sign*wa1[i]*di2)
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ch3.set(i, k, 1, wa1[i-1]*di2+sign*wa1[i]*dr2)
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ch3.set(i-1, k, 2, wa2[i-1]*dr3-sign*wa2[i]*di3)
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ch3.set(i, k, 2, wa2[i-1]*di3+sign*wa2[i]*dr3)
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ch3.set(i-1, k, 3, wa3[i-1]*dr4-sign*wa3[i]*di4)
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ch3.set(i, k, 3, wa3[i-1]*di4+sign*wa3[i]*dr4)
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ch3.set(i-1, k, 4, wa4[i-1]*dr5-sign*wa4[i]*di5)
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ch3.set(i, k, 4, wa4[i-1]*di5+sign*wa4[i]*dr5)
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}
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}
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}
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// pass implements passf and passb depending on sign.
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func pass(ido, ip, l1, idl1 int, cc, c1, c2, ch, ch2, wa []float64, sign float64) (nac bool) {
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cc3 := newThreeArray(ido, ip, l1, cc)
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c13 := newThreeArray(ido, l1, ip, c1)
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ch3 := newThreeArray(ido, l1, ip, ch)
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c2m := newTwoArray(idl1, ip, c2)
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ch2m := newTwoArray(idl1, ip, ch2)
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idot := ido / 2
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ipph := (ip + 1) / 2
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idp := ip * ido
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if ido < l1 {
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for j := 1; j < ipph; j++ {
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jc := ip - j
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for i := 0; i < ido; i++ {
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for k := 0; k < l1; k++ {
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ch3.set(i, k, j, cc3.at(i, j, k)+cc3.at(i, jc, k))
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ch3.set(i, k, jc, cc3.at(i, j, k)-cc3.at(i, jc, k))
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}
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}
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}
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for i := 0; i < ido; i++ {
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for k := 0; k < l1; k++ {
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ch3.set(i, k, 0, cc3.at(i, 0, k))
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}
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}
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} else {
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for j := 1; j < ipph; j++ {
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jc := ip - j
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for k := 0; k < l1; k++ {
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for i := 0; i < ido; i++ {
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ch3.set(i, k, j, cc3.at(i, j, k)+cc3.at(i, jc, k))
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ch3.set(i, k, jc, cc3.at(i, j, k)-cc3.at(i, jc, k))
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}
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}
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}
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for k := 0; k < l1; k++ {
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for i := 0; i < ido; i++ {
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ch3.set(i, k, 0, cc3.at(i, 0, k))
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}
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}
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}
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idl := 1 - ido
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inc := 0
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for l := 1; l < ipph; l++ {
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lc := ip - l
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idl += ido
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for ik := 0; ik < idl1; ik++ {
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c2m.set(ik, l, ch2m.at(ik, 0)+wa[idl-1]*ch2m.at(ik, 1))
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c2m.set(ik, lc, sign*wa[idl]*ch2m.at(ik, ip-1))
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}
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idlj := idl
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inc += ido
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for j := 2; j < ipph; j++ {
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jc := ip - j
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idlj += inc
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if idlj > idp {
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idlj -= idp
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}
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war := wa[idlj-1]
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wai := wa[idlj]
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for ik := 0; ik < idl1; ik++ {
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c2m.add(ik, l, war*ch2m.at(ik, j))
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c2m.add(ik, lc, sign*wai*ch2m.at(ik, jc))
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}
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}
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}
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for j := 1; j < ipph; j++ {
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for ik := 0; ik < idl1; ik++ {
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ch2m.add(ik, 0, ch2m.at(ik, j))
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}
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}
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for j := 1; j < ipph; j++ {
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jc := ip - j
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for ik := 1; ik < idl1; ik += 2 {
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ch2m.set(ik-1, j, c2m.at(ik-1, j)-c2m.at(ik, jc))
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ch2m.set(ik-1, jc, c2m.at(ik-1, j)+c2m.at(ik, jc))
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ch2m.set(ik, j, c2m.at(ik, j)+c2m.at(ik-1, jc))
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ch2m.set(ik, jc, c2m.at(ik, j)-c2m.at(ik-1, jc))
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}
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}
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if ido == 2 {
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return true
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}
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for ik := 0; ik < idl1; ik++ {
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c2m.set(ik, 0, ch2m.at(ik, 0))
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}
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for j := 1; j < ip; j++ {
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for k := 0; k < l1; k++ {
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c13.set(0, k, j, ch3.at(0, k, j))
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c13.set(1, k, j, ch3.at(1, k, j))
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}
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}
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if idot > l1 {
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idj := 1 - ido
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for j := 1; j < ip; j++ {
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idj += ido
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for k := 0; k < l1; k++ {
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idij := idj
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for i := 3; i < ido; i += 2 {
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idij += 2
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c13.set(i-1, k, j, wa[idij-1]*ch3.at(i-1, k, j)-sign*wa[idij]*ch3.at(i, k, j))
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c13.set(i, k, j, wa[idij-1]*ch3.at(i, k, j)+sign*wa[idij]*ch3.at(i-1, k, j))
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}
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}
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}
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return false
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}
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idij := -1
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for j := 1; j < ip; j++ {
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idij += 2
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for i := 3; i < ido; i += 2 {
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idij += 2
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for k := 0; k < l1; k++ {
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c13.set(i-1, k, j, wa[idij-1]*ch3.at(i-1, k, j)-sign*wa[idij]*ch3.at(i, k, j))
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c13.set(i, k, j, wa[idij-1]*ch3.at(i, k, j)+sign*wa[idij]*ch3.at(i-1, k, j))
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}
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}
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}
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return false
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}
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