/
cfft.go
652 lines (611 loc) · 17.6 KB
/
cfft.go
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// Copyright ©2018 The Gonum Authors. All rights reserved.
// Use of this source code is governed by a BSD-style
// license that can be found in the LICENSE file.
// This is a translation of the FFTPACK cfft functions by
// Paul N Swarztrauber, placed in the public domain at
// http://www.netlib.org/fftpack/.
package fftpack
import "math"
// Cffti initializes the array work which is used in both Cfftf
// and Cfftb. the prime factorization of n together with a
// tabulation of the trigonometric functions are computed and
// stored in work.
//
// input parameter
//
// n The length of the sequence to be transformed.
//
// Output parameters:
//
// work A work array which must be dimensioned at least 4*n.
// the same work array can be used for both Cfftf and Cfftb
// as long as n remains unchanged. Different work arrays
// are required for different values of n. The contents of
// work must not be changed between calls of Cfftf or Cfftb.
//
// ifac A work array containing the factors of n. ifac must have
// length 15.
func Cffti(n int, work []float64, ifac []int) {
if len(work) < 4*n {
panic("fourier: short work")
}
if len(ifac) < 15 {
panic("fourier: short ifac")
}
if n == 1 {
return
}
cffti1(n, work[2*n:4*n], ifac[:15])
}
func cffti1(n int, wa []float64, ifac []int) {
ntryh := [4]int{3, 4, 2, 5}
nl := n
nf := 0
outer:
for j, ntry := 0, 0; ; j++ {
if j < 4 {
ntry = ntryh[j]
} else {
ntry += 2
}
for {
if nl%ntry != 0 {
continue outer
}
ifac[nf+2] = ntry
nl /= ntry
nf++
if ntry == 2 && nf != 1 {
for i := 1; i < nf; i++ {
ib := nf - i + 1
ifac[ib+1] = ifac[ib]
}
ifac[2] = 2
}
if nl == 1 {
break outer
}
}
}
ifac[0] = n
ifac[1] = nf
argh := 2 * math.Pi / float64(n)
i := 1
l1 := 1
for k1 := 0; k1 < nf; k1++ {
ip := ifac[k1+2]
ld := 0
l2 := l1 * ip
ido := n / l2
idot := 2*ido + 2
for j := 0; j < ip-1; j++ {
i1 := i
wa[i-1] = 1
wa[i] = 0
ld += l1
var fi float64
argld := float64(ld) * argh
for ii := 3; ii < idot; ii += 2 {
i += 2
fi++
arg := fi * argld
wa[i-1] = math.Cos(arg)
wa[i] = math.Sin(arg)
}
if ip > 5 {
wa[i1-1] = wa[i-1]
wa[i1] = wa[i]
}
}
l1 = l2
}
}
// Cfftf computes the forward complex Discrete Fourier transform
// (the Fourier analysis). Equivalently, Cfftf computes the
// Fourier coefficients of a complex periodic sequence. The
// transform is defined below at output parameter c.
//
// Input parameters:
//
// n The length of the array c to be transformed. The method
// is most efficient when n is a product of small primes.
// n may change so long as different work arrays are provided.
//
// c A complex array of length n which contains the sequence
// to be transformed.
//
// work A real work array which must be dimensioned at least 4*n.
// in the program that calls Cfftf. The work array must be
// initialized by calling subroutine Cffti(n,work,ifac) and a
// different work array must be used for each different
// value of n. This initialization does not have to be
// repeated so long as n remains unchanged thus subsequent
// transforms can be obtained faster than the first.
// the same work array can be used by Cfftf and Cfftb.
//
// ifac A work array containing the factors of n. ifac must have
// length of at least 15.
//
// Output parameters:
//
// c for j=0, ..., n-1
// c[j]=the sum from k=0, ..., n-1 of
// c[k]*exp(-i*j*k*2*pi/n)
//
// where i=sqrt(-1)
//
// This transform is unnormalized since a call of Cfftf
// followed by a call of Cfftb will multiply the input
// sequence by n.
//
// The n elements of c are represented in n pairs of real
// values in r where c[j] = r[j*2]+r[j*2+1]i.
//
// work Contains results which must not be destroyed between
// calls of Cfftf or Cfftb.
// ifac Contains results which must not be destroyed between
// calls of Cfftf or Cfftb.
func Cfftf(n int, r, work []float64, ifac []int) {
if len(r) < 2*n {
panic("fourier: short sequence")
}
if len(work) < 4*n {
panic("fourier: short work")
}
if len(ifac) < 15 {
panic("fourier: short ifac")
}
if n == 1 {
return
}
cfft1(n, r[:2*n], work[:2*n], work[2*n:4*n], ifac[:15], -1)
}
// Cfftb computes the backward complex Discrete Fourier Transform
// (the Fourier synthesis). Equivalently, Cfftf computes the computes
// a complex periodic sequence from its Fourier coefficients. The
// transform is defined below at output parameter c.
//
// Input parameters:
//
// n The length of the array c to be transformed. The method
// is most efficient when n is a product of small primes.
// n may change so long as different work arrays are provided.
//
// c A complex array of length n which contains the sequence
// to be transformed.
//
// work A real work array which must be dimensioned at least 4*n.
// in the program that calls Cfftb. The work array must be
// initialized by calling subroutine Cffti(n,work,ifac) and a
// different work array must be used for each different
// value of n. This initialization does not have to be
// repeated so long as n remains unchanged thus subsequent
// transforms can be obtained faster than the first.
// The same work array can be used by Cfftf and Cfftb.
//
// ifac A work array containing the factors of n. ifac must have
// length of at least 15.
//
// Output parameters:
//
// c for j=0, ..., n-1
// c[j]=the sum from k=0, ..., n-1 of
// c[k]*exp(i*j*k*2*pi/n)
//
// where i=sqrt(-1)
//
// This transform is unnormalized since a call of Cfftf
// followed by a call of Cfftb will multiply the input
// sequence by n.
//
// The n elements of c are represented in n pairs of real
// values in r where c[j] = r[j*2]+r[j*2+1]i.
//
// work Contains results which must not be destroyed between
// calls of Cfftf or Cfftb.
// ifac Contains results which must not be destroyed between
// calls of Cfftf or Cfftb.
func Cfftb(n int, r, work []float64, ifac []int) {
if len(r) < 2*n {
panic("fourier: short sequence")
}
if len(work) < 4*n {
panic("fourier: short work")
}
if len(ifac) < 15 {
panic("fourier: short ifac")
}
if n == 1 {
return
}
cfft1(n, r[:2*n], work[:2*n], work[2*n:4*n], ifac[:15], 1)
}
// cfft1 implements cfftf1 and cfftb1 depending on sign.
func cfft1(n int, c, ch, wa []float64, ifac []int, sign float64) {
nf := ifac[1]
na := false
l1 := 1
iw := 0
for k1 := 1; k1 <= nf; k1++ {
ip := ifac[k1+1]
l2 := ip * l1
ido := n / l2
idot := 2 * ido
idl1 := idot * l1
switch ip {
case 4:
ix2 := iw + idot
ix3 := ix2 + idot
if na {
pass4(idot, l1, ch, c, wa[iw:], wa[ix2:], wa[ix3:], sign)
} else {
pass4(idot, l1, c, ch, wa[iw:], wa[ix2:], wa[ix3:], sign)
}
na = !na
case 2:
if na {
pass2(idot, l1, ch, c, wa[iw:], sign)
} else {
pass2(idot, l1, c, ch, wa[iw:], sign)
}
na = !na
case 3:
ix2 := iw + idot
if na {
pass3(idot, l1, ch, c, wa[iw:], wa[ix2:], sign)
} else {
pass3(idot, l1, c, ch, wa[iw:], wa[ix2:], sign)
}
na = !na
case 5:
ix2 := iw + idot
ix3 := ix2 + idot
ix4 := ix3 + idot
if na {
pass5(idot, l1, ch, c, wa[iw:], wa[ix2:], wa[ix3:], wa[ix4:], sign)
} else {
pass5(idot, l1, c, ch, wa[iw:], wa[ix2:], wa[ix3:], wa[ix4:], sign)
}
na = !na
default:
var nac bool
if na {
nac = pass(idot, ip, l1, idl1, ch, ch, ch, c, c, wa[iw:], sign)
} else {
nac = pass(idot, ip, l1, idl1, c, c, c, ch, ch, wa[iw:], sign)
}
if nac {
na = !na
}
}
l1 = l2
iw += (ip - 1) * idot
}
if na {
for i := 0; i < 2*n; i++ {
c[i] = ch[i]
}
}
}
// pass2 implements passf2 and passb2 depending on sign.
func pass2(ido, l1 int, cc, ch, wa1 []float64, sign float64) {
cc3 := newThreeArray(ido, 2, l1, cc)
ch3 := newThreeArray(ido, l1, 2, ch)
if ido <= 2 {
for k := 0; k < l1; k++ {
ch3.set(0, k, 0, cc3.at(0, 0, k)+cc3.at(0, 1, k))
ch3.set(0, k, 1, cc3.at(0, 0, k)-cc3.at(0, 1, k))
ch3.set(1, k, 0, cc3.at(1, 0, k)+cc3.at(1, 1, k))
ch3.set(1, k, 1, cc3.at(1, 0, k)-cc3.at(1, 1, k))
}
return
}
for k := 0; k < l1; k++ {
for i := 1; i < ido; i += 2 {
ch3.set(i-1, k, 0, cc3.at(i-1, 0, k)+cc3.at(i-1, 1, k))
tr2 := cc3.at(i-1, 0, k) - cc3.at(i-1, 1, k)
ch3.set(i, k, 0, cc3.at(i, 0, k)+cc3.at(i, 1, k))
ti2 := cc3.at(i, 0, k) - cc3.at(i, 1, k)
ch3.set(i, k, 1, wa1[i-1]*ti2+sign*wa1[i]*tr2)
ch3.set(i-1, k, 1, wa1[i-1]*tr2-sign*wa1[i]*ti2)
}
}
}
// pass3 implements passf3 and passb3 depending on sign.
func pass3(ido, l1 int, cc, ch, wa1, wa2 []float64, sign float64) {
const (
taur = -0.5
taui = 0.866025403784439 // sqrt(3)/2
)
cc3 := newThreeArray(ido, 3, l1, cc)
ch3 := newThreeArray(ido, l1, 3, ch)
if ido == 2 {
for k := 0; k < l1; k++ {
tr2 := cc3.at(0, 1, k) + cc3.at(0, 2, k)
cr2 := cc3.at(0, 0, k) + taur*tr2
ch3.set(0, k, 0, cc3.at(0, 0, k)+tr2)
ti2 := cc3.at(1, 1, k) + cc3.at(1, 2, k)
ci2 := cc3.at(1, 0, k) + taur*ti2
ch3.set(1, k, 0, cc3.at(1, 0, k)+ti2)
cr3 := sign * taui * (cc3.at(0, 1, k) - cc3.at(0, 2, k))
ci3 := sign * taui * (cc3.at(1, 1, k) - cc3.at(1, 2, k))
ch3.set(0, k, 1, cr2-ci3)
ch3.set(0, k, 2, cr2+ci3)
ch3.set(1, k, 1, ci2+cr3)
ch3.set(1, k, 2, ci2-cr3)
}
return
}
for k := 0; k < l1; k++ {
for i := 1; i < ido; i += 2 {
tr2 := cc3.at(i-1, 1, k) + cc3.at(i-1, 2, k)
cr2 := cc3.at(i-1, 0, k) + taur*tr2
ch3.set(i-1, k, 0, cc3.at(i-1, 0, k)+tr2)
ti2 := cc3.at(i, 1, k) + cc3.at(i, 2, k)
ci2 := cc3.at(i, 0, k) + taur*ti2
ch3.set(i, k, 0, cc3.at(i, 0, k)+ti2)
cr3 := sign * taui * (cc3.at(i-1, 1, k) - cc3.at(i-1, 2, k))
ci3 := sign * taui * (cc3.at(i, 1, k) - cc3.at(i, 2, k))
dr2 := cr2 - ci3
dr3 := cr2 + ci3
di2 := ci2 + cr3
di3 := ci2 - cr3
ch3.set(i, k, 1, wa1[i-1]*di2+sign*wa1[i]*dr2)
ch3.set(i-1, k, 1, wa1[i-1]*dr2-sign*wa1[i]*di2)
ch3.set(i, k, 2, wa2[i-1]*di3+sign*wa2[i]*dr3)
ch3.set(i-1, k, 2, wa2[i-1]*dr3-sign*wa2[i]*di3)
}
}
}
// pass4 implements passf4 and passb4 depending on sign.
func pass4(ido, l1 int, cc, ch, wa1, wa2, wa3 []float64, sign float64) {
cc3 := newThreeArray(ido, 4, l1, cc)
ch3 := newThreeArray(ido, l1, 4, ch)
if ido == 2 {
for k := 0; k < l1; k++ {
ti1 := cc3.at(1, 0, k) - cc3.at(1, 2, k)
ti2 := cc3.at(1, 0, k) + cc3.at(1, 2, k)
tr4 := sign * (cc3.at(1, 3, k) - cc3.at(1, 1, k))
ti3 := cc3.at(1, 1, k) + cc3.at(1, 3, k)
tr1 := cc3.at(0, 0, k) - cc3.at(0, 2, k)
tr2 := cc3.at(0, 0, k) + cc3.at(0, 2, k)
ti4 := sign * (cc3.at(0, 1, k) - cc3.at(0, 3, k))
tr3 := cc3.at(0, 1, k) + cc3.at(0, 3, k)
ch3.set(0, k, 0, tr2+tr3)
ch3.set(0, k, 2, tr2-tr3)
ch3.set(1, k, 0, ti2+ti3)
ch3.set(1, k, 2, ti2-ti3)
ch3.set(0, k, 1, tr1+tr4)
ch3.set(0, k, 3, tr1-tr4)
ch3.set(1, k, 1, ti1+ti4)
ch3.set(1, k, 3, ti1-ti4)
}
return
}
for k := 0; k < l1; k++ {
for i := 1; i < ido; i += 2 {
ti1 := cc3.at(i, 0, k) - cc3.at(i, 2, k)
ti2 := cc3.at(i, 0, k) + cc3.at(i, 2, k)
ti3 := cc3.at(i, 1, k) + cc3.at(i, 3, k)
tr4 := sign * (cc3.at(i, 3, k) - cc3.at(i, 1, k))
tr1 := cc3.at(i-1, 0, k) - cc3.at(i-1, 2, k)
tr2 := cc3.at(i-1, 0, k) + cc3.at(i-1, 2, k)
ti4 := sign * (cc3.at(i-1, 1, k) - cc3.at(i-1, 3, k))
tr3 := cc3.at(i-1, 1, k) + cc3.at(i-1, 3, k)
ch3.set(i-1, k, 0, tr2+tr3)
cr3 := tr2 - tr3
ch3.set(i, k, 0, ti2+ti3)
ci3 := ti2 - ti3
cr2 := tr1 + tr4
cr4 := tr1 - tr4
ci2 := ti1 + ti4
ci4 := ti1 - ti4
ch3.set(i-1, k, 1, wa1[i-1]*cr2-sign*wa1[i]*ci2)
ch3.set(i, k, 1, wa1[i-1]*ci2+sign*wa1[i]*cr2)
ch3.set(i-1, k, 2, wa2[i-1]*cr3-sign*wa2[i]*ci3)
ch3.set(i, k, 2, wa2[i-1]*ci3+sign*wa2[i]*cr3)
ch3.set(i-1, k, 3, wa3[i-1]*cr4-sign*wa3[i]*ci4)
ch3.set(i, k, 3, wa3[i-1]*ci4+sign*wa3[i]*cr4)
}
}
}
// pass5 implements passf5 and passb5 depending on sign.
func pass5(ido, l1 int, cc, ch, wa1, wa2, wa3, wa4 []float64, sign float64) {
const (
tr11 = 0.309016994374947
ti11 = 0.951056516295154
tr12 = -0.809016994374947
ti12 = 0.587785252292473
)
cc3 := newThreeArray(ido, 5, l1, cc)
ch3 := newThreeArray(ido, l1, 5, ch)
if ido == 2 {
for k := 0; k < l1; k++ {
ti5 := cc3.at(1, 1, k) - cc3.at(1, 4, k)
ti2 := cc3.at(1, 1, k) + cc3.at(1, 4, k)
ti4 := cc3.at(1, 2, k) - cc3.at(1, 3, k)
ti3 := cc3.at(1, 2, k) + cc3.at(1, 3, k)
tr5 := cc3.at(0, 1, k) - cc3.at(0, 4, k)
tr2 := cc3.at(0, 1, k) + cc3.at(0, 4, k)
tr4 := cc3.at(0, 2, k) - cc3.at(0, 3, k)
tr3 := cc3.at(0, 2, k) + cc3.at(0, 3, k)
ch3.set(0, k, 0, cc3.at(0, 0, k)+tr2+tr3)
ch3.set(1, k, 0, cc3.at(1, 0, k)+ti2+ti3)
cr2 := cc3.at(0, 0, k) + tr11*tr2 + tr12*tr3
ci2 := cc3.at(1, 0, k) + tr11*ti2 + tr12*ti3
cr3 := cc3.at(0, 0, k) + tr12*tr2 + tr11*tr3
ci3 := cc3.at(1, 0, k) + tr12*ti2 + tr11*ti3
cr5 := sign * (ti11*tr5 + ti12*tr4)
ci5 := sign * (ti11*ti5 + ti12*ti4)
cr4 := sign * (ti12*tr5 - ti11*tr4)
ci4 := sign * (ti12*ti5 - ti11*ti4)
ch3.set(0, k, 1, cr2-ci5)
ch3.set(0, k, 4, cr2+ci5)
ch3.set(1, k, 1, ci2+cr5)
ch3.set(1, k, 2, ci3+cr4)
ch3.set(0, k, 2, cr3-ci4)
ch3.set(0, k, 3, cr3+ci4)
ch3.set(1, k, 3, ci3-cr4)
ch3.set(1, k, 4, ci2-cr5)
}
return
}
for k := 0; k < l1; k++ {
for i := 1; i < ido; i += 2 {
ti5 := cc3.at(i, 1, k) - cc3.at(i, 4, k)
ti2 := cc3.at(i, 1, k) + cc3.at(i, 4, k)
ti4 := cc3.at(i, 2, k) - cc3.at(i, 3, k)
ti3 := cc3.at(i, 2, k) + cc3.at(i, 3, k)
tr5 := cc3.at(i-1, 1, k) - cc3.at(i-1, 4, k)
tr2 := cc3.at(i-1, 1, k) + cc3.at(i-1, 4, k)
tr4 := cc3.at(i-1, 2, k) - cc3.at(i-1, 3, k)
tr3 := cc3.at(i-1, 2, k) + cc3.at(i-1, 3, k)
ch3.set(i-1, k, 0, cc3.at(i-1, 0, k)+tr2+tr3)
ch3.set(i, k, 0, cc3.at(i, 0, k)+ti2+ti3)
cr2 := cc3.at(i-1, 0, k) + tr11*tr2 + tr12*tr3
ci2 := cc3.at(i, 0, k) + tr11*ti2 + tr12*ti3
cr3 := cc3.at(i-1, 0, k) + tr12*tr2 + tr11*tr3
ci3 := cc3.at(i, 0, k) + tr12*ti2 + tr11*ti3
cr5 := sign * (ti11*tr5 + ti12*tr4)
ci5 := sign * (ti11*ti5 + ti12*ti4)
cr4 := sign * (ti12*tr5 - ti11*tr4)
ci4 := sign * (ti12*ti5 - ti11*ti4)
dr3 := cr3 - ci4
dr4 := cr3 + ci4
di3 := ci3 + cr4
di4 := ci3 - cr4
dr5 := cr2 + ci5
dr2 := cr2 - ci5
di5 := ci2 - cr5
di2 := ci2 + cr5
ch3.set(i-1, k, 1, wa1[i-1]*dr2-sign*wa1[i]*di2)
ch3.set(i, k, 1, wa1[i-1]*di2+sign*wa1[i]*dr2)
ch3.set(i-1, k, 2, wa2[i-1]*dr3-sign*wa2[i]*di3)
ch3.set(i, k, 2, wa2[i-1]*di3+sign*wa2[i]*dr3)
ch3.set(i-1, k, 3, wa3[i-1]*dr4-sign*wa3[i]*di4)
ch3.set(i, k, 3, wa3[i-1]*di4+sign*wa3[i]*dr4)
ch3.set(i-1, k, 4, wa4[i-1]*dr5-sign*wa4[i]*di5)
ch3.set(i, k, 4, wa4[i-1]*di5+sign*wa4[i]*dr5)
}
}
}
// 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.set(ik-1, j, c2m.at(ik-1, j)-c2m.at(ik, jc))
ch2m.set(ik-1, jc, c2m.at(ik-1, j)+c2m.at(ik, jc))
ch2m.set(ik, j, c2m.at(ik, j)+c2m.at(ik-1, jc))
ch2m.set(ik, jc, c2m.at(ik, j)-c2m.at(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.set(0, k, j, ch3.at(0, k, j))
c13.set(1, k, j, ch3.at(1, 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.set(i-1, k, j, wa[idij-1]*ch3.at(i-1, k, j)-sign*wa[idij]*ch3.at(i, k, j))
c13.set(i, k, j, wa[idij-1]*ch3.at(i, k, j)+sign*wa[idij]*ch3.at(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.set(i-1, k, j, wa[idij-1]*ch3.at(i-1, k, j)-sign*wa[idij]*ch3.at(i, k, j))
c13.set(i, k, j, wa[idij-1]*ch3.at(i, k, j)+sign*wa[idij]*ch3.at(i-1, k, j))
}
}
}
return false
}