cmplxs/cmplxs.go - third_party/github.com/gonum/gonum - Git at Google

 // Copyright ©2013 The Gonum Authors. All rights reserved.
 // Use of this code is governed by a BSD-style
 // license that can be found in the LICENSE file.

 package cmplxs

 import (
 	"errors"
 	"math"
 	"math/cmplx"

 	"gonum.org/v1/gonum/cmplxs/cscalar"
 	"gonum.org/v1/gonum/internal/asm/c128"
 )

 const (
 	zeroLength   = "cmplxs: zero length slice"
 	shortSpan    = "cmplxs: slice length less than 2"
 	badLength    = "cmplxs: slice lengths do not match"
 	badDstLength = "cmplxs: destination slice length does not match input"
 )

 // Abs calculates the absolute values of the elements of s, and stores them in dst.
 // It panics if the argument lengths do not match.
 func Abs(dst []float64, s []complex128) {
 	if len(dst) != len(s) {
 		panic(badDstLength)
 	}
 	for i, v := range s {
 		dst[i] = cmplx.Abs(v)
 	}
 }

 // Add adds, element-wise, the elements of s and dst, and stores the result in dst.
 // It panics if the argument lengths do not match.
 func Add(dst, s []complex128) {
 	if len(dst) != len(s) {
 		panic(badLength)
 	}
 	c128.AxpyUnitaryTo(dst, 1, s, dst)
 }

 // AddTo adds, element-wise, the elements of s and t and
 // stores the result in dst.
 // It panics if the argument lengths do not match.
 func AddTo(dst, s, t []complex128) []complex128 {
 	if len(s) != len(t) {
 		panic(badLength)
 	}
 	if len(dst) != len(s) {
 		panic(badDstLength)
 	}
 	c128.AxpyUnitaryTo(dst, 1, s, t)
 	return dst
 }

 // AddConst adds the scalar c to all of the values in dst.
 func AddConst(c complex128, dst []complex128) {
 	c128.AddConst(c, dst)
 }

 // AddScaled performs dst = dst + alpha * s.
 // It panics if the slice argument lengths do not match.
 func AddScaled(dst []complex128, alpha complex128, s []complex128) {
 	if len(dst) != len(s) {
 		panic(badLength)
 	}
 	c128.AxpyUnitaryTo(dst, alpha, s, dst)
 }

 // AddScaledTo performs dst = y + alpha * s, where alpha is a scalar,
 // and dst, y and s are all slices.
 // It panics if the slice argument lengths do not match.
 //
 // At the return of the function, dst[i] = y[i] + alpha * s[i]
 func AddScaledTo(dst, y []complex128, alpha complex128, s []complex128) []complex128 {
 	if len(s) != len(y) {
 		panic(badLength)
 	}
 	if len(dst) != len(y) {
 		panic(badDstLength)
 	}
 	c128.AxpyUnitaryTo(dst, alpha, s, y)
 	return dst
 }

 // Count applies the function f to every element of s and returns the number
 // of times the function returned true.
 func Count(f func(complex128) bool, s []complex128) int {
 	var n int
 	for _, val := range s {
 		if f(val) {
 			n++
 		}
 	}
 	return n
 }

 // Complex fills each of the elements of dst with the complex number
 // constructed from the corresponding elements of real and imag.
 // It panics if the argument lengths do not match.
 func Complex(dst []complex128, real, imag []float64) []complex128 {
 	if len(real) != len(imag) {
 		panic(badLength)
 	}
 	if len(dst) != len(real) {
 		panic(badDstLength)
 	}
 	if len(dst) == 0 {
 		return dst
 	}
 	for i, r := range real {
 		dst[i] = complex(r, imag[i])
 	}
 	return dst
 }

 // CumProd finds the cumulative product of elements of s and store it in
 // place into dst so that
 //  dst[i] = s[i] * s[i-1] * s[i-2] * ... * s[0]
 // It panics if the argument lengths do not match.
 func CumProd(dst, s []complex128) []complex128 {
 	if len(dst) != len(s) {
 		panic(badDstLength)
 	}
 	if len(dst) == 0 {
 		return dst
 	}
 	return c128.CumProd(dst, s)
 }

 // CumSum finds the cumulative sum of elements of s and stores it in place
 // into dst so that
 //  dst[i] = s[i] + s[i-1] + s[i-2] + ... + s[0]
 // It panics if the argument lengths do not match.
 func CumSum(dst, s []complex128) []complex128 {
 	if len(dst) != len(s) {
 		panic(badDstLength)
 	}
 	if len(dst) == 0 {
 		return dst
 	}
 	return c128.CumSum(dst, s)
 }

 // Distance computes the L-norm of s - t. See Norm for special cases.
 // It panics if the slice argument lengths do not match.
 func Distance(s, t []complex128, L float64) float64 {
 	if len(s) != len(t) {
 		panic(badLength)
 	}
 	if len(s) == 0 {
 		return 0
 	}

 	var norm float64
 	switch {
 	case L == 2:
 		return c128.L2DistanceUnitary(s, t)
 	case L == 1:
 		for i, v := range s {
 			norm += cmplx.Abs(t[i] - v)
 		}
 		return norm
 	case math.IsInf(L, 1):
 		for i, v := range s {
 			absDiff := cmplx.Abs(t[i] - v)
 			if absDiff > norm {
 				norm = absDiff
 			}
 		}
 		return norm
 	default:
 		for i, v := range s {
 			norm += math.Pow(cmplx.Abs(t[i]-v), L)
 		}
 		return math.Pow(norm, 1/L)
 	}
 }

 // Div performs element-wise division dst / s
 // and stores the result in dst.
 // It panics if the argument lengths do not match.
 func Div(dst, s []complex128) {
 	if len(dst) != len(s) {
 		panic(badLength)
 	}
 	c128.Div(dst, s)
 }

 // DivTo performs element-wise division s / t
 // and stores the result in dst.
 // It panics if the argument lengths do not match.
 func DivTo(dst, s, t []complex128) []complex128 {
 	if len(s) != len(t) {
 		panic(badLength)
 	}
 	if len(dst) != len(s) {
 		panic(badDstLength)
 	}
 	return c128.DivTo(dst, s, t)
 }

 // Dot computes the dot product of s1 and s2, i.e.
 // sum_{i = 1}^N conj(s1[i])*s2[i].
 // It panics if the argument lengths do not match.
 func Dot(s1, s2 []complex128) complex128 {
 	if len(s1) != len(s2) {
 		panic(badLength)
 	}
 	return c128.DotUnitary(s1, s2)
 }

 // Equal returns true when the slices have equal lengths and
 // all elements are numerically identical.
 func Equal(s1, s2 []complex128) bool {
 	if len(s1) != len(s2) {
 		return false
 	}
 	for i, val := range s1 {
 		if s2[i] != val {
 			return false
 		}
 	}
 	return true
 }

 // EqualApprox returns true when the slices have equal lengths and
 // all element pairs have an absolute tolerance less than tol or a
 // relative tolerance less than tol.
 func EqualApprox(s1, s2 []complex128, tol float64) bool {
 	if len(s1) != len(s2) {
 		return false
 	}
 	for i, a := range s1 {
 		if !cscalar.EqualWithinAbsOrRel(a, s2[i], tol, tol) {
 			return false
 		}
 	}
 	return true
 }

 // EqualFunc returns true when the slices have the same lengths
 // and the function returns true for all element pairs.
 func EqualFunc(s1, s2 []complex128, f func(complex128, complex128) bool) bool {
 	if len(s1) != len(s2) {
 		return false
 	}
 	for i, val := range s1 {
 		if !f(val, s2[i]) {
 			return false
 		}
 	}
 	return true
 }

 // EqualLengths returns true when all of the slices have equal length,
 // and false otherwise. It also eturns true when there are no input slices.
 func EqualLengths(slices ...[]complex128) bool {
 	// This length check is needed: http://play.golang.org/p/sdty6YiLhM
 	if len(slices) == 0 {
 		return true
 	}
 	l := len(slices[0])
 	for i := 1; i < len(slices); i++ {
 		if len(slices[i]) != l {
 			return false
 		}
 	}
 	return true
 }

 // Find applies f to every element of s and returns the indices of the first
 // k elements for which the f returns true, or all such elements
 // if k < 0.
 // Find will reslice inds to have 0 length, and will append
 // found indices to inds.
 // If k > 0 and there are fewer than k elements in s satisfying f,
 // all of the found elements will be returned along with an error.
 // At the return of the function, the input inds will be in an undetermined state.
 func Find(inds []int, f func(complex128) bool, s []complex128, k int) ([]int, error) {
 	// inds is also returned to allow for calling with nil.

 	// Reslice inds to have zero length.
 	inds = inds[:0]

 	// If zero elements requested, can just return.
 	if k == 0 {
 		return inds, nil
 	}

 	// If k < 0, return all of the found indices.
 	if k < 0 {
 		for i, val := range s {
 			if f(val) {
 				inds = append(inds, i)
 			}
 		}
 		return inds, nil
 	}

 	// Otherwise, find the first k elements.
 	nFound := 0
 	for i, val := range s {
 		if f(val) {
 			inds = append(inds, i)
 			nFound++
 			if nFound == k {
 				return inds, nil
 			}
 		}
 	}
 	// Finished iterating over the loop, which means k elements were not found.
 	return inds, errors.New("cmplxs: insufficient elements found")
 }

 // HasNaN returns true when the slice s has any values that are NaN and false
 // otherwise.
 func HasNaN(s []complex128) bool {
 	for _, v := range s {
 		if cmplx.IsNaN(v) {
 			return true
 		}
 	}
 	return false
 }

 // Imag places the imaginary components of src into dst.
 // It panics if the argument lengths do not match.
 func Imag(dst []float64, src []complex128) []float64 {
 	if len(dst) != len(src) {
 		panic(badDstLength)
 	}
 	if len(dst) == 0 {
 		return dst
 	}
 	for i, z := range src {
 		dst[i] = imag(z)
 	}
 	return dst
 }

 // LogSpan returns a set of n equally spaced points in log space between,
 // l and u where N is equal to len(dst). The first element of the
 // resulting dst will be l and the final element of dst will be u.
 // Panics if len(dst) < 2
 // Note that this call will return NaNs if either l or u are negative, and
 // will return all zeros if l or u is zero.
 // Also returns the mutated slice dst, so that it can be used in range, like:
 //
 //     for i, x := range LogSpan(dst, l, u) { ... }
 func LogSpan(dst []complex128, l, u complex128) []complex128 {
 	Span(dst, cmplx.Log(l), cmplx.Log(u))
 	for i := range dst {
 		dst[i] = cmplx.Exp(dst[i])
 	}
 	return dst
 }

 // MaxAbs returns the maximum absolute value in the input slice.
 // It panics if s is zero length.
 func MaxAbs(s []complex128) complex128 {
 	return s[MaxAbsIdx(s)]
 }

 // MaxAbsIdx returns the index of the maximum absolute value in the input slice.
 // If several entries have the maximum absolute value, the first such index is
 // returned.
 // It panics if s is zero length.
 func MaxAbsIdx(s []complex128) int {
 	if len(s) == 0 {
 		panic(zeroLength)
 	}
 	max := math.NaN()
 	var ind int
 	for i, v := range s {
 		if cmplx.IsNaN(v) {
 			continue
 		}
 		if a := cmplx.Abs(v); a > max || math.IsNaN(max) {
 			max = a
 			ind = i
 		}
 	}
 	return ind
 }

 // MinAbs returns the minimum absolute value in the input slice.
 // It panics if s is zero length.
 func MinAbs(s []complex128) complex128 {
 	return s[MinAbsIdx(s)]
 }

 // MinAbsIdx returns the index of the minimum absolute value in the input slice. If several
 // entries have the minimum absolute value, the first such index is returned.
 // It panics if s is zero length.
 func MinAbsIdx(s []complex128) int {
 	if len(s) == 0 {
 		panic(zeroLength)
 	}
 	min := math.NaN()
 	var ind int
 	for i, v := range s {
 		if cmplx.IsNaN(v) {
 			continue
 		}
 		if a := cmplx.Abs(v); a < min || math.IsNaN(min) {
 			min = a
 			ind = i
 		}
 	}
 	return ind
 }

 // Mul performs element-wise multiplication between dst
 // and s and stores the result in dst.
 // It panics if the argument lengths do not match.
 func Mul(dst, s []complex128) {
 	if len(dst) != len(s) {
 		panic(badLength)
 	}
 	for i, val := range s {
 		dst[i] *= val
 	}
 }

 // MulTo performs element-wise multiplication between s
 // and t and stores the result in dst.
 // It panics if the argument lengths do not match.
 func MulTo(dst, s, t []complex128) []complex128 {
 	if len(s) != len(t) {
 		panic(badLength)
 	}
 	if len(dst) != len(s) {
 		panic(badDstLength)
 	}
 	for i, val := range t {
 		dst[i] = val * s[i]
 	}
 	return dst
 }

 // NearestIdx returns the index of the element in s
 // whose value is nearest to v. If several such
 // elements exist, the lowest index is returned.
 // It panics if s is zero length.
 func NearestIdx(s []complex128, v complex128) int {
 	if len(s) == 0 {
 		panic(zeroLength)
 	}
 	switch {
 	case cmplx.IsNaN(v):
 		return 0
 	case cmplx.IsInf(v):
 		return MaxAbsIdx(s)
 	}
 	var ind int
 	dist := math.NaN()
 	for i, val := range s {
 		newDist := cmplx.Abs(v - val)
 		// A NaN distance will not be closer.
 		if math.IsNaN(newDist) {
 			continue
 		}
 		if newDist < dist || math.IsNaN(dist) {
 			dist = newDist
 			ind = i
 		}
 	}
 	return ind
 }

 // Norm returns the L-norm of the slice S, defined as
 // (sum_{i=1}^N abs(s[i])^L)^{1/L}
 // Special cases:
 // L = math.Inf(1) gives the maximum absolute value.
 // Does not correctly compute the zero norm (use Count).
 func Norm(s []complex128, L float64) float64 {
 	// Should this complain if L is not positive?
 	// Should this be done in log space for better numerical stability?
 	//	would be more cost
 	//	maybe only if L is high?
 	if len(s) == 0 {
 		return 0
 	}
 	var norm float64
 	switch {
 	case L == 2:
 		return c128.L2NormUnitary(s)
 	case L == 1:
 		for _, v := range s {
 			norm += cmplx.Abs(v)
 		}
 		return norm
 	case math.IsInf(L, 1):
 		for _, v := range s {
 			norm = math.Max(norm, cmplx.Abs(v))
 		}
 		return norm
 	default:
 		for _, v := range s {
 			norm += math.Pow(cmplx.Abs(v), L)
 		}
 		return math.Pow(norm, 1/L)
 	}
 }

 // Prod returns the product of the elements of the slice.
 // Returns 1 if len(s) = 0.
 func Prod(s []complex128) complex128 {
 	prod := 1 + 0i
 	for _, val := range s {
 		prod *= val
 	}
 	return prod
 }

 // Real places the real components of src into dst.
 // It panics if the argument lengths do not match.
 func Real(dst []float64, src []complex128) []float64 {
 	if len(dst) != len(src) {
 		panic(badDstLength)
 	}
 	if len(dst) == 0 {
 		return dst
 	}
 	for i, z := range src {
 		dst[i] = real(z)
 	}
 	return dst
 }

 // Reverse reverses the order of elements in the slice.
 func Reverse(s []complex128) {
 	for i, j := 0, len(s)-1; i < j; i, j = i+1, j-1 {
 		s[i], s[j] = s[j], s[i]
 	}
 }

 // Same returns true when the input slices have the same length and all
 // elements have the same value with NaN treated as the same.
 func Same(s, t []complex128) bool {
 	if len(s) != len(t) {
 		return false
 	}
 	for i, v := range s {
 		w := t[i]
 		if v != w && !(cmplx.IsNaN(v) && cmplx.IsNaN(w)) {
 			return false
 		}
 	}
 	return true
 }

 // Scale multiplies every element in dst by the scalar c.
 func Scale(c complex128, dst []complex128) {
 	if len(dst) > 0 {
 		c128.ScalUnitary(c, dst)
 	}
 }

 // ScaleTo multiplies the elements in s by c and stores the result in dst.
 // It panics if the slice argument lengths do not match.
 func ScaleTo(dst []complex128, c complex128, s []complex128) []complex128 {
 	if len(dst) != len(s) {
 		panic(badDstLength)
 	}
 	if len(dst) > 0 {
 		c128.ScalUnitaryTo(dst, c, s)
 	}
 	return dst
 }

 // Span returns a set of N equally spaced points between l and u, where N
 // is equal to the length of the destination. The first element of the destination
 // is l, the final element of the destination is u.
 // It panics if the length of dst is less than 2.
 //
 // Span also returns the mutated slice dst, so that it can be used in range expressions,
 // like:
 //
 //     for i, x := range Span(dst, l, u) { ... }
 func Span(dst []complex128, l, u complex128) []complex128 {
 	n := len(dst)
 	if n < 2 {
 		panic(shortSpan)
 	}

 	// Special cases for Inf and NaN.
 	switch {
 	case cmplx.IsNaN(l):
 		for i := range dst[:len(dst)-1] {
 			dst[i] = cmplx.NaN()
 		}
 		dst[len(dst)-1] = u
 		return dst
 	case cmplx.IsNaN(u):
 		for i := range dst[1:] {
 			dst[i+1] = cmplx.NaN()
 		}
 		dst[0] = l
 		return dst
 	case cmplx.IsInf(l) && cmplx.IsInf(u):
 		for i := range dst {
 			dst[i] = cmplx.Inf()
 		}
 		return dst
 	case cmplx.IsInf(l):
 		for i := range dst[:len(dst)-1] {
 			dst[i] = l
 		}
 		dst[len(dst)-1] = u
 		return dst
 	case cmplx.IsInf(u):
 		for i := range dst[1:] {
 			dst[i+1] = u
 		}
 		dst[0] = l
 		return dst
 	}

 	step := (u - l) / complex(float64(n-1), 0)
 	for i := range dst {
 		dst[i] = l + step*complex(float64(i), 0)
 	}
 	return dst
 }

 // Sub subtracts, element-wise, the elements of s from dst.
 // It panics if the argument lengths do not match.
 func Sub(dst, s []complex128) {
 	if len(dst) != len(s) {
 		panic(badLength)
 	}
 	c128.AxpyUnitaryTo(dst, -1, s, dst)
 }

 // SubTo subtracts, element-wise, the elements of t from s and
 // stores the result in dst.
 // It panics if the argument lengths do not match.
 func SubTo(dst, s, t []complex128) []complex128 {
 	if len(s) != len(t) {
 		panic(badLength)
 	}
 	if len(dst) != len(s) {
 		panic(badDstLength)
 	}
 	c128.AxpyUnitaryTo(dst, -1, t, s)
 	return dst
 }

 // Sum returns the sum of the elements of the slice.
 func Sum(s []complex128) complex128 {
 	return c128.Sum(s)
 }
	// Copyright ©2013 The Gonum Authors. All rights reserved.
	// Use of this code is governed by a BSD-style
	// license that can be found in the LICENSE file.

	package cmplxs

	import (
	"errors"
	"math"
	"math/cmplx"

	"gonum.org/v1/gonum/cmplxs/cscalar"
	"gonum.org/v1/gonum/internal/asm/c128"
	)

	const (
	zeroLength = "cmplxs: zero length slice"
	shortSpan = "cmplxs: slice length less than 2"
	badLength = "cmplxs: slice lengths do not match"
	badDstLength = "cmplxs: destination slice length does not match input"
	)

	// Abs calculates the absolute values of the elements of s, and stores them in dst.
	// It panics if the argument lengths do not match.
	func Abs(dst []float64, s []complex128) {
	if len(dst) != len(s) {
	panic(badDstLength)
	}
	for i, v := range s {
	dst[i] = cmplx.Abs(v)
	}
	}

	// Add adds, element-wise, the elements of s and dst, and stores the result in dst.
	// It panics if the argument lengths do not match.
	func Add(dst, s []complex128) {
	if len(dst) != len(s) {
	panic(badLength)
	}
	c128.AxpyUnitaryTo(dst, 1, s, dst)
	}

	// AddTo adds, element-wise, the elements of s and t and
	// stores the result in dst.
	// It panics if the argument lengths do not match.
	func AddTo(dst, s, t []complex128) []complex128 {
	if len(s) != len(t) {
	panic(badLength)
	}
	if len(dst) != len(s) {
	panic(badDstLength)
	}
	c128.AxpyUnitaryTo(dst, 1, s, t)
	return dst
	}

	// AddConst adds the scalar c to all of the values in dst.
	func AddConst(c complex128, dst []complex128) {
	c128.AddConst(c, dst)
	}

	// AddScaled performs dst = dst + alpha * s.
	// It panics if the slice argument lengths do not match.
	func AddScaled(dst []complex128, alpha complex128, s []complex128) {
	if len(dst) != len(s) {
	panic(badLength)
	}
	c128.AxpyUnitaryTo(dst, alpha, s, dst)
	}

	// AddScaledTo performs dst = y + alpha * s, where alpha is a scalar,
	// and dst, y and s are all slices.
	// It panics if the slice argument lengths do not match.
	//
	// At the return of the function, dst[i] = y[i] + alpha * s[i]
	func AddScaledTo(dst, y []complex128, alpha complex128, s []complex128) []complex128 {
	if len(s) != len(y) {
	panic(badLength)
	}
	if len(dst) != len(y) {
	panic(badDstLength)
	}
	c128.AxpyUnitaryTo(dst, alpha, s, y)
	return dst
	}

	// Count applies the function f to every element of s and returns the number
	// of times the function returned true.
	func Count(f func(complex128) bool, s []complex128) int {
	var n int
	for _, val := range s {
	if f(val) {
	n++
	}
	}
	return n
	}

	// Complex fills each of the elements of dst with the complex number
	// constructed from the corresponding elements of real and imag.
	// It panics if the argument lengths do not match.
	func Complex(dst []complex128, real, imag []float64) []complex128 {
	if len(real) != len(imag) {
	panic(badLength)
	}
	if len(dst) != len(real) {
	panic(badDstLength)
	}
	if len(dst) == 0 {
	return dst
	}
	for i, r := range real {
	dst[i] = complex(r, imag[i])
	}
	return dst
	}

	// CumProd finds the cumulative product of elements of s and store it in
	// place into dst so that
	// dst[i] = s[i] * s[i-1] * s[i-2] * ... * s[0]
	// It panics if the argument lengths do not match.
	func CumProd(dst, s []complex128) []complex128 {
	if len(dst) != len(s) {
	panic(badDstLength)
	}
	if len(dst) == 0 {
	return dst
	}
	return c128.CumProd(dst, s)
	}

	// CumSum finds the cumulative sum of elements of s and stores it in place
	// into dst so that
	// dst[i] = s[i] + s[i-1] + s[i-2] + ... + s[0]
	// It panics if the argument lengths do not match.
	func CumSum(dst, s []complex128) []complex128 {
	if len(dst) != len(s) {
	panic(badDstLength)
	}
	if len(dst) == 0 {
	return dst
	}
	return c128.CumSum(dst, s)
	}

	// Distance computes the L-norm of s - t. See Norm for special cases.
	// It panics if the slice argument lengths do not match.
	func Distance(s, t []complex128, L float64) float64 {
	if len(s) != len(t) {
	panic(badLength)
	}
	if len(s) == 0 {
	return 0
	}

	var norm float64
	switch {
	case L == 2:
	return c128.L2DistanceUnitary(s, t)
	case L == 1:
	for i, v := range s {
	norm += cmplx.Abs(t[i] - v)
	}
	return norm
	case math.IsInf(L, 1):
	for i, v := range s {
	absDiff := cmplx.Abs(t[i] - v)
	if absDiff > norm {
	norm = absDiff
	}
	}
	return norm
	default:
	for i, v := range s {
	norm += math.Pow(cmplx.Abs(t[i]-v), L)
	}
	return math.Pow(norm, 1/L)
	}
	}

	// Div performs element-wise division dst / s
	// and stores the result in dst.
	// It panics if the argument lengths do not match.
	func Div(dst, s []complex128) {
	if len(dst) != len(s) {
	panic(badLength)
	}
	c128.Div(dst, s)
	}

	// DivTo performs element-wise division s / t
	// and stores the result in dst.
	// It panics if the argument lengths do not match.
	func DivTo(dst, s, t []complex128) []complex128 {
	if len(s) != len(t) {
	panic(badLength)
	}
	if len(dst) != len(s) {
	panic(badDstLength)
	}
	return c128.DivTo(dst, s, t)
	}

	// Dot computes the dot product of s1 and s2, i.e.
	// sum_{i = 1}^N conj(s1[i])*s2[i].
	// It panics if the argument lengths do not match.
	func Dot(s1, s2 []complex128) complex128 {
	if len(s1) != len(s2) {
	panic(badLength)
	}
	return c128.DotUnitary(s1, s2)
	}

	// Equal returns true when the slices have equal lengths and
	// all elements are numerically identical.
	func Equal(s1, s2 []complex128) bool {
	if len(s1) != len(s2) {
	return false
	}
	for i, val := range s1 {
	if s2[i] != val {
	return false
	}
	}
	return true
	}

	// EqualApprox returns true when the slices have equal lengths and
	// all element pairs have an absolute tolerance less than tol or a
	// relative tolerance less than tol.
	func EqualApprox(s1, s2 []complex128, tol float64) bool {
	if len(s1) != len(s2) {
	return false
	}
	for i, a := range s1 {
	if !cscalar.EqualWithinAbsOrRel(a, s2[i], tol, tol) {
	return false
	}
	}
	return true
	}

	// EqualFunc returns true when the slices have the same lengths
	// and the function returns true for all element pairs.
	func EqualFunc(s1, s2 []complex128, f func(complex128, complex128) bool) bool {
	if len(s1) != len(s2) {
	return false
	}
	for i, val := range s1 {
	if !f(val, s2[i]) {
	return false
	}
	}
	return true
	}

	// EqualLengths returns true when all of the slices have equal length,
	// and false otherwise. It also eturns true when there are no input slices.
	func EqualLengths(slices ...[]complex128) bool {
	// This length check is needed: http://play.golang.org/p/sdty6YiLhM
	if len(slices) == 0 {
	return true
	}
	l := len(slices[0])
	for i := 1; i < len(slices); i++ {
	if len(slices[i]) != l {
	return false
	}
	}
	return true
	}

	// Find applies f to every element of s and returns the indices of the first
	// k elements for which the f returns true, or all such elements
	// if k < 0.
	// Find will reslice inds to have 0 length, and will append
	// found indices to inds.
	// If k > 0 and there are fewer than k elements in s satisfying f,
	// all of the found elements will be returned along with an error.
	// At the return of the function, the input inds will be in an undetermined state.
	func Find(inds []int, f func(complex128) bool, s []complex128, k int) ([]int, error) {
	// inds is also returned to allow for calling with nil.

	// Reslice inds to have zero length.
	inds = inds[:0]

	// If zero elements requested, can just return.
	if k == 0 {
	return inds, nil
	}

	// If k < 0, return all of the found indices.
	if k < 0 {
	for i, val := range s {
	if f(val) {
	inds = append(inds, i)
	}
	}
	return inds, nil
	}

	// Otherwise, find the first k elements.
	nFound := 0
	for i, val := range s {
	if f(val) {
	inds = append(inds, i)
	nFound++
	if nFound == k {
	return inds, nil
	}
	}
	}
	// Finished iterating over the loop, which means k elements were not found.
	return inds, errors.New("cmplxs: insufficient elements found")
	}

	// HasNaN returns true when the slice s has any values that are NaN and false
	// otherwise.
	func HasNaN(s []complex128) bool {
	for _, v := range s {
	if cmplx.IsNaN(v) {
	return true
	}
	}
	return false
	}

	// Imag places the imaginary components of src into dst.
	// It panics if the argument lengths do not match.
	func Imag(dst []float64, src []complex128) []float64 {
	if len(dst) != len(src) {
	panic(badDstLength)
	}
	if len(dst) == 0 {
	return dst
	}
	for i, z := range src {
	dst[i] = imag(z)
	}
	return dst
	}

	// LogSpan returns a set of n equally spaced points in log space between,
	// l and u where N is equal to len(dst). The first element of the
	// resulting dst will be l and the final element of dst will be u.
	// Panics if len(dst) < 2
	// Note that this call will return NaNs if either l or u are negative, and
	// will return all zeros if l or u is zero.
	// Also returns the mutated slice dst, so that it can be used in range, like:
	//
	// for i, x := range LogSpan(dst, l, u) { ... }
	func LogSpan(dst []complex128, l, u complex128) []complex128 {
	Span(dst, cmplx.Log(l), cmplx.Log(u))
	for i := range dst {
	dst[i] = cmplx.Exp(dst[i])
	}
	return dst
	}

	// MaxAbs returns the maximum absolute value in the input slice.
	// It panics if s is zero length.
	func MaxAbs(s []complex128) complex128 {
	return s[MaxAbsIdx(s)]
	}

	// MaxAbsIdx returns the index of the maximum absolute value in the input slice.
	// If several entries have the maximum absolute value, the first such index is
	// returned.
	// It panics if s is zero length.
	func MaxAbsIdx(s []complex128) int {
	if len(s) == 0 {
	panic(zeroLength)
	}
	max := math.NaN()
	var ind int
	for i, v := range s {
	if cmplx.IsNaN(v) {
	continue
	}
	if a := cmplx.Abs(v); a > max \|\| math.IsNaN(max) {
	max = a
	ind = i
	}
	}
	return ind
	}

	// MinAbs returns the minimum absolute value in the input slice.
	// It panics if s is zero length.
	func MinAbs(s []complex128) complex128 {
	return s[MinAbsIdx(s)]
	}

	// MinAbsIdx returns the index of the minimum absolute value in the input slice. If several
	// entries have the minimum absolute value, the first such index is returned.
	// It panics if s is zero length.
	func MinAbsIdx(s []complex128) int {
	if len(s) == 0 {
	panic(zeroLength)
	}
	min := math.NaN()
	var ind int
	for i, v := range s {
	if cmplx.IsNaN(v) {
	continue
	}
	if a := cmplx.Abs(v); a < min \|\| math.IsNaN(min) {
	min = a
	ind = i
	}
	}
	return ind
	}

	// Mul performs element-wise multiplication between dst
	// and s and stores the result in dst.
	// It panics if the argument lengths do not match.
	func Mul(dst, s []complex128) {
	if len(dst) != len(s) {
	panic(badLength)
	}
	for i, val := range s {
	dst[i] *= val
	}
	}

	// MulTo performs element-wise multiplication between s
	// and t and stores the result in dst.
	// It panics if the argument lengths do not match.
	func MulTo(dst, s, t []complex128) []complex128 {
	if len(s) != len(t) {
	panic(badLength)
	}
	if len(dst) != len(s) {
	panic(badDstLength)
	}
	for i, val := range t {
	dst[i] = val * s[i]
	}
	return dst
	}

	// NearestIdx returns the index of the element in s
	// whose value is nearest to v. If several such
	// elements exist, the lowest index is returned.
	// It panics if s is zero length.
	func NearestIdx(s []complex128, v complex128) int {
	if len(s) == 0 {
	panic(zeroLength)
	}
	switch {
	case cmplx.IsNaN(v):
	return 0
	case cmplx.IsInf(v):
	return MaxAbsIdx(s)
	}
	var ind int
	dist := math.NaN()
	for i, val := range s {
	newDist := cmplx.Abs(v - val)
	// A NaN distance will not be closer.
	if math.IsNaN(newDist) {
	continue
	}
	if newDist < dist \|\| math.IsNaN(dist) {
	dist = newDist
	ind = i
	}
	}
	return ind
	}

	// Norm returns the L-norm of the slice S, defined as
	// (sum_{i=1}^N abs(s[i])^L)^{1/L}
	// Special cases:
	// L = math.Inf(1) gives the maximum absolute value.
	// Does not correctly compute the zero norm (use Count).
	func Norm(s []complex128, L float64) float64 {
	// Should this complain if L is not positive?
	// Should this be done in log space for better numerical stability?
	// would be more cost
	// maybe only if L is high?
	if len(s) == 0 {
	return 0
	}
	var norm float64
	switch {
	case L == 2:
	return c128.L2NormUnitary(s)
	case L == 1:
	for _, v := range s {
	norm += cmplx.Abs(v)
	}
	return norm
	case math.IsInf(L, 1):
	for _, v := range s {
	norm = math.Max(norm, cmplx.Abs(v))
	}
	return norm
	default:
	for _, v := range s {
	norm += math.Pow(cmplx.Abs(v), L)
	}
	return math.Pow(norm, 1/L)
	}
	}

	// Prod returns the product of the elements of the slice.
	// Returns 1 if len(s) = 0.
	func Prod(s []complex128) complex128 {
	prod := 1 + 0i
	for _, val := range s {
	prod *= val
	}
	return prod
	}

	// Real places the real components of src into dst.
	// It panics if the argument lengths do not match.
	func Real(dst []float64, src []complex128) []float64 {
	if len(dst) != len(src) {
	panic(badDstLength)
	}
	if len(dst) == 0 {
	return dst
	}
	for i, z := range src {
	dst[i] = real(z)
	}
	return dst
	}

	// Reverse reverses the order of elements in the slice.
	func Reverse(s []complex128) {
	for i, j := 0, len(s)-1; i < j; i, j = i+1, j-1 {
	s[i], s[j] = s[j], s[i]
	}
	}

	// Same returns true when the input slices have the same length and all
	// elements have the same value with NaN treated as the same.
	func Same(s, t []complex128) bool {
	if len(s) != len(t) {
	return false
	}
	for i, v := range s {
	w := t[i]
	if v != w && !(cmplx.IsNaN(v) && cmplx.IsNaN(w)) {
	return false
	}
	}
	return true
	}

	// Scale multiplies every element in dst by the scalar c.
	func Scale(c complex128, dst []complex128) {
	if len(dst) > 0 {
	c128.ScalUnitary(c, dst)
	}
	}

	// ScaleTo multiplies the elements in s by c and stores the result in dst.
	// It panics if the slice argument lengths do not match.
	func ScaleTo(dst []complex128, c complex128, s []complex128) []complex128 {
	if len(dst) != len(s) {
	panic(badDstLength)
	}
	if len(dst) > 0 {
	c128.ScalUnitaryTo(dst, c, s)
	}
	return dst
	}

	// Span returns a set of N equally spaced points between l and u, where N
	// is equal to the length of the destination. The first element of the destination
	// is l, the final element of the destination is u.
	// It panics if the length of dst is less than 2.
	//
	// Span also returns the mutated slice dst, so that it can be used in range expressions,
	// like:
	//
	// for i, x := range Span(dst, l, u) { ... }
	func Span(dst []complex128, l, u complex128) []complex128 {
	n := len(dst)
	if n < 2 {
	panic(shortSpan)
	}

	// Special cases for Inf and NaN.
	switch {
	case cmplx.IsNaN(l):
	for i := range dst[:len(dst)-1] {
	dst[i] = cmplx.NaN()
	}
	dst[len(dst)-1] = u
	return dst
	case cmplx.IsNaN(u):
	for i := range dst[1:] {
	dst[i+1] = cmplx.NaN()
	}
	dst[0] = l
	return dst
	case cmplx.IsInf(l) && cmplx.IsInf(u):
	for i := range dst {
	dst[i] = cmplx.Inf()
	}
	return dst
	case cmplx.IsInf(l):
	for i := range dst[:len(dst)-1] {
	dst[i] = l
	}
	dst[len(dst)-1] = u
	return dst
	case cmplx.IsInf(u):
	for i := range dst[1:] {
	dst[i+1] = u
	}
	dst[0] = l
	return dst
	}

	step := (u - l) / complex(float64(n-1), 0)
	for i := range dst {
	dst[i] = l + step*complex(float64(i), 0)
	}
	return dst
	}

	// Sub subtracts, element-wise, the elements of s from dst.
	// It panics if the argument lengths do not match.
	func Sub(dst, s []complex128) {
	if len(dst) != len(s) {
	panic(badLength)
	}
	c128.AxpyUnitaryTo(dst, -1, s, dst)
	}

	// SubTo subtracts, element-wise, the elements of t from s and
	// stores the result in dst.
	// It panics if the argument lengths do not match.
	func SubTo(dst, s, t []complex128) []complex128 {
	if len(s) != len(t) {
	panic(badLength)
	}
	if len(dst) != len(s) {
	panic(badDstLength)
	}
	c128.AxpyUnitaryTo(dst, -1, t, s)
	return dst
	}

	// Sum returns the sum of the elements of the slice.
	func Sum(s []complex128) complex128 {
	return c128.Sum(s)
	}