/
runtests.jl
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/
runtests.jl
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# This file is part of TaylorSeries.jl, MIT licensed
#
# Tests for TaylorSeries implementation
using TaylorSeries
using FactCheck
using Compat
import Compat.String
FactCheck.setstyle(:compact)
# FactCheck.onlystats(true)
facts("Tests for Taylor1 expansions") do
ta(a) = Taylor1([a,one(a)],15)
t = Taylor1(Int,15)
tim = im*t
zt = zero(t)
ot = 1.0*one(t)
tol1 = eps(1.0)
@fact Taylor1([0,1,0,0]) == Taylor1(3) --> true
@fact get_coeff(Taylor1(Complex128,3),1) == complex(1.0,0.0) --> true
@fact eltype(convert(Taylor1{Complex128},ot)) == Complex128 --> true
@fact eltype(convert(Taylor1{Complex128},1)) == Complex128 --> true
@fact convert(Taylor1{Complex{Int}},[0,2]) == (2+0im)*t --> true
@fact convert(Taylor1{BigFloat},[0.0, 1.0]) == ta(big(0.0)) --> true
@fact promote(0,Taylor1(1.0,0)) == (zt,ot) --> true
@fact eltype(promote(ta(0.0),zeros(Int,2))[2]) == Float64 --> true
@fact eltype(promote(0,Taylor1(ot))[1]) == Float64 --> true
@fact eltype(promote(1.0+im, zt)[1]) == Complex{Float64} --> true
@fact eltype(TaylorSeries.fixshape(zt,ot)[1]) == Float64 --> true
@fact length(Taylor1(10)) == 10 --> true
@fact length(TaylorSeries.fixshape(zt,Taylor1([1.0]))[2]) == 15 --> true
@fact eltype(TaylorSeries.fixshape(zt,Taylor1([1.0]))[1]) == Float64 --> true
@fact TaylorSeries.firstnonzero(t) == 1 --> true
@fact TaylorSeries.firstnonzero(zt) == zt.order+1 --> true
@fact t == Taylor1(ta(0),15) --> true
@fact ot == 1 --> true
@fact 0.0 == zt --> true
@fact get_coeff(tim,1) == complex(0,1) --> true
@fact zt+1.0 == ot --> true
@fact 1.0-ot == zt --> true
@fact t+t == 2t --> true
@fact t-t == zt --> true
@fact +t == -(-t) --> true
tsquare = Taylor1([0,0,1],15)
@fact t * true == t --> true
@fact false * t == zero(t) --> true
@fact t^0 == t^0.0 == one(t) --> true
@fact t*t == tsquare --> true
@fact t*1 == t --> true
@fact 0*t == zt --> true
@fact (-t)^2 == tsquare --> true
@fact t^3 == tsquare*t --> true
@fact tsquare/t == t --> true
@fact t/(t*3) == (1/3)*ot --> true
@fact t/3im == -tim/3 --> true
@fact 1/(1-t) == Taylor1(ones(t.order+1)) --> true
@fact Taylor1([0,1,1])/t == t+1 --> true
@fact (t+im)^2 == tsquare+2im*t-1 --> true
@fact (t+im)^3 == Taylor1([-1im,-3,3im,1],15) --> true
@fact (t+im)^4 == Taylor1([1,-4im,-6,4im,1],15) --> true
@fact imag(tsquare+2im*t-1) == 2t --> true
@fact (Rational(1,2)*tsquare).coeffs[3] == 1//2 --> true
@fact t^2/tsquare == ot --> true
@fact ((1+t)^(1/3)).coeffs[3]+1/9 <= tol1 --> true
@fact 1-tsquare == (1+t)-t*(1+t) --> true
@fact (1-tsquare)^2 == (1+t)^2.0 * (1-t)^2.0 --> true
@fact (sqrt(1+t)).coeffs[3] == -1/8 --> true
@fact ((1-tsquare)^(1//2))^2 == 1-tsquare --> true
@fact ((1-t)^(1//4)).coeffs[15] == -4188908511//549755813888 --> true
@fact abs(((1+t)^3.2).coeffs[14] + 5.4021062656e-5) < tol1 --> true
@fact isapprox( rem(4.1 + t,4).coeffs[1], (0.1 + t).coeffs[1] ) --> true
@fact isapprox( mod(4.1 + t,4).coeffs[1], (0.1 + t).coeffs[1] ) --> true
@fact isapprox( mod2pi(2pi+0.1+t).coeffs[1],(0.1 + t).coeffs[1]) --> true
@fact abs(ta(1)) --> ta(1)
@fact abs(ta(-1.0)) --> -ta(-1.0)
@fact log(exp(tsquare)) == tsquare --> true
@fact exp(log(1-tsquare)) == 1-tsquare --> true
@fact log((1-t)^2) == 2*log(1-t) --> true
@fact real(exp(tim)) == cos(t) --> true
@fact imag(exp(tim)) == sin(t) --> true
@fact exp(conj(tim)) == cos(t)-im*sin(t) == exp(tim') --> true
@fact (exp(t))^(2im) == cos(2t)+im*sin(2t) --> true
@fact (exp(t))^Taylor1([-5.2im]) == cos(5.2t)-im*sin(5.2t) --> true
@fact get_coeff(convert(Taylor1{Rational{Int}},cos(t)),8) ==
1//factorial(8) --> true
@fact abs((tan(t)).coeffs[8]- 17/315) < tol1 --> true
@fact abs((tan(t)).coeffs[14]- 21844/6081075) < tol1 --> true
@fact evaluate(exp(Taylor1([0,1],17)),1.0) == 1.0*e --> true
@fact evaluate(exp(Taylor1([0,1],1))) == 1.0 --> true
@fact evaluate(exp(t),t^2) == exp(t^2) --> true
@fact sin(asin(tsquare)) == tsquare --> true
@fact tan(atan(tsquare)) == tsquare --> true
@fact atan(tan(tsquare)) == tsquare --> true
@fact asin(t) + acos(t) == pi/2 --> true
@fact derivative(acos(t)) == - 1/sqrt(1-t^2) --> true
@fact_throws ArgumentError asin(ta(1.0))
@fact_throws ArgumentError acos(ta(big(1.0)))
@fact_throws ArgumentError atan(ta(1//1*im))
@fact derivative(5, exp(ta(1.0))) == exp(1.0) --> true
@fact derivative(3, exp(ta(1.0pi))) == exp(1.0pi) --> true
@fact isapprox(derivative(10, exp(ta(1.0pi))) , exp(1.0pi) ) --> true
@fact integrate(derivative(exp(t)),1) == exp(t) --> true
@fact integrate(cos(t)) == sin(t) --> true
@fact promote(ta(0.0), t) == (ta(0.0),ta(0.0)) --> true
@fact_throws ArgumentError abs(ta(big(0)))
@fact_throws ArgumentError 1/t
@fact_throws ArgumentError zt/zt
@fact_throws ArgumentError t^1.5
@fact_throws DomainError t^(-2)
@fact_throws ArgumentError sqrt(t)
@fact_throws ArgumentError log(t)
@fact_throws ArgumentError cos(t)/sin(t)
@fact_throws AssertionError derivative(30, exp(ta(1.0pi)))
@fact string(ta(-3)) == " - 3 + 1 t + 𝒪(t¹⁶)" --> true
@fact TaylorSeries.pretty_print(ta(3im)) ==
" ( 3 im ) + ( 1 ) t + 𝒪(t¹⁶)" --> true
end
facts("Matrix multiplication for Taylor1") do
order = 30
n1 = 100
k1 = 90
order = max(n1,k1)
B1 = randn(n1,order)
Y1 = randn(k1,order)
A1 = randn(k1,n1)
for A in (A1,sparse(A1))
# B and Y contain elements of different orders
B = Taylor1{Float64}[Taylor1(collect(B1[i,1:i]),i) for i=1:n1]
Y = Taylor1{Float64}[Taylor1(collect(Y1[k,1:k]),k) for k=1:k1]
Bcopy = deepcopy(B)
A_mul_B!(Y,A,B)
# do we get the same result when using the `A*B` form?
@fact A*B==Y -->true
# Y should be extended after the multilpication
@fact reduce(&, [y1.order for y1 in Y] .== Y[1].order) --> true
# B should be unchanged
@fact B==Bcopy --> true
# is the result compatible with the matrix multiplication? We
# only check the zeroth order of the Taylor series.
y1=sum(Y).coeffs[1]
Y=A*B1[:,1]
y2=sum(Y)
# There is a small numerical error when comparing the generic
# multiplication and the specialized version
@fact abs(y1-y2) < n1*(eps(y1)+eps(y2)) --> true
@fact_throws DimensionMismatch A_mul_B!(Y,A[:,1:end-1],B)
@fact_throws DimensionMismatch A_mul_B!(Y,A[1:end-1,:],B)
@fact_throws DimensionMismatch A_mul_B!(Y,A,B[1:end-1])
@fact_throws DimensionMismatch A_mul_B!(Y[1:end-1],A,B)
end
end
facts("Tests for HomogeneousPolynomial and TaylorN") do
@fact eltype(set_variables(Int, "x", numvars=2, order=6)) --> TaylorN{Int}
@fact eltype(set_variables("x", numvars=2, order=6)) --> TaylorN{Float64}
@fact eltype(set_variables(BigInt, "x y", order=6)) --> TaylorN{BigInt}
@fact eltype(set_variables("x y", order=6)) --> TaylorN{Float64}
@fact TaylorSeries.coeff_table[2][1] == [1,0] --> true
@fact TaylorSeries.index_table[2][1] == 7 --> true
@fact TaylorSeries.in_base(get_order(),[2,1]) == 15 --> true
@fact TaylorSeries.pos_table[4][15] == 2 --> true
@fact get_order() == 6 --> true
@fact get_numvars() == 2 --> true
x, y = set_variables("x y", order=6)
@fact x.order == 6 --> true
@fact TaylorSeries.set_variable_names(["x","y"]) == ["x", "y"] --> true
@fact TaylorSeries.get_variable_names() == ["x", "y"] --> true
set_variables("x", numvars=2, order=17)
xH = HomogeneousPolynomial([1,0])
yH = HomogeneousPolynomial([0,1],1)
xT = TaylorN(xH,17)
# yT = taylorN_variable(Int64, 2, 17)
yT = TaylorN(Int64, 2, order=17)
zeroT = zero( TaylorN([xH],1) )
uT = one(convert(TaylorN{Float64},yT))
@fact ones(xH,1) == [HomogeneousPolynomial(1), xH+yH] --> true
@fact ones(HomogeneousPolynomial{Complex{Int}},0) ==
[HomogeneousPolynomial(1+0im)] --> true
@fact get_order(zeroT) == 1 --> true
@fact get_coeff(xT,[1,0]) == 1 --> true
@fact get_coeff(yH,[1,0]) == 0 --> true
@fact convert(HomogeneousPolynomial{Int64},[1,1]) == xH+yH --> true
@fact convert(HomogeneousPolynomial{Float64},[2,-1]) == 2.0xH-yH --> true
@fact convert(TaylorN{Float64}, yH) == 1.0*yT --> true
@fact convert(TaylorN{Float64}, [xH,yH]) == xT+1.0*yT --> true
@fact convert(TaylorN{Int}, [xH,yH]) == xT+yT --> true
@fact promote(xH, [1,1])[2] == xH+yH --> true
@fact promote(xH, yT)[1] == xT --> true
@fact promote(xT, [xH,yH])[2] == xT+yT --> true
@fact typeof(promote(im*xT,[xH,yH])[2]) == TaylorN{Complex{Int64}} --> true
# @fact TaylorSeries.fixorder(taylorN_variable(1,1),17) == xT --> true
@fact TaylorSeries.fixorder(TaylorN(1, order=1),17) == xT --> true
@fact TaylorSeries.iszero(zeroT.coeffs[2]) --> true
@fact HomogeneousPolynomial(xH,1) == HomogeneousPolynomial(xH) --> true
@fact eltype(xH) == Int --> true
@fact length(xH) == 2 --> true
@fact zero(xH) == 0*xH --> true
@fact one(yH) == xH+yH --> true
@fact xH * true == xH --> true
@fact false * yH == zero(yH) --> true
@fact get_order(yH) == 1 --> true
@fact get_order(xT) == 17 --> true
@fact xT * true == xT --> true
@fact false * yT == zero(yT) --> true
@fact xT == TaylorN([xH]) --> true
@fact one(xT) == TaylorN(1,5) --> true
@fact TaylorN(zeroT,5) == 0 --> true
@fact TaylorN(uT) == convert(TaylorN{Complex},1) --> true
@fact get_numvars() == 2 --> true
@fact length(uT) == get_order()+1 --> true
@fact eltype(convert(TaylorN{Complex128},1)) == Complex128 --> true
@fact 1+xT+yT == TaylorN(1,1) + TaylorN([xH,yH],1) --> true
@fact xT-yT-1 == TaylorN([-1,xH-yH]) --> true
@fact xT*yT == TaylorN([HomogeneousPolynomial([0,1,0],2)]) --> true
@fact (1/(1-xT)).coeffs[4] == HomogeneousPolynomial(1.0,3) --> true
@fact xH^20 == HomogeneousPolynomial([0],get_order()) --> true
@fact (yT/(1-xT)).coeffs[5] == xH^3 * yH --> true
@fact mod(1+xT,1) == +xT --> true
@fact (rem(1+xT,1)).coeffs[1] == 0 --> true
@fact abs(1-xT) --> 1-xT
@fact abs(-1-xT) --> 1+xT
@fact derivative(mod2pi(2pi+yT^3),2) == derivative(yT^3,2) --> true
@fact derivative(yT) == zeroT --> true
@fact -xT/3im == im*xT/3 --> true
@fact (xH/3im)' == im*xH/3 --> true
@fact derivative(2xT*yT^2,1) == 2yT^2 --> true
@fact xT*xT^3 == xT^4 --> true
txy = 1.0 + xT*yT - 0.5*xT^2*yT + (1/3)*xT^3*yT + 0.5*xT^2*yT^2
# @fact (1+taylorN_variable(1,4))^taylorN_variable(2,4) == txy --> true
@fact (1+TaylorN(1,order=4))^TaylorN(2,order=4) == txy --> true
@fact_throws DomainError yT^(-2)
@fact_throws DomainError yT^(-2.0)
@fact (1+xT)^(3//2) == ((1+xT)^0.5)^3 --> true
@fact real(xH) == xH --> true
@fact imag(xH) == zero(xH) --> true
@fact conj(im*yH) == (im*yH)' --> true
@fact conj(im*yT) == (im*yT)' --> true
@fact real( exp(1im * xT)) == cos(xT) --> true
@fact get_coeff(convert(TaylorN{Rational{Int}},cos(xT)),[4,0]) ==
1//factorial(4) --> true
cr = convert(TaylorN{Rational{Int}},cos(xT))
@fact get_coeff(cr,[4,0]) == 1//factorial(4) --> true
@fact imag((exp(yT))^(-1im)') == sin(yT) --> true
exy = exp( xT+yT )
@fact evaluate(exy) == 1 --> true
@fact evaluate(exy,[0.1im,0.01im]) == exp(0.11im) --> true
@fact isapprox(evaluate(exy, [1,1]), e^2) --> true
txy = tan(xT+yT)
@fact get_coeff(txy,[8,7]) == 929569/99225 --> true
ptxy = xT + yT + (1/3)*( xT^3 + yT^3 ) + xT^2*yT + xT*yT^2
# @fact tan(taylorN_variable(1,4)+taylorN_variable(2,4)) == ptxy --> true
@fact tan(TaylorN(1,order=4)+TaylorN(2,order=4)) == ptxy --> true
@fact evaluate(xH*yH,[1.0,2.0]) == 2.0 --> true
g1(xT,yT) = xT^3 + 3yT^2 - 2xT^2 * yT - 7xT + 2
g2(xT,yT) = yT + xT^2 - xT^4
f1 = g1(xT,yT)
f2 = g2(xT,yT)
@fact gradient(f1) == [ 3*xT^2-4*xT*yT-TaylorN(7,0), 6*yT-2*xT^2 ] --> true
@fact ∇(f2) == [2*xT - 4*xT^3, TaylorN(1,0)] --> true
@fact jacobian([f1,f2], [2,1]) ==
jacobian( [g1(xT+2,yT+1), g2(xT+2,yT+1)] ) --> true
@fact [xT yT]*hessian(f1*f2)*[xT, yT] ==
[ 2*TaylorN((f1*f2).coeffs[3]) ] --> true
@fact hessian(f1^2)/2 == [ [49,0] [0,12] ] --> true
@fact hessian(f1-f2-2*f1*f2) == (hessian(f1-f2-2*f1*f2))' --> true
@fact hessian(f1-f2,[1,-1]) == hessian(g1(xT+1,yT-1)-g2(xT+1,yT-1)) --> true
@fact string(-xH) == " - 1 x₁" --> true
@fact string(xT^2) == " 1 x₁² + 𝒪(‖x‖¹⁸)" --> true
@fact string(1im*yT) == " ( 1 im ) x₂ + 𝒪(‖x‖¹⁸)" --> true
@fact string(xT-im*yT) == " ( 1 ) x₁ - ( 1 im ) x₂ + 𝒪(‖x‖¹⁸)" --> true
@fact_throws ArgumentError abs(xT)
@fact_throws AssertionError 1/x
@fact_throws AssertionError zero(x)/zero(x)
@fact_throws ArgumentError sqrt(x)
@fact_throws AssertionError x^(-2)
@fact_throws ArgumentError log(x)
@fact_throws AssertionError cos(x)/sin(y)
end
facts("Testing an identity proved by Euler (8 variables)") do
make_variable(name, index::Int) = string(name, TaylorSeries.subscriptify(index))
@compat variable_names = String[make_variable("α", i) for i in 1:4]
append!(variable_names, [make_variable("β", i) for i in 1:4])
a1, a2, a3, a4, b1, b2, b3, b4 = set_variables(variable_names, order=4)
lhs1 = a1^2 + a2^2 + a3^2 + a4^2
lhs2 = b1^2 + b2^2 + b3^2 + b4^2
lhs = lhs1 * lhs2
rhs1 = (a1*b1 - a2*b2 - a3*b3 - a4*b4)^2
rhs2 = (a1*b2 + a2*b1 + a3*b4 - a4*b3)^2
rhs3 = (a1*b3 - a2*b4 + a3*b1 + a4*b2)^2
rhs4 = (a1*b4 + a2*b3 - a3*b2 + a4*b1)^2
rhs = rhs1 + rhs2 + rhs3 + rhs4
@fact lhs == rhs --> true
v = randn(8)
@fact evaluate( rhs, v) == evaluate( lhs, v) --> true
end
facts("High order polynomials test inspired by Fateman (takes a few seconds))") do
x, y, z, w = set_variables(Int128, "x", numvars=4, order=40)
function fateman2(degree::Int)
T = Int128
oneH = HomogeneousPolynomial(one(T), 0)
# s = 1 + x + y + z + w
s = TaylorN(
[oneH, HomogeneousPolynomial([one(T),one(T),one(T),one(T)],1)], degree)
s = s^degree
# s is converted to order 2*ndeg
s = TaylorN(s, 2*degree)
return s^2 + s
end
function fateman3(degree::Int)
s = x + y + z + w + 1
s = s^degree
s * (s+1)
end
f2 = fateman2(20)
f3 = fateman3(20)
c = get_coeff(f2,[1,6,7,20])
@fact c == 128358585324486316800 --> true
@fact get_coeff(f2,[1,6,7,20]) == c --> true
end
exitstatus()