July 20, 2013
Chapter 12: Best Approximation
Uri M. Ascher and Chen GreifDepartment of Computer ScienceThe University of British Columbia
{ascher,greif}@cs.ubc.ca
Slides for the bookA First Course in Numerical Methods (published by SIAM, 2011)
http://www.ec-securehost.com/SIAM/CS07.html
Best Approximation Goals
Goals of this chapter
• To develop some elegant, classical theory of best approximation using leastsquares and weighted least squares norms;
• to develop important families of orthogonal polynomials;
• to examine in particular Legendre and Chebyshev polynomials.
Uri Ascher & Chen Greif (UBC Computer Science) A First Course in Numerical Methods July 20, 2013 1 / 1
Best Approximation Outline
Outline
• Best least squares approximation
• Orthogonal basis functions
• Weighted least squares
• Chebyshev polynomials
Uri Ascher & Chen Greif (UBC Computer Science) A First Course in Numerical Methods July 20, 2013 2 / 1
Best Approximation Motivation
Approximation, not just interpolation
• As in Chapters 10 and 11, we are given a complicated (or implicit) functionf(x) which may be evaluated anywhere on the interval [a, b], and we look fora simpler approximation v(x) =
∑nj=0 cjϕj(x), where ϕj(x) are known basis
functions with some useful properties.
• Unlike Chapters 10 and 11, we no longer seek to impose v(xi)− f(xi) = 0 atany n + 1 specific interpolation points.
• One reason for shunning interpolation: if there is significant noise in themeasured values of f then measuring the approximation error atm + 1 > n + 1 points may yield more plausible or meaningful results. Thisleads to overdetermined linear systems which are most easily solved usingdiscrete least squares methods; see Section 6.1.
• Here, however, we stick to the continuous, rather than discrete level, and useintegrals rather than vectors. Typically we may seek a function v(x), i.e.,coefficients cj , so as to minimize
∥v − f∥22 =
∫ b
a
(v(x) − f(x))2dx.
Uri Ascher & Chen Greif (UBC Computer Science) A First Course in Numerical Methods July 20, 2013 3 / 1
Best Approximation Motivation
Aside: function norms
• For all integrable functions g(x) and f(x) on an interval [a, b], a norm ∥ · ∥ isa scalar function satisfying
• ∥g∥ ≥ 0; ∥g∥ = 0 iff g(x) ≡ 0,• ∥αg∥ = |α|∥g∥ for all scalars α,• ∥f + g∥ ≤ ∥f∥ + ∥g∥.
The set of all functions whose norm is finite forms a function spaceassociated with that particular norm.
• Some popular function norms and corresponding function spaces are:
L2 : ∥g∥2 =
(∫ b
a
g(x)2dx
)1/2
, (least squares),
L1 : ∥g∥1 =∫ b
a
|g(x)|dx,
L∞ : ∥g∥∞ = maxa≤x≤b
|g(x)|, (max).
• We have seen the L∞ norm in action when estimating errors in Chapters 10and 11. Here we concentrate on the L2 norm, occasionally in weighted form.
Uri Ascher & Chen Greif (UBC Computer Science) A First Course in Numerical Methods July 20, 2013 4 / 1
Best Approximation Motivation
Aside: function norms
• For all integrable functions g(x) and f(x) on an interval [a, b], a norm ∥ · ∥ isa scalar function satisfying
• ∥g∥ ≥ 0; ∥g∥ = 0 iff g(x) ≡ 0,• ∥αg∥ = |α|∥g∥ for all scalars α,• ∥f + g∥ ≤ ∥f∥ + ∥g∥.
The set of all functions whose norm is finite forms a function spaceassociated with that particular norm.
• Some popular function norms and corresponding function spaces are:
L2 : ∥g∥2 =
(∫ b
a
g(x)2dx
)1/2
, (least squares),
L1 : ∥g∥1 =∫ b
a
|g(x)|dx,
L∞ : ∥g∥∞ = maxa≤x≤b
|g(x)|, (max).
• We have seen the L∞ norm in action when estimating errors in Chapters 10and 11. Here we concentrate on the L2 norm, occasionally in weighted form.
Uri Ascher & Chen Greif (UBC Computer Science) A First Course in Numerical Methods July 20, 2013 4 / 1
Best Approximation Motivation
Aside: function norms
• For all integrable functions g(x) and f(x) on an interval [a, b], a norm ∥ · ∥ isa scalar function satisfying
• ∥g∥ ≥ 0; ∥g∥ = 0 iff g(x) ≡ 0,• ∥αg∥ = |α|∥g∥ for all scalars α,• ∥f + g∥ ≤ ∥f∥ + ∥g∥.
The set of all functions whose norm is finite forms a function spaceassociated with that particular norm.
• Some popular function norms and corresponding function spaces are:
L2 : ∥g∥2 =
(∫ b
a
g(x)2dx
)1/2
, (least squares),
L1 : ∥g∥1 =∫ b
a
|g(x)|dx,
L∞ : ∥g∥∞ = maxa≤x≤b
|g(x)|, (max).
• We have seen the L∞ norm in action when estimating errors in Chapters 10and 11. Here we concentrate on the L2 norm, occasionally in weighted form.
Uri Ascher & Chen Greif (UBC Computer Science) A First Course in Numerical Methods July 20, 2013 4 / 1
Best Approximation Motivation
Why continuous approximation?
• Discrete least squares (Chapter 6) is certainly efficiently manageable in manysituations and readily applicable. Indeed, in this chapter we don’t directlyconcentrate on efficient programs or algorithms.
• However, sticking to the continuous allows developing a mathematicallyelegant theory.
• Moreover, classical families of orthogonal basis functions are developed whichprove highly useful in Chapters 13, 15, 16 and others.
Uri Ascher & Chen Greif (UBC Computer Science) A First Course in Numerical Methods July 20, 2013 5 / 1
Best Approximation Continuous least squares
Outline
• Best least squares approximation
• Orthogonal basis functions
• Weighted least squares
• Chebyshev polynomials
Uri Ascher & Chen Greif (UBC Computer Science) A First Course in Numerical Methods July 20, 2013 6 / 1
Best Approximation Continuous least squares
Best least squares approximation
• We are still approximating a given f(x) on interval [a, b] using
v(x) =n∑
j=0
cjϕj(x),
but we are no longer necessarily interpolating.
• Determine the coefficients cj as those that solve the minimization problem
minc
∥f − v∥2 ≡ minc
∫ b
a
f(x) −n∑
j=0
cjϕj(x)
2
dx.
• Taking first derivative with respect to each ck in turn and equating to 0 gives
∫ b
a
f(x) −n∑
j=0
cjϕj(x)
ϕk(x)dx = 0, k = 0, 1, . . . , n.
Uri Ascher & Chen Greif (UBC Computer Science) A First Course in Numerical Methods July 20, 2013 7 / 1
Best Approximation Continuous least squares
Best least squares approximation
• We are still approximating a given f(x) on interval [a, b] using
v(x) =n∑
j=0
cjϕj(x),
but we are no longer necessarily interpolating.
• Determine the coefficients cj as those that solve the minimization problem
minc
∥f − v∥2 ≡ minc
∫ b
a
f(x) −n∑
j=0
cjϕj(x)
2
dx.
• Taking first derivative with respect to each ck in turn and equating to 0 gives
∫ b
a
f(x) −n∑
j=0
cjϕj(x)
ϕk(x)dx = 0, k = 0, 1, . . . , n.
Uri Ascher & Chen Greif (UBC Computer Science) A First Course in Numerical Methods July 20, 2013 7 / 1
Best Approximation Continuous least squares
Best least squares approximation cont.
• This yields the following construction algorithm:...1 Calculate
B̃j,k =
Z b
a
ϕj(x)ϕk(x)dx, j, k = 0, 1, . . . , n
b̃j =
Z b
a
f(x)ϕj(x)dx j = 0, 1, . . . , n.
...2 Solve linear (n + 1) × (n + 1) system
B̃c = b̃
for the coefficients c = (c0, c1, . . . , cn)T .
• Example: using a monomial basis ϕj(x) = xj for a polynomialapproximation on [0, 1], obtain
B̃j,k =∫ 1
0
xj+kdx = 1/(j + k + 1), 0 ≤ j, k ≤ n.
Also, b̃j =∫ 1
0f(x)xjdx can be approximated by Chapter 15 techniques.
Uri Ascher & Chen Greif (UBC Computer Science) A First Course in Numerical Methods July 20, 2013 8 / 1
Best Approximation Continuous least squares
Best least squares approximation cont.
• This yields the following construction algorithm:...1 Calculate
B̃j,k =
Z b
a
ϕj(x)ϕk(x)dx, j, k = 0, 1, . . . , n
b̃j =
Z b
a
f(x)ϕj(x)dx j = 0, 1, . . . , n.
...2 Solve linear (n + 1) × (n + 1) system
B̃c = b̃
for the coefficients c = (c0, c1, . . . , cn)T .
• Example: using a monomial basis ϕj(x) = xj for a polynomialapproximation on [0, 1], obtain
B̃j,k =∫ 1
0
xj+kdx = 1/(j + k + 1), 0 ≤ j, k ≤ n.
Also, b̃j =∫ 1
0f(x)xjdx can be approximated by Chapter 15 techniques.
Uri Ascher & Chen Greif (UBC Computer Science) A First Course in Numerical Methods July 20, 2013 8 / 1
Best Approximation Orthogonal polynomials
Outline
• Best least squares approximation
• Orthogonal basis functions
• Weighted least squares
• Chebyshev polynomials
Uri Ascher & Chen Greif (UBC Computer Science) A First Course in Numerical Methods July 20, 2013 9 / 1
Best Approximation Orthogonal polynomials
Orthogonal basis functions
• The simple example using monomials leads to stability problems (badconditioning in B̃).
• Using monomial and other simple-minded bases, must evaluate manyintegrals when n is large.
• Idea: construct basis functions such that B̃ is diagonal!This means∫ b
a
ϕj(x)ϕk(x)dx = 0, j ̸= k.
• Take advantage of the fact that here, unlike interpolation, there are no inputabscissae: can construct families of orthogonal basis functions in advance!
Uri Ascher & Chen Greif (UBC Computer Science) A First Course in Numerical Methods July 20, 2013 10 / 1
Best Approximation Orthogonal polynomials
Orthogonal basis functions
• The simple example using monomials leads to stability problems (badconditioning in B̃).
• Using monomial and other simple-minded bases, must evaluate manyintegrals when n is large.
• Idea: construct basis functions such that B̃ is diagonal!This means∫ b
a
ϕj(x)ϕk(x)dx = 0, j ̸= k.
• Take advantage of the fact that here, unlike interpolation, there are no inputabscissae: can construct families of orthogonal basis functions in advance!
Uri Ascher & Chen Greif (UBC Computer Science) A First Course in Numerical Methods July 20, 2013 10 / 1
Best Approximation Orthogonal polynomials
Orthogonal basis functions
• The simple example using monomials leads to stability problems (badconditioning in B̃).
• Using monomial and other simple-minded bases, must evaluate manyintegrals when n is large.
• Idea: construct basis functions such that B̃ is diagonal!This means∫ b
a
ϕj(x)ϕk(x)dx = 0, j ̸= k.
• Take advantage of the fact that here, unlike interpolation, there are no inputabscissae: can construct families of orthogonal basis functions in advance!
Uri Ascher & Chen Greif (UBC Computer Science) A First Course in Numerical Methods July 20, 2013 10 / 1
Best Approximation Orthogonal polynomials
Best least squares using orthogonal basis functions
• The construction algorithm becomes simply:...1 For j = 0, 1, . . . , n, set dj = B̃j,j =
R b
aϕ2
j (x)dx in advance, and calculate
b̃j =
Z b
a
f(x)ϕj(x)dx.
...2 The sought coefficients are
cj = b̃j/dj , j = 0, 1, . . . , n.
• Next, restrict to polynomial best approximation and ask, how to construct afamily of orthogonal polynomials?
• ASIDE: Two square-integrable functions g ∈ L2 and h ∈ L2 are orthogonalto each other if their inner product vanishes, i.e., they satisfy∫ b
a
g(x)h(x)dx = 0.
Uri Ascher & Chen Greif (UBC Computer Science) A First Course in Numerical Methods July 20, 2013 11 / 1
Best Approximation Orthogonal polynomials
Best least squares using orthogonal basis functions
• The construction algorithm becomes simply:...1 For j = 0, 1, . . . , n, set dj = B̃j,j =
R b
aϕ2
j (x)dx in advance, and calculate
b̃j =
Z b
a
f(x)ϕj(x)dx.
...2 The sought coefficients are
cj = b̃j/dj , j = 0, 1, . . . , n.
• Next, restrict to polynomial best approximation and ask, how to construct afamily of orthogonal polynomials?
• ASIDE: Two square-integrable functions g ∈ L2 and h ∈ L2 are orthogonalto each other if their inner product vanishes, i.e., they satisfy∫ b
a
g(x)h(x)dx = 0.
Uri Ascher & Chen Greif (UBC Computer Science) A First Course in Numerical Methods July 20, 2013 11 / 1
Best Approximation Orthogonal polynomials
Best least squares using orthogonal basis functions
• The construction algorithm becomes simply:...1 For j = 0, 1, . . . , n, set dj = B̃j,j =
R b
aϕ2
j (x)dx in advance, and calculate
b̃j =
Z b
a
f(x)ϕj(x)dx.
...2 The sought coefficients are
cj = b̃j/dj , j = 0, 1, . . . , n.
• Next, restrict to polynomial best approximation and ask, how to construct afamily of orthogonal polynomials?
• ASIDE: Two square-integrable functions g ∈ L2 and h ∈ L2 are orthogonalto each other if their inner product vanishes, i.e., they satisfy∫ b
a
g(x)h(x)dx = 0.
Uri Ascher & Chen Greif (UBC Computer Science) A First Course in Numerical Methods July 20, 2013 11 / 1
Best Approximation Orthogonal polynomials
Legendre polynomials
Defined on the interval [−1, 1] by
ϕ0(x) = 1,
ϕ1(x) = x,
ϕj+1(x) =2j + 1j + 1
xϕj(x) − j
j + 1ϕj−1(x), j ≥ 1.
So,
ϕ2(x) =12(3x2 − 1),
ϕ3(x) =56x(3x2 − 1) − 2
3x =
12(5x3 − 3x).
Uri Ascher & Chen Greif (UBC Computer Science) A First Course in Numerical Methods July 20, 2013 12 / 1
Best Approximation Orthogonal polynomials
Legendre polynomials
Defined on the interval [−1, 1] by
ϕ0(x) = 1,
ϕ1(x) = x,
ϕj+1(x) =2j + 1j + 1
xϕj(x) − j
j + 1ϕj−1(x), j ≥ 1.
So,
ϕ2(x) =12(3x2 − 1),
ϕ3(x) =56x(3x2 − 1) − 2
3x =
12(5x3 − 3x).
Uri Ascher & Chen Greif (UBC Computer Science) A First Course in Numerical Methods July 20, 2013 12 / 1
Best Approximation Orthogonal polynomials
Legendre polynomials: properties
• Orthogonality:∫ 1
−1
ϕj(x)ϕk(x)dx =
{0 j ̸= k
22j+1 j = k.
• Calibration: |ϕj(x)| ≤ 1, −1 ≤ x ≤ 1, and ϕj(1) = 1.
• Oscillation: ϕj(x) has degree j (not less). All its j zeros are simple and lieinside the interval (−1, 1). Hence the polynomial oscillates j times in thisinterval.
Uri Ascher & Chen Greif (UBC Computer Science) A First Course in Numerical Methods July 20, 2013 13 / 1
Best Approximation Orthogonal polynomials
Legendre polynomials: the picture
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
x
φ j
You can determine which curve corresponds to which ϕj by the number ofoscillations.
Uri Ascher & Chen Greif (UBC Computer Science) A First Course in Numerical Methods July 20, 2013 14 / 1
Best Approximation Orthogonal polynomials
Example
• Find polynomial best approximations for f(x) = cos(2πx) over [0, 1] using northogonal polynomials.
• Using Legendre polynomials (note interval change from [−1, 1] to [0, 1]),b̃j =
∫ 1
0cos(2πt)ϕj(2t − 1)dt, and cj = (2j + 1)b̃j .
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−1
−0.5
0
0.5
1
1.5
2
x
p n
exactn=0 or 1n=2 or 3n=4
Here f is in green. Obtain the same approximation for n = 2l and n = 2l + 1due to symmetry of f . Note improvement in error as l increases.
Uri Ascher & Chen Greif (UBC Computer Science) A First Course in Numerical Methods July 20, 2013 15 / 1
Best Approximation Orthogonal polynomials
Example
• Find polynomial best approximations for f(x) = cos(2πx) over [0, 1] using northogonal polynomials.
• Using Legendre polynomials (note interval change from [−1, 1] to [0, 1]),b̃j =
∫ 1
0cos(2πt)ϕj(2t − 1)dt, and cj = (2j + 1)b̃j .
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−1
−0.5
0
0.5
1
1.5
2
x
p n
exactn=0 or 1n=2 or 3n=4
Here f is in green. Obtain the same approximation for n = 2l and n = 2l + 1due to symmetry of f . Note improvement in error as l increases.
Uri Ascher & Chen Greif (UBC Computer Science) A First Course in Numerical Methods July 20, 2013 15 / 1
Best Approximation Orthogonal polynomials
Example
• Find polynomial best approximations for f(x) = cos(2πx) over [0, 1] using northogonal polynomials.
• Using Legendre polynomials (note interval change from [−1, 1] to [0, 1]),b̃j =
∫ 1
0cos(2πt)ϕj(2t − 1)dt, and cj = (2j + 1)b̃j .
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−1
−0.5
0
0.5
1
1.5
2
x
p n
exactn=0 or 1n=2 or 3n=4
Here f is in green. Obtain the same approximation for n = 2l and n = 2l + 1due to symmetry of f . Note improvement in error as l increases.
Uri Ascher & Chen Greif (UBC Computer Science) A First Course in Numerical Methods July 20, 2013 15 / 1
Best Approximation Orthogonal trigonometric polynomials
Trigonometric polynomials
• Not all interesting orthogonal basis functions are polynomials.
• Of particular interest are the periodic basis functions
ϕ0(x) =1√2π
, ϕ2l−1(x) =1√π
sin(lx), ϕ2l(x) =1√π
cos(lx) l = 1, 2, . . . n/2.
• These are orthonormal on [−π, π]:∫ π
−π
ϕi(x)ϕj(x)dx =
{0, i ̸= j
1, i = j.
• Lead to the famous Fourier transform (Chapter 13).
Uri Ascher & Chen Greif (UBC Computer Science) A First Course in Numerical Methods July 20, 2013 16 / 1
Best Approximation Orthogonal trigonometric polynomials
Trigonometric polynomials
• Not all interesting orthogonal basis functions are polynomials.
• Of particular interest are the periodic basis functions
ϕ0(x) =1√2π
, ϕ2l−1(x) =1√π
sin(lx), ϕ2l(x) =1√π
cos(lx) l = 1, 2, . . . n/2.
• These are orthonormal on [−π, π]:∫ π
−π
ϕi(x)ϕj(x)dx =
{0, i ̸= j
1, i = j.
• Lead to the famous Fourier transform (Chapter 13).
Uri Ascher & Chen Greif (UBC Computer Science) A First Course in Numerical Methods July 20, 2013 16 / 1
Best Approximation Weighted least squares
Outline
• Best least squares approximation
• Orthogonal basis functions
• Weighted least squares
• Chebyshev polynomials
Uri Ascher & Chen Greif (UBC Computer Science) A First Course in Numerical Methods July 20, 2013 17 / 1
Best Approximation Weighted least squares
Weighted least squares
• An important generalization of best least squares approximation: with aweight function w(x) that is never negative, has bounded L2 norm, and mayequal 0 only at isolated points, seek coefficients cj for v(x) =
∑nj=0 cjϕj(x)
so as to minimize∫ b
a
w(x)(v(x) − f(x))2dx.
• The orthogonality condition correspondingly generalizes to∫ b
a
w(x)ϕj(x)ϕk(x)dx = 0, j ̸= k.
• If this holds then the best approximation solution is given by cj = b̃j/dj ,
where dj =∫ b
aw(x)ϕ2
j (x)dx > 0 and
b̃j =∫ b
a
w(x)f(x)ϕj(x)dx, j = 0, 1, . . . , n.
• Important families of orthogonal polynomials may be obtained for goodchoices of the weight function.
Uri Ascher & Chen Greif (UBC Computer Science) A First Course in Numerical Methods July 20, 2013 18 / 1
Best Approximation Weighted least squares
Weighted least squares
• An important generalization of best least squares approximation: with aweight function w(x) that is never negative, has bounded L2 norm, and mayequal 0 only at isolated points, seek coefficients cj for v(x) =
∑nj=0 cjϕj(x)
so as to minimize∫ b
a
w(x)(v(x) − f(x))2dx.
• The orthogonality condition correspondingly generalizes to∫ b
a
w(x)ϕj(x)ϕk(x)dx = 0, j ̸= k.
• If this holds then the best approximation solution is given by cj = b̃j/dj ,
where dj =∫ b
aw(x)ϕ2
j (x)dx > 0 and
b̃j =∫ b
a
w(x)f(x)ϕj(x)dx, j = 0, 1, . . . , n.
• Important families of orthogonal polynomials may be obtained for goodchoices of the weight function.
Uri Ascher & Chen Greif (UBC Computer Science) A First Course in Numerical Methods July 20, 2013 18 / 1
Best Approximation Weighted least squares
Gram-Schmidt process for constructing orthogonalpolynomials
• Given an interval [a, b] and a weight function w(x), this algorithm producesan orthogonal polynomial family via a short three-term recurrence!
• Set
ϕ0(x) = 1,
ϕ1(x) = x − β1,
ϕj(x) = (x − βj)ϕj−1(x) − γjϕj−2(x), j ≥ 2,
where
βj =
∫ b
axw(x)[ϕj−1(x)]2dx∫ b
aw(x)[ϕj−1(x)]2dx
, j ≥ 1,
γj =
∫ b
axw(x)ϕj−1(x)ϕj−2(x)dx∫ b
aw(x)[ϕj−2(x)]2dx
j ≥ 2.
• Intimately connected also to the derivation of Krylov subspace iterativemethods for symmetric linear systems (CG, MINRES); see Section 7.5.
Uri Ascher & Chen Greif (UBC Computer Science) A First Course in Numerical Methods July 20, 2013 19 / 1
Best Approximation Chebyshev polynomials
Outline
• Best least squares approximation
• Orthogonal basis functions
• Weighted least squares
• Chebyshev polynomials
Uri Ascher & Chen Greif (UBC Computer Science) A First Course in Numerical Methods July 20, 2013 20 / 1
Best Approximation Chebyshev polynomials
Chebyshev polynomials
• This is a family of orthogonal polynomials on [−1, 1] with respect to theweight function
w(x) =1√
1 − x2.
• Define
ϕj(x) = Tj(x) = cos(j arccos x) = cos(jθ),
where x = cos(θ).• So, these polynomials are naturally defined in terms of an angle, θ. Clearly,
the larger j the more oscillatory Tj .
• Easy to verify orthogonality:∫ 1
−1
w(x)Tj(x)Tk(x)dx =∫ 1
−1
Tj(x)Tk(x)√1 − x2
dx
=∫ π
0
cos(jθ) cos(kθ)dθ =
{0 j ̸= kπ2 j = k > 0
.
Uri Ascher & Chen Greif (UBC Computer Science) A First Course in Numerical Methods July 20, 2013 21 / 1
Best Approximation Chebyshev polynomials
Chebyshev polynomials: derivation and properties
• These polynomials obey the simple 3-term recursion
T0(x) = 1,
T1(x) = x,
Tj+1(x) = 2xTj(x) − Tj−1(x), j ≥ 1,
and obviously satisfy
|Tj(x)| ≤ 1, ∀x.
• The roots (zeros) of Tn are the Chebyshev points
xk = cos(
2k − 12n
π
), k = 1, . . . , n.
See Section 10.6 for their magic properties.
• The also-important interleaving extremal points (where Tn(x) = ±1) are
ξk = cos(
k
nπ
), k = 0, 1, . . . , n.
Uri Ascher & Chen Greif (UBC Computer Science) A First Course in Numerical Methods July 20, 2013 22 / 1
Best Approximation Chebyshev polynomials
Chebyshev polynomials: the picture
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
x
Tj
T4
T6
T5
You can determine which curve corresponds to which ϕj = Tj by the number ofoscillations. Note the perfect calibration.
Uri Ascher & Chen Greif (UBC Computer Science) A First Course in Numerical Methods July 20, 2013 23 / 1
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