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Accelerating Convergence
Zeros of Polynomials Deflation, M¨
uller’s Method Polynomial Approximation
Numerical Analysis and Computing
Lecture Notes #04 — Solutions of Equations in One Variable,Interpolation and Polynomial Approximation — AcceleratingConvergence; Zeros of Polynomials; Deflation; M¨
uller’s Method;
Lagrange Polynomials; Neville’s Method
Peter Blomgren,
〈[email protected]
Department of Mathematics and Statistics
Dynamical Systems Group Computational Sciences Research Center^ San Diego State UniversitySan Diego, CA 92182-7720^ http://terminus.sdsu.edu/
Fall 2009
Peter Blomgren,
〈[email protected]
〉^
#4: Solutions of Equations in One Variable
— (1/57)
Accelerating Convergence
Zeros of Polynomials Deflation, M¨
uller’s Method Polynomial Approximation
Outline
1
Accelerating Convergence^ Review^ Aitken’s ∆
2 Method
Steffensen’s Method 2
Zeros of Polynomials^ Fundamentals^ Horner’s Method 3
Deflation, M¨
uller’s Method
Deflation: Finding All the Zeros of a Polynomial M¨uller’s Method — Finding Complex Roots 4
Polynomial Approximation^ Fundamentals^ Moving Beyond Taylor Polynomials^ Lagrange Interpolating Polynomials^ Neville’s Method^ Peter Blomgren,
〈[email protected]
〉^
#4: Solutions of Equations in One Variable
— (2/57)
Accelerating Convergence
Zeros of Polynomials Deflation, M¨
uller’s Method Polynomial Approximation
ReviewAitken’s
(^2) ∆ Method
Steffensen’s Method
Introduction
“It is rare to have the luxury of quadratic convergence.”
(Burden-Faires, p.83)
There are a number of methods for squeezing faster convergence out ofan^
already computed sequence
of numbers.
We here explore one method which seems the have been around since thebeginning of numerical analysis...
Aitken’s ∆
2 method
. It can be used
to accelerate convergence of a sequence that is linearly convergent,regardless of its origin or application.A review of modern extrapolation methods can be found in:^ “Practical Extrapolation Methods:
Theory and Applications,”
Avram
Sidi, Number 10 in Cambridge Monographs on Applied and Compu-tational Mathematics, Cambridge University Press, June 2003.
ISBN:
0-521-66159-5^ Peter Blomgren,
〈[email protected]
〉^
#4: Solutions of Equations in One Variable
— (3/57)
Accelerating Convergence
Zeros of Polynomials Deflation, M¨
uller’s Method Polynomial Approximation
ReviewAitken’s
(^2) ∆ Method
Steffensen’s Method
Recall: Convergence of a Sequence
Definition Suppose the sequence
{p
∞}n n
converges to
p, with
pn
p^
for all
n. If positive constants
λ^
and
α
exists with
lim n→∞
|pn
+^
−^ p
|pn
p|
α^
=^
λ
then
{p
}n ∞ n=
converges to
p^
of order
α, with asymptotic error
constant
λ.
Linear convergence means that
α^
= 1, and
|λ
|^ <
Peter Blomgren,
〈[email protected]
〉^
#4: Solutions of Equations in One Variable
— (4/57)
Accelerating Convergence
Zeros of Polynomials Deflation, M¨
uller’s Method Polynomial Approximation
ReviewAitken’s
(^2) ∆ Method
Steffensen’s Method
Aitken’s ∆
2 Method
I/II
Assume
{p
}n ∞ n=
is a
linearly convergent sequence
with limit
p.
Further, assume we are far out into the tail of the sequence (
n
large), and the signs of the successive errors agree,
i.e.
sign
(pn
p) =
sign
(pn
p) =
sign
(pn
p) =
and that
pn+
p
pn+
p^
≈^
pn+
p pn^
−^
p^
≈^
λ^
(the asymptotic limit)
This would indicate
(pn
p)
(p
n+
p)(
pn^
−^
p)
(^2) p n+
2 p
n+
p^ +
(^2) p
pn
pn^
−^
(pn
+^
+^ p
)pn
(^2) p
We solve for
p^
and get...
Peter Blomgren,
〈[email protected]
〉^
#4: Solutions of Equations in One Variable
— (5/57)
Accelerating Convergence
Zeros of Polynomials Deflation, M¨
uller’s Method Polynomial Approximation
ReviewAitken’s
(^2) ∆ Method
Steffensen’s Method
Aitken’s ∆
2 Method
II/II
We solve for
p^
and get...
p^ ≈
pn+
pn^
−^
(^2) p n+
pn+
2 p
n+
pn
A little bit of algebraic manipulation put this into the classicalAitken form:
ˆpn^
=^
p^ =
pn
(pn
pn
pn+
2 p
n+
pn
Aitken’s ∆
2 Method is based on the assumption that the ˆ
pn^
we
compute from
pn
,^ p
n+
and
pn
is a better approximation to the
real limit
p.
The analysis needed to
prove
this is beyond the scope of this class, see
e.g.
Sidi’s book. Peter Blomgren,
〈[email protected]
〉^
#4: Solutions of Equations in One Variable
— (6/57)
Accelerating Convergence
Zeros of Polynomials Deflation, M¨
uller’s Method Polynomial Approximation
ReviewAitken’s
(^2) ∆ Method
Steffensen’s Method
Aitken’s ∆
2 Method
The Recipe
Given a sequence finite
{p
}n Nn=
or infinite
{q
∞}n n
sequence
which converges linearly to some limit.Define the new sequences
ˆpn^
=^
pn^
−^
(pn
pn
pn+
2 p
n+
pn
,^
n^ = 0
,^1 ,... ,
N^
−^
or
ˆqn^
=^
qn^
−^
(qn
qn
qn+
2 q
n+
qn
,^
n^ = 0
,^1 ,... ,
Peter Blomgren,
〈[email protected]
〉^
#4: Solutions of Equations in One Variable
— (7/57)
Accelerating Convergence
Zeros of Polynomials Deflation, M¨
uller’s Method Polynomial Approximation
ReviewAitken’s
(^2) ∆ Method
Steffensen’s Method
Aitken’s ∆
2 Method
Example
Consider the sequence
{p
∞}n n
, where the sequence is generated by the=
fixed point iteration
pn
+^
= cos(
p),n
p^0
= 0.
Iteration
pn^
ˆpn
0
1
28010361467617
2
3665164585231
3
6906294340474
4
8050421371664
5
93480358742566
8636096881655
6
01368773622757
8876582817136
7
63959682900654
8992243027034
8
22102425026708
42511328159
9
50417761763761
65949599941
10
1404042422510
76383318956
11
44237354900557
1177259563
∗
12
5604740436347
3333909684
∗
Note:
Bold digits are correct; ˆ
p^11
needs
p^13
, and ˆ
p^12
additionally needs
p^14
.
Peter Blomgren,
〈[email protected]
〉^
#4: Solutions of Equations in One Variable
— (8/57)
Accelerating Convergence
Zeros of Polynomials Deflation, M¨
uller’s Method Polynomial Approximation
FundamentalsHorner’s Method
Zeros of Polynomials
Definition: Degree of a Polynomial A^
polynomial of degree
n^
has the form
P(
x) =
an
n^ x
+^
an−
x 1 n−
a^1
x^ +
a^0
,^
an^
where the
ai
’s are constants (either real, or complex) called the
coefficients
of
P
Why look at polynomials? — We’ll be looking at the problem P(
x) = 0 (
i.e. f
(x
) = 0 for a special class of functions.)
Polynomials are the basis for many approximation methods, hencebeing able to solve polynomial equations fast is valuable.We’d like to use Newton’s method, so we need to compute
P
(x)
and
P
′(x
) as efficiently as possible. Peter Blomgren,
〈[email protected]
〉^
#4: Solutions of Equations in One Variable
— (13/57)
Accelerating Convergence
Zeros of Polynomials Deflation, M¨
uller’s Method Polynomial Approximation
FundamentalsHorner’s Method
Fundamentals
Theorem (The Fundamental Theorem of Algebra) If P
(x)
is a polynomial of degree n
with real or complex
coefficients, then P
(x) = 0
has at least one (possibly complex)
root. The proof is surprisingly(?) difficult and requires understanding ofcomplex analysis... We leave it as an exercise for the motivatedstudent!
Peter Blomgren,
〈[email protected]
〉^
#4: Solutions of Equations in One Variable
— (14/57)
Accelerating Convergence
Zeros of Polynomials Deflation, M¨
uller’s Method Polynomial Approximation
FundamentalsHorner’s Method
Key Consequences of the Fundamental Theorem of Algebra
1 of 2
Corollary If^ P
(x) is a polynomial of degree
n^
≥^
1 with real or complex
coefficients then there exists unique constants
x^1
,^ x
,^2
,^ x
k
(possibly complex) and unique positive integers
m
,^1
m^2
,^...
,^ m
k
such that
ki=
m
=i n^
and
P(
x) =
an
(x^
−^ x
m) 1 1 (x
x^2
m)
(x
xk
m) k
The collection of zeros is unique. —
m
are the multiplicities of the individual zeros.i
—
A
polynomial
of
degree
n
has
exactly
n
zeros,
counting
multiplicity. Peter Blomgren,
〈[email protected]
〉^
#4: Solutions of Equations in One Variable
— (15/57)
Accelerating Convergence
Zeros of Polynomials Deflation, M¨
uller’s Method Polynomial Approximation
FundamentalsHorner’s Method
Key Consequences of the Fundamental Theorem of Algebra
2 of 2
Corollary Let
P
(x) and
Q
(x) be polynomials of degree at most
n. If
x^1
,^ x
,^ x
, withk
k^
>^
n^ are
distinct
numbers with
P
(xi
Q
(xi
) for
i^ = 1
,^2 ,... ,
k, then
P
(x) =
Q
(x) for all values of
x.
If two polynomials of degree
n^
agree at at least (
n^ + 1) points,
then they must be the same. Peter Blomgren,
〈[email protected]
〉^
#4: Solutions of Equations in One Variable
— (16/57)
Accelerating Convergence
Zeros of Polynomials Deflation, M¨
uller’s Method Polynomial Approximation
FundamentalsHorner’s Method
Horner’s Method: Evaluating Polynomials Quickly
1 of 3
Let
P(
x) =
an
n^ x +^ a
n−
x 1 n−
a^1
x^ +
a^0
If we are looking to evaluate
P
(x^0
) for any
x^0
, let
bn^
=^
a,n
bk^
=^
ak^
+^ b
k+
x,^0
k^ = (
n^ −
,^ (n
1 ,^
then
b^0
P
(x^0
). We have only used
n^
multiplications and
n
additions.At the same time we have computed the decomposition
P(
x) = (
x^ −
x^0
)Q
(x) +
b^0
where
Q(
x) =
n−
1 ∑ k=
bk+
k^ x
Peter Blomgren,
〈[email protected]
〉^
#4: Solutions of Equations in One Variable
— (17/57)
Accelerating Convergence
Zeros of Polynomials Deflation, M¨
uller’s Method Polynomial Approximation
FundamentalsHorner’s Method
Horner’s Method: Evaluating Polynomials Quickly
2 of 3
Huh?!? Where did the expression come from? — Consider
P(
x)^
=^
axn
n^ +
an
−^1
nx −^1
a^1
x^ +
a^0
=^
(an
n−x
an
−^1
n−x
+^
a)^1
x^ +
a^0
=^
((a
xn n−
an
−^1
nx −^3
+^ · · ·
)x
a^1
)x^
+^
a^0
=^
︸^ ︷︷
n−
1
axn
an
−^1
︸^
︷︷^
bn−
1
)x^
+^
)x
a^1
)x^
+^
a^0
Horner’s method is “simply” the computation of this parenthesizedexpression from the inside-out...
Peter Blomgren,
〈[email protected]
〉^
#4: Solutions of Equations in One Variable
— (18/57)
Accelerating Convergence
Zeros of Polynomials Deflation, M¨
uller’s Method Polynomial Approximation
FundamentalsHorner’s Method
Horner’s Method: Evaluating Polynomials Quickly
3 of 3
Now, if we need to compute
P
′(x
) we have 0
′ P
(x)
∣∣∣∣x=
x^0
x^ −
x^0
)Q
′(x
Q
(x)
∣∣∣∣x=
x^0
=^
Q(
x)^0
Which we can compute (again using Horner’s method) in (
n^ −
multiplications and (
n^ −
- additions.
Proof?
We really ought to prove that Horner’s method works. It
basically boils down to lots of algebra which shows that thecoefficients of
P
(x) and (
x^ −
x^0
)Q
(x) +
b^0
are the same...
A couple of examples may be more instructive...
Peter Blomgren,
〈[email protected]
〉^
#4: Solutions of Equations in One Variable
— (19/57)
Accelerating Convergence
Zeros of Polynomials Deflation, M¨
uller’s Method Polynomial Approximation
FundamentalsHorner’s Method
Example#1: Horner’s Method
For
P
(x) =
(^4) x
(^3) x
(^2) x
x^
−^
1, compute
P
x^0
a^4
a^3
=^
−^1
a^2
a^1
a^0
=^
−^1
b^4
x^0
bx^3
bx^2
b^1
x^0
=^
b^4
b^3
b^2
b^1
b^0
=^
Hence,
P
(^5 ) =
, and
P(
x) = (
x^ −
(^3) x
(^2) x
x^ + 106) + 529
Similarly we get
P
Q
x^0
a^3
a^2
a^1
a^0
bx^3
b^2
x^0
bx^1
b^3
b^2
b^1
b^0
=^
Peter Blomgren,
〈[email protected]
〉^
#4: Solutions of Equations in One Variable
— (20/57)
Accelerating Convergence
Zeros of Polynomials Deflation, M¨
uller’s Method Polynomial Approximation
Deflation: Finding All the Zeros of a PolynomialM¨uller’s Method — Finding Complex Roots
Quality of Deflation
Now, the big question is
“are the approximate roots
ˆr^1
,^ ˆr
,^2
ˆrn^
good approximations of the roots of
P
(x)
Unfortunately, sometimes,
no
In each step we solve the equation to some tolerance,
i.e.
( |b k)| 0
tol
Even though we may solve to a tight tolerance (
−^8
), the errors
accumulate and the inaccuracies increase iteration-by-iteration... Question:
Is deflation therefore useless???
Peter Blomgren,
〈[email protected]
〉^
#4: Solutions of Equations in One Variable
— (25/57)
Accelerating Convergence
Zeros of Polynomials Deflation, M¨
uller’s Method Polynomial Approximation
Deflation: Finding All the Zeros of a PolynomialM¨uller’s Method — Finding Complex Roots
Improving the Accuracy of Deflation
The problem with deflation is that the zeros of
Q
(xk ) are not good
representatives of the zeros of
P
(x), especially for high
k’s.
As
k^
increases, the quality of the root ˆ
rk^
decreases.
Maybe there is a way to get all the zeros with the same quality?The idea is quite simple... in each step of deflation, instead of justaccepting ˆ
rk^
as a root of
P
(x), we re-run Newton’s method on the
full polynomial
P
(x), with ˆ
rk^
as the starting point — a couple of
Newton iterations should quickly converge to the root of the fullpolynomial.
Peter Blomgren,
〈[email protected]
〉^
#4: Solutions of Equations in One Variable
— (26/57)
Accelerating Convergence
Zeros of Polynomials Deflation, M¨
uller’s Method Polynomial Approximation
Deflation: Finding All the Zeros of a PolynomialM¨uller’s Method — Finding Complex Roots
Improved Deflation — Algorithm Outline
Algorithm Outline: Improved Deflation 1.
Apply Newton’s method to
P
(x)
→^
ˆr,^1
Q
(x 1
For
k^
,^3 ,... ,
(n
- do
Apply Newton’s method to
Q
k−^1
ˆr
∗,^ k
∗Q k (x).
Apply Newton’s method to
P
(x) with
∗ ˆr k as the initial point
ˆrk Apply Horner’s method to
Q
k−
(x 1 ) with
x^
=^
ˆrk^
Q
(xk
Use the quadratic formula on
Q
n−
(x 2 ) to get the two remaining
roots. Note:
“Inside” Newton’s method, the evaluations of polynomi-als and their derivatives are also performed using Horner’smethod. Peter Blomgren,
〈[email protected]
〉^
#4: Solutions of Equations in One Variable
— (27/57)
Accelerating Convergence
Zeros of Polynomials Deflation, M¨
uller’s Method Polynomial Approximation
Deflation: Finding All the Zeros of a PolynomialM¨uller’s Method — Finding Complex Roots
Deflation & Improvement
Wilkinson Polynomials
The Wilkinson Polynomials
W P n (x
n∏^ ( k=
x^ −
k)
have the roots
{^1
,^2 ,... ,
n}
, but provide surprisingly tough
numerical root-finding problems.
(Additional details in
Math 543
In the next few slides we show the results of Deflation andImproved Deflation applied to Wilkinson polynomials of degree 9,10, 12, and 13.
Peter Blomgren,
〈[email protected]
〉^
#4: Solutions of Equations in One Variable
— (28/57)
Accelerating Convergence
Zeros of Polynomials Deflation, M¨
uller’s Method Polynomial Approximation
Deflation: Finding All the Zeros of a PolynomialM¨uller’s Method — Finding Complex Roots
Deflation & Improvement
WP 9
x) and
P
W( 10
x)
0
2
4
6
8
10
−8^10 −10 10 −12 10 −14 10 −16 10
Relative Error of the Computed Roots DeflationImproved
0
2
4
6
8
10
−8^10 −10 10 −12 10 −14 10
Relative Error of the Computed Roots DeflationImproved
Figure:
[Left]
The result of the two algorithms on the Wilkinson polynomial of degree
9; in this case the roots are computed so that
( |b k)|^0
<^10
−^6.
[Right]
The result of
the two algorithms on the Wilkinson polynomial of degree 10; in this case the roots arecomputed so that
( |b k)|^0
<^10
−^6.
In both cases the
lower line
corresponds to
improved
deflation
and we see that we get an improvement in the relative error of several orders of magnitude.
Peter Blomgren,
〈[email protected]
〉^
#4: Solutions of Equations in One Variable
— (29/57)
Accelerating Convergence
Zeros of Polynomials Deflation, M¨
uller’s Method Polynomial Approximation
Deflation: Finding All the Zeros of a PolynomialM¨uller’s Method — Finding Complex Roots
Deflation & Improvement
WP 12
(x
) and
P
W( 13
x)
0
2
4
6
8
10
12
−6 10 −8 10 −10 10 −12 10 −14 10 −16 10
Relative Error of the Computed Roots DeflationImproved
0
2
4
6
8
10
12
14
−4^10 −6 10 −8 10 −10 10 −12 10 −14 10 −16 10
Relative Error of the Computed Roots DeflationImproved
Figure:
[Left]
The result of the two algorithms on the Wilkinson polynomial of degree
12; in this case the roots are computed so that
( |b k)|^0
<^10
−^4.
[Right]
The result of
the two algorithms on the Wilkinson polynomial of degree 13; in this case the roots arecomputed so that
( |b k)|^0
<^10
−^3.
In both cases the
lower line
corresponds to
improved
deflation
and we see that we get an improvement in the relative error of several orders of magnitude.
Peter Blomgren,
〈[email protected]
〉^
#4: Solutions of Equations in One Variable
— (30/57)
Accelerating Convergence
Zeros of Polynomials Deflation, M¨
uller’s Method Polynomial Approximation
Deflation: Finding All the Zeros of a PolynomialM¨uller’s Method — Finding Complex Roots
What About Complex Roots???
One interesting / annoying feature of polynomials with realcoefficients is that they may have complex roots,
e.g.
P(
x) =
(^2) x
i,^
i}. Where by definition
i^ =
If the initial approximation given to Newton’s method is real, allthe successive iterates will be real... which means we will not findcomplex roots.One way to overcome this is to start with a complex initialapproximation and do all the computations in complex arithmetic.Another solution is
M¨
uller’s Method
Peter Blomgren,
〈[email protected]
〉^
#4: Solutions of Equations in One Variable
— (31/57)
Accelerating Convergence
Zeros of Polynomials Deflation, M¨
uller’s Method Polynomial Approximation
Deflation: Finding All the Zeros of a PolynomialM¨uller’s Method — Finding Complex Roots
M¨uller’s Method
M¨uller’s method is an extension of the Secant method...Recall that the secant method uses two points
xk
and
xk
−^1
and
the function values in those two points
f^ (
x) andk^
f^ (
xk−
). The 1
zero-crossing of the linear interpolant (the secant line) is used asthe next iterate
xk
M¨uller’s method takes the next logical step: it uses
three points
x,k^
xk
−^1
and
xk
−^2
, the function values in those points
f^ (
x),k^
f^ (x
k−
) and 1
f^ (
xk−
); a second degree polynomial fitting these 2
three points is found, and its zero-crossing is the next iterate
xk
Next slide:
f^ (
x) =
(^4) x
3 x
xk
−^2
xk
−^1
xk
Peter Blomgren,
〈[email protected]
〉^
#4: Solutions of Equations in One Variable
— (32/57)
Accelerating Convergence
Zeros of Polynomials Deflation, M¨
uller’s Method Polynomial Approximation
Deflation: Finding All the Zeros of a PolynomialM¨uller’s Method — Finding Complex Roots
Now We Know “Everything” About Solving
f^ (
x) = 0 !?
Let’s recap... Things to remember...The relation between
root finding
(f
(x
) = 0) and
fixed point
(g^
(x) =
x).
Key algorithms for root finding: Bisection, Secant Method, and Newton’s Method
. — Know what they are (the updates), how to
start (one or two points? bracketing or not bracketing the root?),can the method break, can breakage be fixed? Convergenceproperties.Also, know the mechanics of the Regula Falsi method, andunderstand why it can run into trouble.Fixed point iteration: Under what conditions do FP-iterationconverge for all starting values in the interval?
Peter Blomgren,
〈[email protected]
〉^
#4: Solutions of Equations in One Variable
— (37/57)
Accelerating Convergence
Zeros of Polynomials Deflation, M¨
uller’s Method Polynomial Approximation
Deflation: Finding All the Zeros of a PolynomialM¨uller’s Method — Finding Complex Roots
Recap, continued...
Basic error analysis: order
α, asymptotic error constant
λ. —
Which one has the most impact on convergence? Convergencerate for general fixed point iterations?Multiplicity of zeros: What does it mean? How do we use thisknowledge to “help” Newton’s method when we’re looking for azero of high multiplicity?Convergence acceleration: Aitken’s ∆
2 -method. Steffensen’s
Method.Zeros of polynomials: Horner’s method, Deflation (withimprovement), M¨
uller’s method.
Peter Blomgren,
〈[email protected]
〉^
#4: Solutions of Equations in One Variable
— (38/57)
Accelerating Convergence
Zeros of Polynomials Deflation, M¨
uller’s Method Polynomial Approximation
Deflation: Finding All the Zeros of a PolynomialM¨uller’s Method — Finding Complex Roots
Homework #3 — Due 10/2/2009, 12-noon
Final Version
(Part-1)Implement Steffensen’s method in matlab (or other language).Apply to
g^ (
x) = 1 + (sin(
x))
2 ,^
p^0
= 1, and compute the first 5
iterates.(Part-2)Implement M¨
uller’s method in matlab (or other language). Use to
solve
BF-2.6.7bce
, but go up to accuracy 10
−^8
. Turn in your
code for M¨
uller’s method, and the iteration tables
{x
,^ k f^ (x
)}k nk=
for all methods.
Peter Blomgren,
〈[email protected]
〉^
#4: Solutions of Equations in One Variable
— (39/57)
Accelerating Convergence
Zeros of Polynomials Deflation, M¨
uller’s Method Polynomial Approximation
FundamentalsMoving Beyond Taylor PolynomialsLagrange Interpolating PolynomialsNeville’s Method
New Favorite Problem:
Interpolation and Polynomial Approximation
Peter Blomgren,
〈[email protected]
〉^
#4: Solutions of Equations in One Variable
— (40/57)
Accelerating Convergence
Zeros of Polynomials Deflation, M¨
uller’s Method Polynomial Approximation
FundamentalsMoving Beyond Taylor PolynomialsLagrange Interpolating PolynomialsNeville’s Method
Weierstrass Approximation Theorem
The following theorem is the basis for polynomial approximation: Theorem (Weierstrass Approximation Theorem) Suppose f
C^
[a,
b]
. Then
∀ǫ >
∃^ a polynomial P
(x) :
|f^ (
x)^
−^ P
(x)
|^ < ǫ,
∀x
[a
,^ b
].
Note:
The bound is
uniform
,^ i.e.
valid for all
x^
in the interval.
Note:
The theorem says nothing about how to find the polyno-mial, or about its order. Peter Blomgren,
〈[email protected]
〉^
#4: Solutions of Equations in One Variable
— (41/57)
Accelerating Convergence
Zeros of Polynomials Deflation, M¨
uller’s Method Polynomial Approximation
FundamentalsMoving Beyond Taylor PolynomialsLagrange Interpolating PolynomialsNeville’s Method
Illustrated: Weierstrass Approximation Theorem
0
2
4
6
8
10
2.5^2 1.5^1 0.5^0 −0.5^ −
f f+ ε f− ε
Figure:
Weierstrass approximation Theorem guarantees that we (maybe with sub- stantial work) can find a polynomial which fits into the “tube” around the function f^ , no matter how thin we make the tube.^ Peter Blomgren,
〈[email protected]
〉^
#4: Solutions of Equations in One Variable
— (42/57)
Accelerating Convergence
Zeros of Polynomials Deflation, M¨
uller’s Method Polynomial Approximation
FundamentalsMoving Beyond Taylor PolynomialsLagrange Interpolating PolynomialsNeville’s Method
Candidates: the Taylor Polynomials???
Natural Question:
Are our old friends, the Taylor Polynomials, good candidatesfor polynomial interpolation? Answer:
No.
The Taylor expansion works very hard to be accurate in the neighborhood of
one point
.^
But we want to fit data at
many points (in an extended interval). [Next slide: The approximation is great near the expansion point x^0
= 0, but get progressively worse at we get further away from the point, even for the higher degree approximations.]
Peter Blomgren,
〈[email protected]
〉^
#4: Solutions of Equations in One Variable
— (43/57)
Accelerating Convergence
Zeros of Polynomials Deflation, M¨
uller’s Method Polynomial Approximation
FundamentalsMoving Beyond Taylor PolynomialsLagrange Interpolating PolynomialsNeville’s Method
Taylor Approximation of
x e on the Interval [
,^ 3]
0
0.^
1
1.^
2
2.^
3
20 15 10 5 0
e^x P_0(x) P_1(x) P_2(x) P_3(x) P_4(x) P_5(x)
Peter Blomgren,
〈[email protected]
〉^
#4: Solutions of Equations in One Variable
— (44/57)
Accelerating Convergence
Zeros of Polynomials Deflation, M¨
uller’s Method Polynomial Approximation
FundamentalsMoving Beyond Taylor PolynomialsLagrange Interpolating PolynomialsNeville’s Method
The
th n Lagrange Interpolating Polynomial We use
Ln
(,k x),
k^
n^
as building blocks for the Lagrange
interpolating polynomial:
P(
x) =
n∑ k=
f^ (x
)Lk n,k
(x
which has the property
P(
x) =i^
f^ (
x)i^
,^
∀i^
n.
This is the unique polynomial passing through the points (xi
,^ f^
(xi
i^ = 0
n.
Peter Blomgren,
〈[email protected]
〉^
#4: Solutions of Equations in One Variable
— (49/57)
Accelerating Convergence
Zeros of Polynomials Deflation, M¨
uller’s Method Polynomial Approximation
FundamentalsMoving Beyond Taylor PolynomialsLagrange Interpolating PolynomialsNeville’s Method
Error bound for the Lagrange interpolating polynomial
Suppose x
,^ ii
n are distinct numbers in the interval
[a
,^ b
],
and f
C^
n+
[a,
b]
. Then
∀x
[a
,^ b
]^ ∃
ξ(
x)^
∈^ (
a,^
b)^
so that:
f^ (x
PLagrange
(x) +
(f (^) n+1)
(ξ(
x))
(n^
n∏^ ( i=
x^ −
xi^
where P
Lagrange
(x
)^ is the n
th^
Lagrange interpolating polynomial.
Compare with the error formula for Taylor polynomials
f^ (x
PTaylor
(x) +
(f (^) n+1)
(ξ(
x)) (n^
(x^
−^
x)^0
n+
Problem:
Applying the error term may be difficult... Peter Blomgren,
〈[email protected]
〉^
#4: Solutions of Equations in One Variable
— (50/57)
Accelerating Convergence
Zeros of Polynomials Deflation, M¨
uller’s Method Polynomial Approximation
FundamentalsMoving Beyond Taylor PolynomialsLagrange Interpolating PolynomialsNeville’s Method
The Lagrange and Taylor Error Terms
Just to get a feeling for the non-constant part of the error terms inthe Lagrange and Taylor approximations, we plot those parts onthe interval [
,^ 4] with interpolation points
xi
i,^
i^ = 0
,^1 ,... ,
0
0.^
1
1.^
2
2.^
3
3.^
4
(^43210) −1 −2 −3 −
0
0.^
1
1.^
2
2.^
3
3.^
4
1200 1000 800 600 400 200 0
Figure:
[Left]
The non-constant error terms for the Lagrange interpolation oscillates in the interval [
−^4 ,
4]
(and takes the value
zero
at the node point
x), andk^
[Right]
the non-constant error term for the Taylor
extrapolation grows in the interval [
,^ 1024].
Peter Blomgren,
〈[email protected]
〉^
#4: Solutions of Equations in One Variable
— (51/57)
Accelerating Convergence
Zeros of Polynomials Deflation, M¨
uller’s Method Polynomial Approximation
FundamentalsMoving Beyond Taylor PolynomialsLagrange Interpolating PolynomialsNeville’s Method
Practical Problems
Applying (estimating) the error term is difficult.The degree of the polynomial needed for some desired accuracy isnot known until after cumbersome calculations — checking theerror term. If we want to increase the degree of the polynomial (to e.g. n^ + 1
) the previous calculations are not of any help... Building block for a fix:
Let
f^
be a function defined at
x^0
xn
and suppose that
m
,^ m 1
m
arek
k^
n) distinct integers,
with 0
m
≤i n^
∀i. The Lagrange polynomial that agrees with
f^ (x
) the
k^
points
xm
,^ x 1 m^2
xm
, is denotedk
Pm
,m 1
,..., 2
mk^
(x
Note:
{m
,^ m 1
mk
0 ,^
n}
Peter Blomgren,
〈[email protected]
〉^
#4: Solutions of Equations in One Variable
— (52/57)
Accelerating Convergence
Zeros of Polynomials Deflation, M¨
uller’s Method Polynomial Approximation
FundamentalsMoving Beyond Taylor PolynomialsLagrange Interpolating PolynomialsNeville’s Method
Increasing the degree of the Lagrange Interpolation
Theorem Let f be defined at x
,^ x 0
xk
, and x
and xi
be two distinctj
points in this set, thenP
(x) =
(x^
−^ x
)Pj
0 ,...,
j−^1
,j+
,...,
(xk
)^ −
(x
xi^
)P
0 ,...,
i−^1
,i+
,...,
(xk
xi^
−^
xj
is the k
th^
Lagrange polynomial that interpolates f at the k
points x
xk
Peter Blomgren,
〈[email protected]
〉^
#4: Solutions of Equations in One Variable
— (53/57)
Accelerating Convergence
Zeros of Polynomials Deflation, M¨
uller’s Method Polynomial Approximation
FundamentalsMoving Beyond Taylor PolynomialsLagrange Interpolating PolynomialsNeville’s Method
Recursive Generation of Higher Degree Lagrange Interpolating Polynomials
x^0
P^0
x^1
P^1
P^0
,^1
x^2
P^2
P^1
,^2
P^0
,^1 ,^2
x^3
P^3
P^2
,^3
P^1
,^2 ,^3
P^0
,^1 ,^2
,^3
x^4
P^4
P^3
,^4
P^2
,^3 ,^4
P^1
,^2 ,^3
,^4
P^0
,^1 ,^2
,^3 ,^4
Peter Blomgren,
〈[email protected]
〉^
#4: Solutions of Equations in One Variable
— (54/57)
Accelerating Convergence
Zeros of Polynomials Deflation, M¨
uller’s Method Polynomial Approximation
FundamentalsMoving Beyond Taylor PolynomialsLagrange Interpolating PolynomialsNeville’s Method
Neville’s Method
The notation in the previous table gets cumbersome... Weintroduce the notation
Q
Last,Degree
PLast–Degree,
...^
,Last
, the table
becomes:
x^0
Q^0
,^0 x^1
Q^1
,^0
Q^1
,^1
x^2
Q^2
,^0
Q^2
,^1
Q^2
,^2
x^3
Q^3
,^0
Q^3
,^1
Q^3
,^2
Q^3
,^3
x^4
Q^4
,^0
Q^4
,^1
Q^4
,^2
Q^4
,^3
Q^4
,^4
Compare with the old notation:
x^0
P^0 x^1
P^1
P^0 ,
1
x^2
P^2
P^1 ,
2
P^0 ,
1 ,^2
x^3
P^3
P^2 ,
3
P^1 ,
2 ,^3
P^0 ,
1 ,^2 ,
3
x^4
P^4
P^3 ,
4
P^2 ,
3 ,^4
P^1 ,
2 ,^3 ,
4
P^0 ,
1 ,^2 ,
3 ,^4
Peter Blomgren,
〈[email protected]
〉^
#4: Solutions of Equations in One Variable
— (55/57)
Accelerating Convergence
Zeros of Polynomials Deflation, M¨
uller’s Method Polynomial Approximation
FundamentalsMoving Beyond Taylor PolynomialsLagrange Interpolating PolynomialsNeville’s Method
Algorithm: Neville’s Method — Iterated Interpolation
Algorithm: Neville’s Method To evaluate the polynomial that interpolates the
n^
(xi
,^ f^ (xi
i^ = 0
n^
at the point
x:
Initialize
Q
i,^0
=^
f^ (x
).i
FOR
i^
n
FOR
j^
i
Qi
= ,j
(x^
−^
xi−
)Qj
i,j−
(x
xi^
)Q
i−^1
,j−
1
xi^
−^
xi−
j
END
END
Output the
Q
-table.
Peter Blomgren,
〈[email protected]
〉^
#4: Solutions of Equations in One Variable
— (56/57)
