Introduction to Scientific Computing

-- A Matrix Vector Approach Using Matlab Written by Charles F.Van Loan

陈 文 斌Wbchen@fudan.edu.cn

复旦大学

Chapter3 Piecewise Polynomial Interpolation

• Piecewise Linear Interpolation• Piecewise Cubic Hermite Interpolation• Cubic Splines

0 1 2 3 4 5 6 7-1

-0.8-0.6-0.4-0.2

00.20.40.60.8

1Piecewise Linear Interpolation

)()( iiii zzbazL ),( ii yx

ii

iiiii xx

yybya

1

1,

nnn xzxzL

xzxzLxzxzL

zL

11

322

211

if)(

if)(if)(

)(

Piecewise Linear

function [a,b] = pwL(x,y)n = length(x); a = y(1:n-1);b = diff(y) ./ diff(x);

z=linspace(0,1,9);

[a,b]=pwL(z,sin(2*pi*z));

for i=1:n-1

b(i)=(y(i+1)- y(i))/ (x(i+1)- x(i));

end

b=(y(2:n)-y(1:n-1))./(x(2:n)-x(1:n-1))

b=diff(y)./diff(x)

Test code

Evalution

],[ z ?)( zL

Problem: ?],,[ 1 ixxz ii

if z==x(n)

i =n-1;

else

i=sum(x<=z);

end

Binary search

mid = floor((Left+Right)/2);

If z<x(mid)

Right=mid;

Else

Left =mid;

end

10000N 13.3 log2 Nn

function i = Locate(x,z,g)% g (1<=g<=n-1) is an optional input parameter% search for i begins, the value i=g is tried.if nargin==3 if (x(g)<=z) & (z<=x(g+1)) i = g; return end;endn = length(x);if z = = x(n) i = n-1;elseLeft = 1; Right = n;while Right > Left+1 Binary_searchend

g: guss

function LVals = pwLEval(a,b,x,zVals)

% Evaluates the piecewise linear polynomial defined by the column %(n-1)-vectors

m = length(zVals);

LVals = zeros(m,1);

g = 1;

for j=1:m

i = Locate(x,zVals(j),g);

LVals(j) = a(i) + b(i)*(zVals(j)-x(i));

g = i;

end

m-vector

604.2)^9.(

101.)3.(

12

xx

A priori determination of breakpoints

],[ 1 ii xxz

))((2

)()()( 1

)2(

ii xzxzfzLzf

8|)()(|

22hMzLzf ],[ z

2

22

2

188|)()(|

nMhMzLzf

8/)(1 2Mn Static

Adaptive Piecewise Linear Interpolation

Problem

Adaptive Piecewise Linear Interpolation

],[ xRxLacceptable

2)()(

2xRfxLfxRxLf

minhxLxR or

Piecewise Cubic Hermit Interpolation

2

3,2

3,,0 4321 xxxx

0

Hermit cubic interpolant

)()()()()( 22RLLL xzxzdxzcxzbazq

RRLL

RRLL

sxqsxqyxqyxq

)(',)('

)(,)(

LR

LRLRL

LLR

LL

LL

xxyy

xyyy

xyssd

xsyc

sbya

'

'2

'

],[)( 4)4(

RL xxzMzf

LR xxhhMzqzf ,384

)()( 44

410,

10errorerrorhh

)()()()()( 122

iiiiiiiii xzxzdxzcxzbazq

nnn xzxzq

xzxzqxzxzq

zC

11

322

211

if)(

if)(if)(

)(

Piecewise cubic polynomial

nisxCyxC iiii :1,)(',)(

For i=1:n-1

[a(i),b(i), c(i), d(i)=Hcubic(x(i),y(i),s(i),x(i+1),y(i+1),s(i+1))

End

function [a,b,c,d] = pwC(x,y,s)

% Piecewise cubic Hermite interpolation.

n = length(x);

a = y(1:n-1);

b = s(1:n-1);

Dx = diff(x);

Dy = diff(y);

yp = Dy ./ Dx;

c = (yp - s(1:n-1)) ./ Dx;

d = (s(2:n) + s(1:n-1) - 2*yp) ./ (Dx.* Dx);

x,y,s: vector

Evaluation

function Cvals = pwCEval(a,b,c,d,x,zVals)

m = length(zVals);

Cvals = zeros(m,1);

g=1;

for j=1:m

i = Locate(x,zVals(j),g);

Cvals(j) = d(i)*(zVals(j)-x(i+1)) + c(i);

Cvals(j) = Cvals(j)*(zVals(j)-x(i)) + b(i);

Cvals(j) = Cvals(j)*(zVals(j)-x(i)) + a(i);

g = i;

end

Locate

Cubic version of HornerN

Cubic spline interpolant

Given with

find a piecewise cubic interpolant with the property that S, S' and S'' are continuous.

),(),...,,( 11 nn yxyx,...1 nxx

Why need spline?

Continuity at the interior knots

)()('2

)(')()(

12

21

2

iii

iii

ii

iiiiii

xzxzx

yss

xzx

syxzsyzq

ii

iii xx

yyy

1

1'

)](2)(4['2'2)('' 12

1

iii

iii

i

iii xzxz

xyss

xsyzq

)](2)(4[22)( 212

1

'121

1

1'

1''1

ii

i

iii

i

iii xzxz

xyss

xsyzq

)32(2)( '11

''iii

iii yss

xxq

)23(2)( 21'

11

1''

1

iiii

ii ssyx

xq

2:1,3

)(2'

1'

1

2111

niyxyx

sxsxxsx

iiii

iiiiiii

)(3)(21 '21

'123122112 yxyxsxsxxsxi

)(3)(22 '32

'234233223 yxyxsxsxxsxi

)(3)(23 '43

'345344334 yxyxsxsxxsxi

)(3)(24 '54

'456455445 yxyxsxsxxsxi

)(3)(25 '65

'567566556 yxyxsxsxxsxi

r

sxyxyxyxyxyxyxyxyx

sxyxyx

sssss

TTs

75'65

'56

'54

'45

'43

'34

'32

'23

12'21

'12

6

5

4

3

2

)(3)(3)(3)(3

)(3

)6:2(

655

4545

3434

2323

121

22

22

2

xxxxxxx

xxxxxxxx

xxx

T

tridiagonal

)2()2(2,23,2

3232

2323

1211

0020

200

nnnnnn

nnnn

ttxxxx

xxxxtt

T

nss ,1choice

2

'23

'32

'32

'23

1

)(3

)(3

n

nnnn

ryxyx

yxyxr

r rTs

n=length(x);

Dx=diff(x);

yp=diff(y)./Dx;

T=zeros(n-2,n-2);

r=zeros(n-2,1);

for i=2:n-3

T(i,i)=2(Dx(i)+Dx(i+1));

T(i,i-1)=Dx(i+1);

T(i,i+1)=Dx(i);

r(i)=3(Dx(i+1)*yp(i)+Dx(i)*yp(i+1));

end

Help diag

The complete spline

RnL ss ,1

'21

'12312212 3)(2 yxyxsxsxxx L

'12

'21

211221

3

)(2

nnnn

Rnnnnnn

yxyx

xsxxsx

Lxyxyxsxsxx 2'21

'1231221 3)(2

Rnnnnn

nnnnn

xyxyx

sxxsx

2'

12'

21

11221

3

)(2

T(1,1)=2*(Dx(1)+Dx(2));

T(1,2)=Dx(1);

r(1)=3*(Dx(2)*yp(1)+Dx(1)*yp(2))-Dx(2)*muL;

T(n-2,n-2)=2*(Dx(n-2)+Dx(n-1));

T(n-2,n-3)=Dx(n-1);

r(n-2)=3*(Dx(n-1)*yp(n-2)+Dx(n-2)*yp(n-1))-Dx(n-2)*muR;

s=[muL;T\r(1:n-2);muR];

The Natural Spline

)( 1''

1 xqL

)](2)(4['2'2)('' 12

1

iii

iii

i

iii xzxz

xyss

xsyzq

1

121

1

11 '22'2x

yssx

syL

12

'11 2

321 xsys L

11

'1 2

321

nR

nnn xsys

)(''1 nnR xq 0 RL

T(1,1) = 2*Dx(1) + 1.5*Dx(2);

T(1,2) = Dx(1);

r(1) = 1.5*Dx(2)*yp(1) + 3*Dx(1)*yp(2) + Dx(1)*Dx(2)*muL/4;

T(n-2,n-2) = 1.5*Dx(n-2)+2*Dx(n-1);

T(n-2,n-3) = Dx(n-1);

r(n-2) = 3*Dx(n-1)*yp(n-2) + 1.5*Dx(n-2)*yp(n-1)- …

Dx(n-1)*Dx(n-2)*muR/4;

stilde = T\r;

s1 = (3*yp(1) - stilde(1) - muL*Dx(1)/2)/2;

sn = (3*yp(n-1) - stilde(n-2) + muR*Dx(n-1)/2)/2;

s = [s1;stilde;sn];

The Not-a-Knot Spline

)()( 2'''

22'''

1 xqxq

22

'232

21

'121 22

xyss

xyss

)2(2 '232

2

2

1'121 yss

xxyss

)2(2 '222

2

2

1'11

nnnn

nnnn yss

xxyss

identical are )( and )( 21 xqxq

q = Dx(1)*Dx(1)/Dx(2); T(1,1) = 2*Dx(1) +Dx(2) + q; T(1,2) = Dx(1) + q; r(1) = Dx(2)*yp(1) + Dx(1)*yp(2)+2*yp(2)*(q+Dx(1)); q = Dx(n-1)*Dx(n-1)/Dx(n-2); T(n-2,n-2) = 2*Dx(n-1) + Dx(n-2)+q; T(n-2,n-3) = Dx(n-1)+q; r(n-2) = Dx(n-1)*yp(n-2) + Dx(n-2)*yp(n-1) … +2*yp(n-2)*(Dx(n-1)+q); stilde = T\r; s1 = -stilde(1)+2*yp(1); s1 = s1 + ((Dx(1)/Dx(2))^2)*(stilde(1)+stilde(2)-2*yp(2)); sn = -stilde(n-2) +2*yp(n-1); sn = sn+((Dx(n-1)/Dx(n-2))^2)*(stilde(n-3) … +stilde(n-2)-2*yp(n-2)); s = [s1;stilde;sn];

function [a,b,c,d] = CubicSpline(x,y,derivative,muL,muR)

% Cubic spline interpolation with prescribed end conditions.

% Usage:

% [a,b,c,d] = CubicSpline(x,y,1,muL,muR)

% S'(x(1)) = muL, S'(x(n)) = muR

% [a,b,c,d] = CubicSpline(x,y,2,muL,muR)

% S''(x(1)) = muL, S''(x(n)) = muR

% [a,b,c,d] = CubicSpline(x,y)

% S'''(z) continuous at x(2) and x(n-1)

0 0.5 110-12

10-10

10-8

10-6

10-4

Knot Spacing = 0.0500 0.5 110-12

10-10

10-8

10-6

10-4

Knot Spacing = 0.005

Not-a-knot spline error

0 1 2 3 4 5 6 7 8 9 10-1

-0.5

0

0.5

1

1.5

Bad end conditions

-5 -4 -3 -2 -1 0 1 2 3 4 5-1.5

-1

-0.5

0

0.5

1

1.5 n = 9 Spline Interpolant of atan(x)

Matlab Spline Tools

z=linspace(-5,5);

x=linspace(-5,5,9);

y=atan(x);

Svals=spline(x,y,z);

plot(z,Svals)

Not-a-knot spline

pp-representation

x=linspace(-5,5,9);

y=atan(x);

S=spline(x,y);

z=linspace(-5,5);

Svals=ppval(S,z)

plot(z,Svals)

pp-representation

[x, rho,L,k]=unmkpp(S)

The coefficients of the local polynomials are assembled in an L-by-k matrix rho

33,

22,3,4, )()()()( iiiiiii xzxzxzzS

1, )(j)rho(i, jk

iji xx

33,

22,3,4, )()()()( iiiiiii xzxzxzzS

23,2,3, )(3)(2)(' iiiii xzxzzS

drho = [3*rho(:,1) 2*rho(:,2) rho(:,3)];

dS = mkpp(x,drho);

V = PPVAL(PP,XX) returns the value at the points XX of the piecewise

polynomial contained in PP, as constructed by SPLINE or the spline utility

z = linspace(-5,5);

Svals = ppval(S,z);

dSvals = ppval(dS,z);

-5 -4 -3 -2 -1 0 1 2 3 4 500.10.20.30.40.50.60.70.80.9

1 Derivative of n = 9 Spline Interpolant of atan(x)

n = 9;x = linspace(-5,5,n);y = atan(x);S = spline(x,y);[x,rho,L,k] = unmkpp(S);drho = [3*rho(:,1) 2*rho(:,2) rho(:,3)];dS = mkpp(x,drho); z = linspace(-5,5);Svals = ppval(S,z);dSvals = ppval(dS,z);atanvals = atan(z);Figure;plot(z,atanvals,z,Svals,x,y,'*');title(sprintf('n = %2.0f Spline Interpolant of atan(x)',n))datanvals = ones(size(z))./(1 + z.*z);figureplot(z,datanvals,z,dSvals)title(sprintf('Derivative of n = %2.0f Spline Interpolant of atan(x)',n))