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MAE 305 Lecture 3/
X I x I
X (^2) y ↑ (^3) y L (^22) Y (^3 ) p(x) = (x-0.5)(X-0.8) (1.9048) + (x- 0.1)(X-0.0 (1.6065) + (X-0.11(1-0.5) (^) (1.4493) 10.1-0.5)(0.1-0.8) 10.5-0.1)^ (0.5^ -0.8)^ 10.8-0.1)(0.8-0.5) -In (^) general for^ the^ degree Polynomial -L (^) is lagrange Pn(x) =(, (^) (x)y, +h2(X)yzt.... that, (X)^ yn+1 = rimy; ricx= (^) ( Eil For (^) example, = inri (e
-golny back^ toearlier^ example
= (815) (^) (8.88) Newton's (^) Divided (^) Difference Interpolating Polynomials xI (^) p(x)= a, + A2(X - x,) + ag(X- x,)(X - xz) + ay(X- x,)(X - xc)(X - xz)+.... xYI
p(X) =a,^
- 0 + 0+ ... = y, + (^) a, = y, x2 (^) YL
p(X2)
= a, + 92(X2 - x,) + 0 + 0 .... (^) = y, + dc(x - x) = Y : (^) Ac = yc-y, In (^) Yn x2 - 4
P(X3)=^ a,^ +^ dz(X3^ -xi)^ + 93(Xz - x)(X) - x2)+^ 0+0...^ = ye
↑ ↑ y I as =^ ys-y,^ - ?!)(Xs-x,)
(N3-Xi)(X3 -Xa)
as-(2-(ii) x 3 -^ Y
FirstDivided^ Difference
f(xi+1,xiJ:Yi+1-y;eslope^ ofthe^ line Xi+1 - xi check i^ =^1
+ (x2, x) =
= Second Divided^ Difference f(xi+2, Xi^ +1,^ Xi]^ =^ f[xi+2,^ xi+1]^ -f[xi+,x] Xitz -^ Xi meck 1.(xx, (^) x2, x.) =+(x3, x,3-+(x--x3 = x3 -^ x,^ ?)-(!) x3 -^ x
Kin (^) Divided Difference f X= (^) +1, XK (^) ,. (^).. x2, (^) x,] = f[XF+1, ...,x2] - f(xx,.^.. ., x] XR+ 1 - XI P(x) = a, + 12(X
1(X - 0)(X (^) -0.1) + 4(X - 0)(X - 0.1)(X - 0.2) + 15(X - 0)(X (^) -0.1)(X-0.2)(x- 0.5)
T
y,=a, X 1stDD -E P 10.4268)
- (-0.81) = 0.3664: 94 i) (^) = Ioni.21=-0. 3 (0.5) - 0.2250-0.121 = 1. 0.2-0. !
=^ -^ 0.
X 3 -^ x2^ 0.5-0.1^ (-0.4405)^ -^1 -0.6260)^
= 0. 0.2 0.^ 0.4650 -^ 0.2258^ = 0.7973 0.8-0. (^13 ) (0.5-0. 0.5330 (^) -0.7973 = -0. 0.0 -0. 0.5 (^) 0. 0.6249 -0.4658 (^) = 0. 0.8-0. 0.8 0. P(x)
= 0 + 1.21)x -0) + 70.87(X -0) (X -0.1)+ 0.3464(X -0)(X - 0.1)(X - 0.2) +
(-0.1254)(X -0)(X^ -0.1)(X-0.2)(x
a, +^ dc(x^ -0.1)^ + 9,1-0.1) (X-0.5)
X (^) y 0.1 1.9048 I^ a 1.3065-1. 0.5 1. 0.5-0. =A (-0.524)
- 110.74575) = 0.317 = 93 0.8-0. 1 = - 8. 1.4493-1. 0.8 -^ 0.5^ P(X) = 1.9048 + (-0.74575)/x -^ 0.1)^ + 0.317/X-0.1) (^) (X-0.5) 0.8 1.4493 =^ - 0.
A - I = b bl (^) C. (^) A2 b2 22 93 by (^23) 2 I (^0000 0 0) a (^) = 0 42 = 2.05 (^) 4) =^ - 2. 3 100000 0
b, =^ -^ 2.7^ b2^ =^ -^15 b^3 =^ 33. O O (^9 3100 0) I (^) c, =10.4 22 = 28.85 23 =^ - 91. 00 15510 0 0 b !
I
(^00 00 0255 )
I
it()
3, =^0 +^ (-2.7)^ x + 10.4^ 21x
0 0 00 056157.51 a3^ S2 = 2.05x2 + 1 - 15)x + 28.85 31X
I 0 - 6 -^10 000 53 =^ - 2.77bX2 + 33.26x - 91.851X- 7. L I S Cubic (^) Spline S,(x) =^ 9,(X^ -^ xi)3^ +^ bi(x^ -^ xi)^
- (i(X- xi) +^ di -> (^3) Spline x4unknowns = (^12) equations needed
d
-take the^8 equations from quadratic butwe need 4 more (?)
S"( xi) = S,"+,(Xi)
I Clamp Boundary condition
SiCX1=P Sin = (X,+n) = 9
·si 2 Cubic
spline WI Free^ boundary conditions
S"(X.)^ = 0, Sh(Xn+1) = 0
I
Chapter 6 Numerical Methods^ and (^) Integration
6.1 Numerical Differentiation
I (^) use offinite (^) difference 2 use ofbest fit curce a p'(X) A (^0) ⑧ pl (^0) O 0 ⑧ ⑧ (^0 ) O (^) - Hi C -
6.2 Finite (^) Difference Formulas for Numerical (^) Differation (^) I ⑤ (^1) IstDerivative
n=^1 I (^2) two (^) pointbackward difference formula
I · I (^) I I f(xi-) = f(xi) -^ hf(xi)^ + hf"(x)
- HO (^) Xi-1 X;Xi+h Remainder
f(xi) = f(xi) -^ f(xi-1)
Xi -Xi-
e
-h b)two point forward (^) diff formula f(xi+) = f(x,) +^ f(xi) + h2+"()+.... ↓ Xi + 1 - x; f(xi) = f(x, + 1) - f(xi) Xi+1 - Xi c) two^ pointcentroid (^) diff formula f(xi-1) = f(xi) - hf(xi) + h2f"(i)-4txi)+.... f(xi+1) = f(xi) + hf'(xi)+ if"(xi)+;h3+"(i)+...
f(Xi+,) -f(X;-1) = 2h f'(xi) + HOT
f'(xi) = f(Xi+1) -^ f(xi-1) 2h a) 3-pointbackward^ difference formula
- ⑲ & (-4) +^ (xi-1) = f(xi)-h+'(xi) +...^ I
f(xi-2) =^ f(Xi)-2h+'(x,)+...^ I ⑤ ⑤
I I
f(xi -^ 2) -^4 + (^) (xi - (^) 1) = - 3f(xi) + 2h (^) f'/xi) + HOT (^) - in " f(x,)= f(xi - 2) - 4f(xi - (^) 1) +^3 +^ (Xi) " I 2n (^) Xi-L
