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Numerical Methods for Engineers and Scientists
11.3. Divided Difference Polynomial + = +
other
Publication date:
October 3 2018
Publisher:
CRC Press
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Author and book information
Book Chapter
Publication date:
October 3 2018
Pages
: 252-261
DOI:
10.1201/9781315274508-122
SO-VID:
2fb96421-02c5-4430-82e0-e0d09e036342
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Book chapters
pp. 15
Chapter O. Introduction
pp. 17
Programs
pp. 18
of of
pp. 19
of error of
pp. 22
f(x) + = = = ...
pp. 23
of of off(x,y)
pp. 25
Part I. Basic Tools of Numerical
pp. 26
aij' of of of of I.la of of
pp. 27
of of of
pp. 31
Systems of Linear Algebraic Equations
pp. 33
= = = =
pp. 37
of of
pp. 38
mayor " },
pp. 39
A+B=B+A
pp. 40
L1,2,3,
pp. 41
of -I. of
pp. 42
(i,j I, ...
pp. 44
of of ' L " . A· ' . ' L "
pp. 46
20 -20 -20 -20 -20 -20
pp. 54
of of of
pp. 56
I0.60] oI o
pp. 57
of -I. of
pp. 60
lux =b'l
pp. 61
2 2 0] [Xl]
pp. 62
of -I. if of -I. of -I br
pp. 63
5 TRIDIAGONAL SYSTEMS EQUATIONS of
pp. 64
of I' of of T, of of
pp. 65
a ' -2.25 = = =
pp. 66
, I' pI ace
pp. 67
0.9999
pp. 69
of +Xl 2
pp. 70
of of
pp. 71
IIA-1 of (l.l75) IIAllllxllllA-
pp. 73
of 1.7 ITERATIVE METHODS of of of If of
pp. 74
= L 0= ... -
pp. 77
of of Ax
pp. 80
of of of of of of
pp. 81
of If
pp. 83
,-1 (i)
pp. 85
(i) (i)=x(i)-a(i,j)*x(j) (i)=x(i)/a(i,i)
pp. 86
Land
pp. 87
(i)=(b(i)-a(i,3)*x(i+1))/a(i,2)
pp. 88
main
pp. 89
end format Solution failed to converge'/' iteration
pp. 90
of of
pp. 91
of of of of
pp. 95
Eigenproblems
pp. 97
K + +
pp. 98
2 2]
pp. 101
of Aare
pp. 102
= = =
pp. 103
O
pp. 105
......................................... ..................
pp. 106
(Ai)k
pp. 107
I -I -I = (I) =
pp. 109
................................................
pp. 110
6, -4,
pp. 111
...................................................
pp. 113
..... ... ..... ..... ....... .............................. 0.50]
pp. 114
= II
pp. 115
= +s =
pp. 123
.....................................
pp. 125
7 OTHER METHODS of of
pp. 127
end end call power (ndim,n,a,x,y,norm,iter,tol,shift,iw,k,ev2) (x(i), 'I' 'I'
pp. 128
format = ',e10.2,' approaching zero, stop')
pp. 129
of -I.
pp. 130
call lufactor then if if
pp. 131
Land in process to b' b'
pp. 138
5 The Method of of
pp. 141
Nonlinear Equations
pp. 146
05.0a = 40 deg
pp. 149
of of
pp. 157
f(¢)
pp. 158
= = = = = =
pp. 159
if of
pp. 160
of of
pp. 164
of I(x) of
pp. 165
= +'
pp. 166
!(¢)
pp. 169
of of
pp. 174
± Jb ±
pp. 176
f(x;)f' f(x)/f'(x) of f(x) = = + =
pp. 177
f(x)
pp. 178
of if
pp. 179
o , of = ...
pp. 183
of of of
pp. 184
vg(X,y)=
pp. 186
= = = =
pp. 189
end data 5, 40.0 f=r1*cos(alpha*rad)-r2*cos(x*radJ+r3-cos((alpha-x)*rad) fp=(r2*sin(x*rad)-sin((alpha-x)*rad))*rad return fi
pp. 192
program main main illustrate
pp. 194
of of
pp. 197
ofj(x)
pp. 198
+ (y /4 +;
pp. 200
andf
pp. 201
Polynomial Approximation and Interpolation
pp. 202
of of of(x,f(x)] f(x) of of of of of x
pp. 208
of x, + + + + + +
pp. 210
P(x) s s -225.0 -90.6250
pp. 211
3 DIRECT FIT POLYNOMIALS of
pp. 212
+ + + + + +
pp. 213
of of
pp. 214
f(x)
pp. 216
x1)
pp. 217
fll)
pp. 218
= = =
pp. 225
+ hf + +
pp. 229
+ + + + 1)(-0.6 + + ...
pp. 230
of of of
pp. 231
of of
pp. 232
000508i
pp. 234
+ = =
pp. 235
of of
pp. 236
of of of of of of of of fj =fi =
pp. 239
= + + + = + +
pp. 240
f(xJ
pp. 244
of mayor of of
pp. 246
+ LX; + + LxT +
pp. 250
,x(i),f(i) 'I' 'I' f'I'
pp. 251
f(a) + f(b) + f(c)
pp. 252
11.3. Divided Difference Polynomial + = +
pp. 262
+ bT + +
pp. 264
band
pp. 265
Numerical Differentiation and Difference Formulas
pp. 270
= + + + + + + + + + ...
pp. 271
(5.lla)
pp. 272
rdid . P,,(x))
pp. 274
+3 + ...
pp. 280
fx,J;,fxl' ofj(x, j(x, M...
pp. 281
of of of of
pp. 283
of of f(x) f(t) Ix.
pp. 284
hli.
pp. 287
of of of
pp. 288
first fxx numerical approximation data 3, 2, 3, ,x(i),f(i) fit 'I' 'I' 'x',llx, 'f'I'
pp. 289
+ f(a) + + + + f(e)
pp. 291
if13fo + .. -)
pp. 292
'f'I' (' '110x,'dx',8x,'0(h**2)',5x,'0(h**4)'I' ') (' (' fxx',3f12.6) format (' ') subroutine deriv (ndim,ndeg,num,n,x,f,dx,fx,fxx) differentiation
pp. 294
2 Unequally Spaced Data
pp. 296
+ i-Ii-I + I, + i-I, + I, +
pp. 298
kA-
pp. 299
Numerical Integration
pp. 300
32258065 3.20.31250000 3.30.30303030 3.49.29411765 3.50.28571429 3.60.27777778 3.70.27027027 3.80.26315789 3.90.25641026
pp. 301
ofdiscrete of of
pp. 303
= f/(X) aOx+al-+a2-+···
pp. 304
of of
pp. 307
+ + ... +
pp. 309
of of
pp. 318
1/./3 -v'G.6
pp. 319
F(__
pp. 320
+a = = + ..,-+
pp. 321
IJy . J _ ...../
pp. 322
+ 0.lr of of of
pp. 324
,21,41,81
pp. 326
sum=sum+2.0*f(iJ end sum=sum*(x(nJ-x(lJJlfloat(n-1JI2.0 return
pp. 328
main
pp. 329
integration
pp. 330
of of Of
pp. 331
+ f + + + + 3 + +
pp. 332
f(x) of f(x)
pp. 334
= of
pp. 337
Ordinary Differential Equations
pp. 338
of of
pp. 341
t[or
pp. 342
E=mCT of of
pp. 343
= = = =
pp. 349
One-Dimensional Initial-Value Ordinary Differential Equations
pp. 351
= = =
pp. 354
of of of
pp. 358
of ofYo o f(t,y)
pp. 359
+ aT' T' = T'
pp. 360
---------
pp. 361
• • • • •
pp. 362
of of
pp. 363
++ + + ... + + + + ...
pp. 365
of of
pp. 368
= = I
pp. 374
if of of
pp. 377
of of
pp. 380
f..t+-
pp. 381
= + =
pp. 383
+ = = + +
pp. 385
+ + + ...
pp. 386
= = =
pp. 387
= hf(t n+ + = = = n+ ,Yn + = h( ]}) 2 + -4- al 2 ---4 -
pp. 389
= z =
pp. 390
= E(M
pp. 394
= = = + + +
pp. 400
= + +
pp. 403
16 -59
pp. 408
11.1
pp. 409
f(t, a(T
pp. 410
!it T;) = of
pp. 411
offirst-order
pp. 412
= ...
pp. 413
of of y'=V
pp. 414
f i1t +2,Yn ,Yn i1V,) +-2- f= i1t +-2- i1tg = + + +
pp. 415
= + + + =
pp. 416
of of if of of of of of of
pp. 418
+ = = + = +
pp. 421
y'=-y, of of of
pp. 424
derivative
pp. 425
main
pp. 426
derivative
pp. 427
main
pp. 428
000 2154.47079576 -86.16753966 2075.24833822 -74.17350708 4.000 2074.55075892 -74.07380486 2005.68816487 -64.71557210 5.000 2005.33468855 -64.66995205 ...................................
pp. 431
= Y'
pp. 436
+ + s= + s
pp. 437
= = =
pp. 446
jm, o
pp. 447
= of
pp. 449
One-Dimensional Boundary-Value Ordinary Differential Equations
pp. 450
of of T T
pp. 451
= cm-=
pp. 452
+ + = = =
pp. 454
of of of
pp. 455
2.4 Systems of Second-Order Boundary-Value ODEs of
pp. 457
x x x
pp. 459
Clem xr =
pp. 460
LU=0.125cm of of
pp. 461
of of of
pp. 467
ill<=~.25
pp. 469
of of of = of of If = If =
pp. 474
of em-=
pp. 477
= + =
pp. 478
() X X
pp. 480
ofX, ofX.
pp. 481
+ + + + +
pp. 483
~ Y, f'li ~ &3 d4V\
pp. 491
Iy(xi
pp. 493
ax:. of of of of i. x a + bi + cY of = = =
pp. 496
-10-
pp. 498
y(i) z(i) side
pp. 499
iter ,i3)
pp. 500
250000 3.645833 19.270833 0.500000 11.241319 46.723090 0.750000 30.957935 124.490922 1.000000 84.036669 336.393870 0.000000 0.000000 14.874459 0.250000 4.338384 22.931458 0.500000 13.376684 55.598456 0.750000 36.838604 148.138810 1.000000 100.000000 400.294148 y(l) = by= = = = =
pp. 501
') (' ') tol, the equilibrium method for nonlinear second-order ode term coefficient of fx,p,q else it=l,iter
pp. 502
Output 8.2. Solution by the second-order equilibrium method Equilibrium method 0.000000 0.000000 0.250000 25.000000 0.500000 50.000000 0.750000 75.000000 1.000000 100.000000 0.000000 0.000000 0.250000 4.761905 0.500000 14.285714 0.750000 38.095238 1.000000 100.000000
pp. 506
= = =
pp. 507
+ + + + + + + +
pp. 512
of of of
pp. 515
Partial Differential Equations
pp. 516
of of
pp. 517
+ ay +
pp. 518
+ bg +
pp. 522
of of
pp. 523
ofj(x, of
pp. 527
of of
pp. 541
Elliptic Partial Differential Equations
pp. 542
of of
pp. 543
= = = =
pp. 544
100 80 60 40 10 20 10
pp. 545
of of of
pp. 549
h-IJ =];J + 4 + . + + + .
pp. 552
0 TTT 12.5 TTT 10.0 TT7.5
pp. 553
4 -4 -4 -4 -4 -4 -4 -4 -4 -4 -4 -4 -4 -4 -4
pp. 554
of of
pp. 560
of of of of
pp. 562
= = x=7.5cm + + + +
pp. 563
00.0 20 30 40
pp. 566
,j+l I-3,jI-2,j l,j
pp. 569
+ 2.
pp. 570
50 -----
pp. 574
of of of 20 -20 -20 -20 -20 -20 -20 -20 -20 -20 -20 -20 -20 -20 -20
pp. 577
y y ~
pp. 580
of Q/k Q= --
pp. 581
1.00 0.2182 T 0.1160 4 0.75 T 1614 3 0.50 TTT \
pp. 585
j(x,
pp. 586
____
pp. 587
= + (T z To) To) + Tsz -
pp. 590
in this imax grid points in the direction jmax grid points in the direction iw intermediate results all x(i,j) y(i,j) f(i,j)
pp. 591
j=2,jmax-1 i=2,imax-1
pp. 592
+ + +
pp. 593
Laplace equation with solution has converged,
pp. 594
df=(f(i+1,j}+beta2*f(i,j+1}+f(i-1,j}+beta2*f(i,j-1} -d*f(i,j}-fxy*dx**2}/d
pp. 598
& = = = = =
pp. 601
Parabolic Partial Differential Equations
pp. 603
90 • 0.51.0 80 • 2 3 . . 0 () • 4 70 o5 . . 0 0
pp. 604
= = = = =
pp. 606
of of ofparabolic
pp. 607
= ak -=
pp. 612
_ _];n _ !It/r) :s :s
pp. 613
of of
pp. 616
: = 0.01
pp. 617
s 0.01 0.4 15 0.4
pp. 619
of of of of
pp. 622
f') of of of of j(x,
pp. 623
of of of
pp. 624
of of
pp. 626
+ :!i +
pp. 629
/'+
pp. 630
](0, = ](L, of
pp. 631
a = /s
pp. 635
of of
pp. 636
n = 0.01 cm /s
pp. 640
n 5 90 1.0
pp. 644
f(x, of off), ofIx),
pp. 650
_ d) + + + _
pp. 651
... 0.51 90 2.55 -
pp. 653
00.20.30.40.50.60.70.80.91.0
pp. 655
t=t1oat (n-1) *dt write t, (f(i,n),i=l,imax,ix) 'I' 'I 'I'
pp. 657
(i, (i)
pp. 658
main
pp. 661
of of
pp. 664
= = = =
pp. 665
Hyperbolic Partial Differential Equations
pp. 667
oo5.0 ()
pp. 668
of f(x, f(x,
pp. 669
offour of
pp. 670
------ooooIE--------f
pp. 671
±a
pp. 672
of of of
pp. 673
for]_I, + ...
pp. 675
= emls
pp. 678
= of
pp. 679
of of 11.5 LAX-WENDROFF TYPE METHODS 0(M + 0(M
pp. 681
+ of
pp. 682
= of
pp. 687
of of of
pp. 692
of of of of of
pp. 693
!it/Ax
pp. 694
+ = + = -p+ +p+ + +h
pp. 696
of of
pp. 697
9 THE WAVE EQUATION of
pp. 698
lex,
pp. 700
a = 1000
pp. 703
+ e-Af +
pp. 704
of of
pp. 705
df + = -a,
pp. 707
n-1, =l, stop solver =',f4.2 n',2x, 'time',3x, 'f(i,n)
pp. 708
1781.31 60.0940.64 5.07.4714.9726.12 1.96 11.10.2 The Lax-Wendroff One-Step Method of :/i -
pp. 710
main main illustrate
pp. 711
0.27 .....................................................................
pp. 712
format (' iu = ' ,' '/' '/'
pp. 714
a(imax,2)=1.0 b(imax) n=l, t=t+dt i=2, imax-1 a(i,l)=-0.5*c b(i)=f(i,n)
pp. 715
10.6 Packages For Solving The Convection Equation
pp. 717
* = = =
pp. 722
of of
pp. 725
The Finite Element Method
pp. 726
.....
pp. 729
of of
pp. 732
aj (2x-l)
pp. 737
+ + + +
pp. 740
of of
pp. 741
0-
pp. 742
... ... + + ... +
pp. 743
= Qf + I
pp. 744
of of
pp. 745
(_1_ __
pp. 748
+ Ly' +
pp. 750
= [-y' +
pp. 751
(i Q(i-Il ( M) ( +y. 1+ =
pp. 754
of of
pp. 758
of of of/(x, of = Vf = + nJ;
pp. 759
(f(x, h(y) + fie-x) + hex) x)]dxdy
pp. 760
...) ~ 2ji+ + i) 0 [ fi ( 3) (-2-3) +13 -x
pp. 761
z= z ,
pp. 762
2 7 4 4 8 0
pp. 763
[-
pp. 764
0 10.0 7.5
pp. 765
= = = =
pp. 768
= Qf
pp. 769
+f+, + (_1 ) , , ,
pp. 770
2X -
pp. 771
• • •
pp. 772
Q FEM of of 0e.
pp. 775
'f'I' the for dimension
pp. 778
the Poisson equation dx,dy x-direction y-direction grid
pp. 779
then j=l, write (6,1000) (f{i,j),i=l,irnax,ix)
pp. 780
000000 6.3. The Diffusion Equation + Qf -
pp. 781
t,iw,ix,a,b,z) stop n,t,
pp. 785
= Q = = = = =
pp. 786
= = = =
pp. 787
of of
pp. 789
References
pp. 790
ofthe ofMechanical
pp. 793
Answers Selected Problems
pp. 794
1.00J 00,
pp. 795
0.5 ,0]
pp. 796
= = = ,x = -0.8 = = = = = = = = = 0.815544,j(x) = = = = = = = -= = =
pp. 798
f'(x)
pp. 799
/'(1.0)
pp. 800
n/2n/4n/8
pp. 801
= = = = = = =
pp. 802
LAY
pp. 803
= = =
pp. 805
xy(x) y(x) y(x) = y(5.0) = = = = = = = =
pp. 809
Index
pp. 822
w.,
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