Ηλεκτρονικοί Υπολογιστές ii - notes
TRANSCRIPT
: . . . . 2011Copyright c 2005 . , ([email protected]), . , . -, . . LaTEX2. (c , -). 25 2011. http://www.edu.physics.uoc.gr/tety2131 11.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 52.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.1.1 . . . . . . . . . . . . . . . . . . . 62.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.2.1 ( , -) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.3 x = g(x) . . . . . . . . . . . . . . . . . . . . . . . . . 92.3.1 . . . . . . . . . . . . . . . . . . . . . 92.4 NewtonRaphson . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 173.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.2 Cramer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.3 Gauss . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.4 . . . . . . . . . . . . . . . . . . . . . . . . . 213.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223.5.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223.5.2 GaussJordan . . . 223.5.3 . . . . . . . . . . . . . . . . . . . . 233.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244 274.1 Lagrange . . . . . . . . . . . . . . . . . . . . . . . . 274.2 . . . . . . . . . . . . . . . . . . 284.2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . 304.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315 335.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335.2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34iii 5.2.2 . . . . . . . . . . . . . . . . . . . . . . . . . 355.2.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355.3 Simpson . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365.3.1 Simpson 3/8 . . . . . . . . . . . . . . . . . . . . . . . . . . . 385.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395.4.1 . . . . . . . . . . . . . . . . . . . . . . . . 395.4.2 . . . . . . . . . . . 395.5 Gauss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 405.5.1 GaussLegendre . . . . . . . . . . . . . . . . . . . . . . . . . . 405.5.2 GaussHermite . . . . . . . . . . . . . . . . . . . . . . . . . . . 425.5.3 GaussLaguerre . . . . . . . . . . . . . . . . . . . . . . . . . . 425.5.4 GaussChebyshev . . . . . . . . . . . . . . . . . . . . . . . . . 425.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 436 476.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 476.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 476.2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 486.3 . . . . . . . . . . . . . . . . . . . 496.3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 496.3.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 496.3.3 . . . . . . . . . . . . 506.4 Taylor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 516.4.1 Euler . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 536.4.2 Taylor . . . . . . . . . . . . . . . . . . . . . . . . . . . 546.5 RungeKutta. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 556.5.1 RungeKutta 2 . . . . . . . . . . . . . . . . . . . . . . 566.5.2 RungeKutta 4 . . . . . . . . . . . . . . . . . . . . . . 576.5.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 586.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 586.6.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 596.6.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 606.6.3 Newton . . . . . . . . . . . . . . . . . . . . . . . . 626.6.4 . . . . . . . . . 626.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 636.7.1 AdamsBashforth . . . . . . . . . . . . . . . . . . . . . . . . . . 646.7.2 AdamsMoulton . . . . . . . . . . . . . . . . . . . . . . . . . . 656.7.3 (PredictorCorrector) . . . . . . . . . . . 666.8 . . . . . . . . . . . . . . . . . . . . . . . . . . 686.9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 706.9.1 . . . . . . . . . . . . . . . . . . . . . . 716.9.2 . . . . . . . . . . . . . . . . . . . . . 716.9.3 . . . . . . . . . . . . . . . . . . . . . . 736.9.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 736.10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 746.11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 776.12 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 11.1 /. , y F ( )y = 0.d1d2 . . . dK 10s,1 d1 9 ,0 di 9 , i 2, 3, . . . , K 0 s M, K,M /. , K= 6, M= 10 = 0.314159 101, K= 5, M= 10 = 0.31416 101,( round-oerror) |y y|. K ( signicant digits).:1. y ( ) / , ( overow). , / -, ( underow). , , , . IEEE innity ( ) 0.12. - . , , .1 +0 ( ) 0 ( ).12 1. : x = 5891.26 y=.0773414 K= 5 . . x, y x = 0.58913 104, y = 0.77341 101. :x = 0.58913 104, y = 0.00001 104., x, y 0.58914 104=5891.4 5891.3373414, - K 5891.3.3. . .. x= 5891.26,y = .0773414, z = 0.07 K= 5 x = +0.58913 104, y = +0.77341 101, z = 0.70000 101. x +y +z (x +y) +z x + (y + z) ( !).4. 1 + x= 1 x |x| < 5 10K. 1. .: / 2, 4 bit man-tissa, 4 bit bits . : (+111)2 2(+111)2= +7 27 (111)2 2(+111)2= 7 27 (+001)2 2(111)2= +1 27 (001)2 2(111)2= 1 27 x x > 7 27 x < 7 27, 27< x < 27 x0.:1. bit bias. (2Ne12) x (2Ne1 1) Ne .2. mantissa Nm bits Nm bits . - 1. f1f2f3 . . . , , 0. f1f2f3 . . . f1, f2, f3,. . . .1.2. 31.2 1. - :() : 1. 1 + 1 /2 .() SLAMCH() DLAMCH() LAPACK.() EPSILON() FORTRAN 90.2. ax2+ bx + c x1,2=b b2 4ac2a, a0. a = 1, b = 3000.001, c = 3.() x1,2 . (x1 = 0.001, x2 = 3000.0).() - x1,2=2cb b2 4ac. ;3. e1 ex=limn
1 +xn
n., , n= 1, 2, 3, . . .. (2.718281828459045 . . .);4. FORTRAN ex ex=n=0xnn!. n x/n. x. ;5. FORTRAN sin x sin x =k=0(1)kx2k+1(2k + 1)!. k x22k(2k+1). x. ;4 1. 6. FORTRAN cos x cos x =k=0(1)kx2k(2k)!. x. ; 2 2.1 () - . f (x) = 0 , x F . (2.1) , x , -f (x). f (x) (, f (x) =ax +b) . . f (x) 4, ., , 3 . , ( ) . (2.1) - x0, x1, . . . , xk, k . : f (x) [a, b]. f (a), f (b) (- ), c [a, b] f (c) = . f (x) x [a, b], (a, b), f(x). c [a, b] f (b) f (a) =f(c)(ba). f (a) =f (b) c [a, b] f(c) = 0 ( Rolle). Taylor f (x), x [a, b], n+1 fn+1(x) [a, b]. x, x0 [a, b], xx0, (x0, x) f (x) =f (x0) + f(x0)(x x0) +f(x0)2!(x x0)2++fn(x0)n!(x x0)n+ Rn(x) ,56 2. Rn(x) =fn+1()(n + 1)!(x x0)n+1.2.1.1 f (x) = 0, - x0, x1, . . . x. , , > 0 limn|xn+1 x||xn x|= . ( ) . , ( ), . - . , .2.2 f (x)ab x0 xx1x2 2.1: (). f (x) [a, b] f (a) f (b)< 0, , c= x (a, b) f ( x)= 0. f (x) (a, b). Weierstrass. [a, b], . [a1, b1], [a2, b2],. . . ,[aN, bN] x1= (a1+ b1)/2, x2= (a2+ b2)/2,. . . ,xN= (aN+ bN)/2. |bN aN| < 2.2. 7 |xN xN1| <
xN xN1xN
< xN 0 | f (xN)| < N N0 f (x) . .: f (x) = 0 :1. a, b f (x) [a, b] f (a) f (b) < 0.2. x a + b2.3. x 6.4. f (a) f (x) < 0 b x. , a x.5. 2.6. .: f (x) = x3+4x210, [1, 2]. f (1)= 5 f (2)= 14, f (1) f (2) < 0 , , [1, 2]. f(x) = 3x2+8x > 0 x ., f (x) [1, 2]. 2.1. 20 |x20 x| 0.5 |b20 a20| 0.95106, 6 . 1.36523 1.36523001361638 . . ..2.2.1 (,-) f (x) [a, b] f (a) f (b) < 0, x1, x2, . . . |xn x| 12n(b a), n 1, x f (x) [a, b]. () , 0.5.: b1 a1= b a, x (a1, b1)b2 a2=12(b1 a1) =12(b a), x (a2, b2)b3 a3=12(b2 a2) =122(b a), x (a3, b3).........bn an=12n1(b a), x (an, bn)8 2. n anbnxnf (xn)1 1.00000000 2.00000000 1.50000000 2.37502 1.00000000 1.50000000 1.25000000 1.79693 1.25000000 1.50000000 1.37500000 0.162114 1.25000000 1.37500000 1.31250000 0.848395 1.31250000 1.37500000 1.34375000 0.350986 1.34375000 1.37500000 1.35937500 0.96409 1017 1.35937500 1.37500000 1.36718750 0.32356 1018 1.35937500 1.36718750 1.36328125 0.32150 1019 1.36328125 1.36718750 1.36523438 0.72025 10410 1.36328125 1.36523438 1.36425781 0.16047 10111 1.36425781 1.36523438 1.36474609 0.79893 10212 1.36474609 1.36523438 1.36499023 0.39591 10213 1.36499023 1.36523438 1.36511230 0.19437 10214 1.36511230 1.36523438 1.36517334 0.93585 10315 1.36517334 1.36523438 1.36520386 0.43192 10316 1.36520386 1.36523438 1.36521912 0.17995 10317 1.36521912 1.36523438 1.36522675 0.53963 10418 1.36522675 1.36523438 1.36523056 0.90310 10519 1.36522675 1.36523056 1.36522865 0.22466 10420 1.36522865 1.36523056 1.36522961 0.67174 105 2.1: , f (x) = x3+ 4x2 10 xn=12(an + bn) xn x bn an x xn, :| x xn| =
x 12(an + bn)
12(bn an) =12n(b a) ., limn xn= x limn12n(b a) = 0.: f (x)=x3+ 4x2 10, [1, 2]. |xn x| = 105; |xn x| 2n(ba) = 2n(21) = 2n 2n . n log2 = log10 log10 2. = 105 n 5log10 2 16.61 17 |xn x| 105.: . .. , 2.2, - . , , ( 2.2).2.3. X= G(X) 9a bf (x) f (x)() ()x0x0 2.2: () , () 2.3 x = g(x) () f (x) = 0 x= g(x) g(x) . g(x) .: ()x0. x0,x1,x2, . . ., xn :x1= g(x0), x2= g(x1), x3= g(x2), . . . , xn= g(xn1) . x g(x) 1 x limn xn=limng(xn1) = g( limn xn1) g( x) .1. x .2. ( ). , 4.3. x g(x) 2.4. .2.3.1 . g(x) [a, b] [a, b] g() = .1limg(xn) = g(lim xn).10 2. . g(x) [a, b], a g(x) b, x [a, b]. g(x) [a, b].: g(a) a, g(b) b. h(x) = g(x) x. h(a) 0, h(b) 0. x h( x) = 0.: g(x) =3x, x [0, 1]. g(0) =1, g(1) =1/3 g(x) =3xln 30 xn NewtonRaphson x, x0 [ x , x + ].: ( x xi)2 - . , . ,f( x) 0 . f( x)0 ( ). g(x) = x f (x)f(x). g(x) =f (x) f(x)[ f(x)]2f( x)0f ( x) = 0 g( x) = 0 . ( f( x) =0) . -xn+1= xn f (xn) f(xn)[ f(xn)]2 f (xn) f(xn) xn+1= xn mf (xn)f(xn), m .2.5. 13: f (x) = x2 6x + 5. xn+1= xn x2n 6xn + 52xn 6, n = 0, 1, 2, . . . 1.0, 5.0 2.0, 6.0 n x(1)nx(2)n0 2.0 6.01 0.5 5.166666666666672 0.95 5.006410256410263 0.999390243902439 5.000010240026224 0.999999907077705 5.000000000026215 0.999999999999998 5.06 1.02.5 , f (x) (xn1, f (xn1)) (xn, f (xn)). xn1, xn . , xn+1, x ( ) . y = y(x) y =f (xn) +f (xn) f (xn1)xn xn1(x xn) .,xn+1= xn f (xn)f (xn) f (xn1)(xn xn1) =xn1f (xn) xnf (xn1)f (xn) f (xn1). , , x0, x1, . , , .: f (x) = 0 :1. a, b.2. x (a, f (a)),(b, f (b)). c.3. c 6.4. a b, b c.5. 2.6. .14 2. 2.6 1. Fortran. f (x) = x3+ 4x2 10 [1, 2], f (x) =x cos x [0, 1].2. g(x) = ln x + 2 [2, 4]. |xn x| 103.3. , -. x1 x2 , f (x1) f (x2). x f (a) f (b). ( ) (a, f (a)) (b, f (b)), ,x, x, ( [a, b] ). 2.() Fortran .() f (x) = 2.0 + 6.2x 4.0x2+ 0.7x3 [0.4, 0.6].() f (x) = x10 0.95[0, 1.4]. < 106; ;4. x= g(x). - f (x) = x2 6x + 5, f (x) = x cos3x 0.6.5. y=ex1x , , x. |x| Taylor ex - .6. 2http://en.wikipedia.org/wiki/False_position_method2.6. 15() 1.5x2+ 13 106x + 0.037 = 0. x1 2.8462 109, x2 8.6667 106.() 1.5x2 37 106x + 0.057 = 0. x1 1.5405 109, x2 2.4667 107.7. NewtonRaphson () f (x) = sin x x2,() f (x) = 3xex 1.8. f (x)= 4 cos x ex 108 , , NewtonRaphson .9. 12 ex, tan(2x) [1, 1].: .10. Fortran M uller3 . ( y = ax2+bx+c) , , . . . f (x) = sin x x2.11. NewtonRaphson, n, pn(x) = 0+1x+2x2++nxn, 0, 1, . . . , n. Horner.12. , - n, pn(x) = 0+1x +2x2+ +nxn, 0, 1, . . . , n. Horner.3http://mathworld.wolfram.com/MullersMethod.html16 2. 3 3.1 n n Annxn1= bn1, A = [ai j] =,a11a12 a1na21a22 a2n............an1an2 ann, x =,x1x2...xn, b =,b1b2...bn.: :1. b, Ax = b .2. A (A1).3. A, det A .4. Ax = 0 x = 0.5. A . (iterative).3.2 Cramer Cramer Ax = b xj=det Bjdet A, j = 1, 2, . . . , n , Bj A j A b. (n+1)! n 4.1718 3. : - 3.5.1.3.3 Gauss: Ax = b1. ,2. ,3. j i j, ( ) Ax = b . , det A0 det A0. Gauss Ax = b Ax = b ai j= 0, i >j. ,a11a12 a1n0 a22 a2n............0 0ann,x1x2...xn=,b1b2...bn. (3.1) :1. a11=0 A a110. ai1=0 det A = 0 (b = 0) (b0).2. a110, a21/a11- . a2j, b2a21= 0 , a2j= a2j a1ja21a11, b2= b2 b1a21a11. . , aj1/a11 j. , Ax = b ,a11a12 a1n0 a22 a2n............0 an2 ann,x1x2...xn=,b1b2...bn., A11(A ) [b2, . . . , bn]T. (3.1).3.3. GAUSS 19 , . xn=bnann, xk=1akk
bk nj=k+1ak jxj
, k= n 1, n 2, . . . , 1 .: aj j0, det A = 0.: ,0 1 25 3 12 2 1,x1x2x3 =,346. : 1. a11= 0 a210 ,5 3 10 1 22 2 1,x1x2x3 =,436. a21=0, . 2/5 ,5 3 10 1 20 3.2 0.6,x1x2x3 =,434.4. 3.2 ,5 3 10 1 20 0 7,x1x2x3 =,4314. x3= 2, x2= 1, x1= 1 .: k= 1, 2, . . . , n 1:1. akk= 0, aik,i =k+ 1, . . . , n. , . k i ( A b).2. i = k + 1, . . . , n,20 3. = aikakk,ai j ai j + ak j, j = k, . . . , nbj bj + bk, j = k, . . . , n k= n, n 1, . . . , 1, xk=1akk
bk nj=k+1ak jxj
. k + 1 > n .: , b = bnm A m b, - b, , b m b . . ( - ) Gauss n1k=1,(n k)2....ai j+2(n k)....bi+nk=1(n k + 1) =n33+ n2 n3, (n + 1)! Cramer. . A n2 (, ). n b. . , akk ( k ) . (k, i >k) aik, i k.: ,0.0003 1.5660.3454 2.436 ,x1x2 =,1.5691.018 x1= 10, x2= 1. , / 0. f1f2 fn 10|s|, |s| 10, n = 5 , Gauss ,0.3 1030.1566 1010 0.1804 101 ,x1x2 =,0.1569 1010.1805 1003.4. 21 , x2= 1.0006, x1= 6.868. x1, x2. , , ,, ,0.3454 2.4360.0003 1.566 ,x1x2 =,1.0181.569 ,0.3454 1000.2436 1010 0.1568 101 ,x1x2 =,0.1018 1010.1568 101., x2= 1, x1= 10. ( ). akk - . x.3.4 A, b ;:,1 31 3.01 ,x1x2 =,44.01 x1= x2= 1. ,1 31 2.99 ,x1x2 =,44.02 x1= 10, x2= 2, .: Ax=b A, b. (well-conditioned system) = ||A||||A1|| A ||||. .. ||A||= max1innj=1
ai j
. > 1 .22 3. 3.5 Gauss , - .3.5.1 () . - Gauss , , , , . (3.2), , :det A =ni=1aii. aii . , (. ) ., det A - ( ), A, (1)ss ( ) ., det A =ni=1(1)i+jai j det Ai j, (3.2) Ai j (n 1) (n 1) A i j.3.5.2 GaussJordan , - : Gauss -. , , Ax = BAx = B, A , ,I x= B. , , - . , x = A1B .3.5. 23, B n (1, 0, . . . , 0)T,(0, 1, . . . , 0)T,. . . , (0, 0, . . . , 1)T A1. Ann GaussJordan n Ax= B . , A , , . , A ,a11a12 a1n| 1 00a21a22 a2n| 0 10............ |............an1an2 ann| 0 01 ,1 00 | a11 a12 a1n0 10 | a21 a22 a2n............ |............0 01 | an1 an2 ann. , A1.3.5.3 A. , , () x, (0, 0, . . . , 0) Ax = x , (3.3) x A . x (3.3) . (3.3) Ax= x Ax= I x (A I)x = 0 . , x =(0, 0, . . . , 0)T, A I. , det(AI) = 0. det(AI) n . n (nnj=1, ji
ai j
, i = 1, . . . , n , : xi=1aii
bi i1j=1ai jxj nj=i+1ai jxj
, i = 1, . . . , n . xi, , . :GaussJacobi. , xi, x(k)i, - , x(k+1)i:x(k+1)i=1aii
bi i1j=1ai jx(k)jnj=i+1ai jx(k)j
, i = 1, . . . , n . x(k+1)i x(k)jj < i, j > i.26 3. GaussSeidel. , xi,x(k+1)i, :x(k+1)i=1aii
bi i1j=1ai jx(k+1)jnj=i+1ai jx(k)j
, i = 1, . . . , n . x(k+1)i x(k)jj >i x(k+1)jj < i. Fortran, JACOBI, SEIDEL . Ax = B A =,12.1 3.9 0.3 4.14.3 11.3 0.8 1.51.0 2.8 14.3 8.12.4 6.1 1.1 12.5, x =,x1x2x3x4, B =,1.22.33.44.5. : Ax B 107. 4 4.1 Lagrange f (x), f0, f1, . . . , f x0, x1,. . ., x, f (x), xxi x ,minxi
h
1 + 2h2
|1| , h 21
n( ) n . n2 - . , .:1. y= 3y, y(0)= 1, y(x)= e3x.(6.54) h =0.1y0=y(0) =1, y1=0.7408 6.2. , xn>1 (1)n (6.55).xnynyn0.0 1.0 1.00.1 0.7408 0.7408180.2 0.555520 0.5488120.3 0.407488 0.4065700.4 0.311027 0.3011940.5 0.220872 0.2231300.6 0.178504 0.1652990.7 0.113769 0.1224560.8 0.110243 0.0907180.9 0.047624 0.0672061.0 0.081669 0.0497871.1 0.001378 0.0368831.2 0.082495 0.0273241.3 0.050875 0.0202421.4 0.113020 0.0149961.5 0.118687 0.011109 6.2: y= 3y, y(0) = 1, (6.54) h = 0.12. AdamsBashforth yn+1= yn=h2(3fn fn1) ,6.11. 77 y= y, y(0) =1, > 0. AdamsBashforth yn+1 1 32h
yn 12hyn1= 0 . (6.56) 1,2=12,1 32h
1 h +942h2.1=12,1 32h
+ 1 +12h +942h2
18h +942h2
+ O
h3
= 1 h +122h2+ O
h3
= eh ,1 + O
h3.,n1= exn,1 + O
h2. h n1 y(xn). , (6.56) yn= (1 + 1)n1 + 2n221=1 32h
1 h +942h21 32h +
1 h +942h2. 1 3/2h 0
21
1. h 23. . , , .6.11 k. yn= c1n1 ++ cknk. , n1 , k 1 . |i| > |1| i1, ni . y= y, |i| < 1 i, |1| > |i| i = 2, . . . , k.78 6. : y= y, >0 Euler . Euler yr+1= (1 h)yr= yr, = 1 h . || 0, h.6.12 1. Euler y= cos x sin y + x2 [1, 1], y(1) = 3.0. h = 0.01.2. Taylor5.3. y1= 2y1 2y2 + 3y3y2= y1 + y2 + y3y3= y1 + 3y2 y3 ( t= 0) y1= 2, y2= 30, y3= 0. ,1 11 11 1 11 14 11=,1/25/61/301/151/151/21/102/5.4. y= y, y(0)= 1 [0, 1] Euler h = 0.2, 0.02, 0.002, 0.0002, 0.00002.5. RungeKutta 3 y=f (x, y),y(x0) = y0 :yr+1= yr+16(k1 + 4k2 + k3)k1= hf (xr, yr)k2= h f (xr+ h/2, yr+ k1/2)k3= h f (xr+ h, yr k1 + 2k2)6.12. 796. Taylor 4 y= y + z2 x3z= z + y3+ cos x ( x= 0) y= 0.3, z= 0.1. y, z [0, 1] 0.1.7. Fortran RungeKutta 2.8. RungeKuttap.h .9. RungeKutta 2 y= x .10. Taylor Fortran. y= y3+ x + y, [0, 0.5], y(0) = 1, h = 0.1.11. y=x2+ x y, y(0) =0 RungeKutta 2. x = 0.6 h = 0.2.12. y= ax + b, y(0) = 0, y(x) =a2x2+ bx . Euler yn=12(axn + 2b ah)xn, xr= rh, , ,y(xn) yn=a2hxn.13. RungeKutta 2 h, y=y,y(0) =1. y(2h) (h)3/3.14. RungeKutta 4 y=yx
1 yx
[1, 3], y(1) =2. h=1/128, y(x) =x0.5 + ln x.15. RungeKutta 2 m = 2 Kg, F(x) = x2+0.01x3. t= 0 , , x = 2.5 cm.80 6. 16. = sin , . , = 45. sin . .17. =(x2 5) (0) = (2)1/2. 100 [2, 2]. , [0, 2] [2, 0].18. yn+1 ayn= 0,yn+2 4yn+1 + 4yn= 0,yn+2 4yn+1 + 4yn= 1.19. : = E 1, E= ehD, =E 1E, = E1/2 E1/2, = 0.5
E1/2+ E1/2
, = coshhD2
, E= 1 + +22 , 2= 1 +24 .20. yr+2 5yr+1 + 6yr= 0, y0= 0, y1= 1,yr+2 4yr+1 + 4yr= 0, y0= 1, y1= 6,yr+2 + 6yr+1 + 25yr= 0, y0= 0, y1= 4.21. yr+2
2 + h2
yr+1 + yr= h2, h > 0 ,yn= c1,1 + h +h22+ O
h3
n+ c2,1 h +h22+ O
h3
n 1 .6.12. 8122. yr+2 + 4hyr+1 yr= 2h , h > 0 ,yn= c1,1 2h + O
h2n+ c2(1)n ,1 + 2h + O
h2n+12.23. h2
yr+ yr+1
= +112311205+
yr+12. sinh1z = z 16z3+34z5+.24. y0 y1 + y2 =r=0(1)rEry0 12y0 14y0 +182y0 .25. AdamsBashforth 3 .26. y(1.0) y= 1 y, y(0) = 0, AdamsBashforth2h =0.2 y(0) =0,y1= y(0.2) = 0.18127. y(x) = 1 ex.27. Fortran h Tc.28. y= y, > 0.() AdamsMoulton 3yn+1= yn +h12
5yn+1 + 8yn yn1
,() MilneSimpsonyn+1= yn1 +h3
yn+1 + 4yn + yn1
,()yn+1= yn1 +h2
yn + 3yn1
. h .29. y= y, > 0. yn 0 n h ( ), 82 6.
1 h +122h2
< 1 RungeKutta 2,h < 1/ AdamsBashforth 2.30. yn+1= yn1 +h2
yn+1 + 2yn + yn1
y= y y= y +, > 0 0.31. Taylor 2, y1= x2y1 y2, y1(0) = s1,y2= y1 + xy2, y2(0) = s2.32. y + 4xyy + 2y2= 0 , y(0) = 1 , y(0) = 0 , (6.57) Euler h = 0.1. y(0.5), y(0.5).33. (6.57) 2. .34. y 2y, y(a)= y0. Euler yk y0. h 0.35. y=xy2,[0, 0.2], y(0) = 0.25 h = 0.1.36. h y= y, y(0) = 1 RungeKutta 3.37. h312y() , (xr, xr+1) .38. AdamsMoulton 3yr+1= yr+h12
5yr+1 + 8yr yr1
.39. .40. y= y. 2.1 , f (x) = x3+ 4x2 10 . . . . . . 86.1 f (x) = ex 6 . . 616.2 y= 3y, y(0) = 1, (6.54) h = 0.1 . . . . . . . 7683 , 2 , 28, 66AdamsBashforth, 64AdamsMoulton, 65 , 21 , 47 , 49 , 49, 47 Euler, 53, 64 RungeKutta, 55, 65, 49, 47 , 48 , 70 , 71 , 71, 70 , 77 , 74 , 77, 70 Gauss, 40 , 33, 63Simpson, 37, 62, 19 Lagrange, 27, 12 , 1, 54, 54, 54, 1, 54 , 9 Lipschitz, 48, 69 , 11, 64, 65, 65, 64, 59 , 60, 59 , 60 , 58 , 60, 18, 1, 184