Appendix A BASIC BASIC Beginner s All-purpose Symbolic Instruction Code FORTRAN COBOL C JAVA PASCAL (NEC N88-BASIC Windows BASIC (1) (2) ( ) BASIC BAS
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1 Appendix A BASIC BASIC Beginner s All-purpose Symbolic Instruction Code FORTRAN COBOL C JAVA PASCAL (NEC N88-BASIC Windows BASIC (1 (2 ( BASIC BASIC download TUTORIAL.PDF ( BASIC download 1
2 2 APPENDIX A BASIC 1(prog1.bas 10 LET A=25 20 LET B=37 30 PRINT A+B 40 PRINT A-B 50 PRINT A*B 60 PRINT A/B 70 END 1 10,20 A, B LET ( BASIC BASIC PRINT (+ ( (* (/ 70 END LET = ( PRINT ( INPUT A 20 INPUT B INPUT (
3 3 2(prog2.bas 10 RANDOMIZE 20 LET X=INT(RND*10 30 INPUT A 40 IF A<X THEN PRINT " " 50 IF A>X THEN PRINT " " 60 IF A=X THEN 70 PRINT " " 80 GOTO END IF 100 GOTO END INT ( INT(Y Y Gauss RND 0 1 INT(RND* IF THEN A<X IF IF then END IF 80,100 GOTO RANDOMIZE ( INT( ( RND (0 1 (
4 4 APPENDIX A BASIC IF THEN ( IF THEN ELSE 1 2 END IF ELSE 2 BASIC GOTO GOTO 2-1(prog2-1.bas 10 RANDOMIZE 20 LET X=INT(RND*10 30 DO 40 INPUT A 50 IF A<X THEN PRINT " " 60 IF A>X THEN PRINT " " 70 LOOP UNTIL A=X 80 PRINT " " 90 END (1 DO WHILE LOOP (2 DO UNTIL LOOP (1 (2 (3 DO LOOP WHILE (4 DO LOOP UNTIL
5 5 (3 (4 3(prog3.bas LET S=0 20 FOR K=1 TO LET S=S+K 40 NEXT K 50 PRINT S 60 END ( K 100 IF NEXT FOR = TO NEXT 1 (1 INPUT (2 ( n 2 (3 ( n (4 (1 n! = n (5 (4 (Napier e
6 6 APPENDIX A BASIC ! + 1 2! + 1 3! n! 4(prog4.bas y = x 3 6x 10 LET left=-5 20 LET right=5 30 LET bottom= LET top=10 50 LET h= SET WINDOW left, right, bottom, top 70 DRAW GRID 80 DRAW AXES 90 def f(x=x^3-6*x 100 for x=-5 to 5-h step h 110 LET x1=x 120 LET y1=f(x1 130 LET x2=x+h 140 LET y2=f(x2 150 plot lines: x1,y1;x2,y2 160 next x 170 END left,right,bottom,top x h f(x = x 3 6x x 5 5 h h x1 y x2 y2 150 (x1, y1 (x2, y2
7 7 160 x h 100 x 5 h 170 SET WINDOW,,, ( for = 1 to 2 step 3 next DRAW GRID DRAW AXES ( ( PLOT LINES: 1, 2; 3, 4 ( 1, 2 ( 3, 4 5(prog5.bas (cycloid LET left=-1 LET right=20 LET bottom=-10 LET top=10 LET h=0.02 SET WINDOW left, right, bottom, top DRAW GRID DRAW AXES def f(t=t-sin(t def g(t=1-cos(t for t=0 to 20-h step h LET x1=f(t LET y1=g(t LET x2=f(t+h LET y2=g(t+h plot lines: x1,y1;x2,y2 next t END
8 8 APPENDIX A BASIC 2 SET WINDOW (1 1 (f(t = cos t, g(t = sin t, 0 < = t < = 2π (2 f(t = cos t, g(t = sin 3t, 0 < = t < = 2π (3 ( cardioid= r = 1 + cos θ ( x = (1 + cos t cos t, y = (1 + cos t sin t, 0 < = t < = 2π 1(example01.bas P n+1 = (1 + a/100p n, a = 3, P 0 = 100 LET A=3 LET P0=100 INPUT N FOR K=1 TO N LET P1=(1+A/100*P0 PRINT K,P1 LET P0=P1 NEXT K END LET A=3 LET P0=100 INPUT N SET WINDOW 0,N,0,2000 FOR K=1 TO N LET P1=(1+A/100*P0 PRINT K,P1 PLOT LINES:K-1,P0;K,P1 LET P0=P1 NEXT K END
9 9 3 (1 2(Fibonacci (example02.bas (2 3( n (example03.bas ( a n+1 = a n + n + 1 (3 4( (example04.bas ( p n+1 = {1 + (a/100(1 p n /C}p n, a = 5, p 0 = 100, C = 700 (4 5( (example05.bas ( x n+1 = x n + kx n (N x n, k = 0.1, x 0 = 100, N = (5 6( 1 (example06.bas [ ] Rn+1 F n+1 [ ] [ ] a b Rn = d/100 1 c/100 a = 5, b = 10, c = 5, d = 5, R 0 = 1000, F 0 = 50 F n (6 7( 2 (example07.bas R n+1 = {1 + A(1 R n /C} R n DR n F n F n+1 = QDR n F n A = 0.5, C = 300, D = 0.025, Q = 0.3, R 0 = 1000, F 0 = 100 (7 8( (example08.bas P 0 n P 0 n P Pn+1 1 Pn+1 2 = P 1 n Pn 2, P 1 0 P0 2 = P 3 n+1 P 3 n (8 9( (example09.bas x n+1 = (1 + rx n b, x 0 = , r = 0.03, b =? (9 10( (example10.bas x n x n x 0 1 y n+1 = y n, y 0 = 0 z n z n z 0 0 P 3 0
10 10 APPENDIX A BASIC (10 12( n k (example12.bas f(n + 1, k = f(n, k 1 + kf(n, k, f(n, 1 = 1, f(n, n = 1 (11 13( n (example13.bas p(n, k = p(n 1, k 1 + p(n k, k, p(n, 1 = 1, p(n, n = 1 n p(n = p(n, k k=1 (12 16(Newton 2 (example16.bas x n+1 = x2 n + 2 2x n, x 0 = 2
11 Appendix B (Jordan A Ax = λx (x 0 λ A x A λ B 1 A : n A : det A = A = 0 rank(a = n A n 1 A n 1 1 Ax = 0 (x = 0 B 2 A : n λ : A f A (λ = det(λi A = λi A = 0 (f A (λ A f A (λ = 0 A B 3 ( n f(z = z n + a n 1 z n 1 + a n 2 z n a 1 z + a 0 (a 0, a 1,, a n 1 f(z = (z α 1 n 1 (z α 2 n2 (z α k n k (α 1, α 2,, α k, n 1 + n n k = n 11
12 12 APPENDIX B (JORDAN f(λ = (λ λ 1 n 1 (λ λ 2 n2 (λ λ k n k n i λ i W λi = {x Ax = λ i x} R n A λ i 1 dim W λi n i B 4 {p i1, p i2,, p imi } W λi {p 11, p 12,, p 1m1, p 21, p 22,, p 2m2,, p k1, p k2,, p kmk } 1 1 P P 1 AP A B 5 A λ i n i λ i dim W λi = n i ( n rank(a λ i I = n i W λi {p i1, p i2,, p ini } n P = [p 11, p 12,, p 1n1, p 21, p 22,, p 2n2,, p k1, p k2,, p knk ] λ 1 λ 2 O P 1. AP = D =.. O... λn B 6 A 1 A B ( ( ( ( ( (
13 13 B - 2 [ 2 1 (1 2 3 ] [ 5 1 (2 1 3 ] [ ] 0 2 ( ( ( ( t A = A A t P P = I ( t P = P 1 P B 7 P : n P : P n R n P n R n B 8 P : (P x, P y = (x, y B 9 A n A A (dim W λi = n i W λi {p i1, p i2,, p ini } (Gram-Schmidt P = [p 11, p 12,, p 1n1, p 21, p 22,, p 2n2,, p k1, p k2,, p knk ] P A P 1 AP = t P AP = D (D Q(x = i,j a ij x i x j = a 11 x a 22 x a nn x 2 n + 2a 12 x 1 x 2 + 2a 13 x 1 x a n 1 n x n 1 x n (x 1, x 2,, x n 2 ( a ij = a ji x 1 a 11 a 12 a 1n x x = 2., A = a 21 a 22 a 2n (A a n1 a n2 a nn x n
14 14 APPENDIX B (JORDAN 2 Q(x Q(x = (Ax, x = (x, Ax = t xax A P ( t P AP = D: x = P y (y = t P x Q(x = t xax = t (P yap y = t y t P AP y = t ydy = λ 1 y1 2 + λ 2 y λ n yn 2 ( λ 1, λ 2,, λ n A. λ 1 y1 2 + λ 2 y λ n yn 2 2 Q(x α 2 Q(x = t xax x Q(x = t xax > = α x 2 Q(x B 10 2 Q(x = t xax A B 11 2 Q(x = t xax a 11 a 12 a 1n a 11 a 12 a 11 > 0, a 21 a 22 > 0,, a 21 a 22 a 2n > 0 a n1 a n2 a nn B - 3 P P 1 AP [ 2 2 (1 2 2 ] ( ( B B-11 (1 Q(x, y, z = x 2 + y 2 + z 2 + xy + yz + zx (2 Q(x, y, z = x 2 + 2y 2 + 3z 2 2xy 2yz 2zx (3 Q(x, y, z = x 2 + 2y 2 + 3z 2 2xy 6yz 8zx
15 15 Jordan A W λi = {x l (A λ i I l x = 0} λ i B 12 (Cayley-Hamilton A f A (λ = λ n + a n 1 λ n 1 + a n 2 λ n a 1 λ + a 0 = (λ λ 1 n 1 (λ λ 2 n2 (λ λ k n k f A (A = A n + a n 1 A n 1 + a n 2 A n a 1 A + a 0 I = (A λ 1 I n 1 (A λ 2 I n2 (A λ k I n k = O A f A (λ 1 (λ λ i n i f A (λ/(λ λ i n i f A,i (λ {f A,i (λ} i=1,2,,k 1 B 13 {f 1 (x, f 2 (x,, f k (x} ( 1 {g 1 (x, g 2 (x,, g k (x} g 1 (xf 1 (x + g 2 (xf 2 (x + + g k (xf k (x 1 {f A,i (λ} i=1,2,,k B - 13 g 1 (λf A,1 (λ + g 2 (λf A,2 (λ + + g k (λf A,k (λ 1 {g i (λ} i=1,2,,k g 1 (Af A,1 (A + g 2 (Af A,2 (A + + g k (Af A,k (A I g i (Af A,i (A = M i M 1 + M M k I, M i M j O (i j, M 2 i = M i B 14 (1 { W λi } M i (image W λi = {x (A λ i I n i x = 0} (2 { W λi } n i (3 R n { W λi } i=1,2,,k R n = W λ1 W λ2 W λk
16 16 APPENDIX B (JORDAN Jordan P W λi λ = λ i W (k λ i = {x (A λ i I k x = 0} W λi = W (1 λ i W (2 λ i W (3 λ i W (ν 1 λ i W (ν λ i = W λi dim W (k λ i = m k, r k = m k m k 1 (k 2, r 1 = m 1 r ν r ν 1 r ν 2 r 2 r 1 = m 1
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