iv 2 6 [1] [2] 2015 11 [1] A.. [ ], [ ]. 3 [M]. :, 2008 [2] R. [ ], [ ]. [M]. :, 2013 [3]. [M]. :, 2005. [4]. 2 [M]. :, 2002. [5]. 2 [M]. :, 2003. [6],,. [M]. :, 2000.
3 a b c d e a c
vi b 1 1 2 1 P 10 i=1 a i a 1 + + a 10 2 1
vii xx T x T x Ax + b 1 c 2 # p 1 p = p + 1 # 1 1.2.13 1.2.9 2
viii p = p + 1 d e a 3
ix 1 Ruby Ruby Ruby Ruby Ruby Ruby Ruby Ruby 1 http://www.ituring.com.cn/book/1239
Ruby Gnuplot Maxima xlispstat
m n A n m n x n m m Ax F
xii A ( ) 1.5 0 A = 0 0.5 A 1.5 0.5 1.5 0.5 = 0.75 0.75 det A =
xiii 0 A 0 ( ) 0 0 A = 0 0.5 A 0 det A = 0
xiv A ( ) 1.5 0 A = 0 0.5 A 0.5 deta < 0
xv A ( ) 1 0.3 A = 0.7 0.6 A A 1 ( ) ( 1 0.7 10 ) ( A 2 0.3 ) ( 0.6 01 )
xvi ( ) 1 0.3 A = 0.7 0.6 1.3 0.3
xvii ( ) 1 0.3 A = 0.7 0.6 1.3 0.3 = 0.39 det A = 0.39 =
xviii A ( ) 0.8 0.6 A = 0.4 0.3 A A Im A 1 (rank A = 1) (rank A < 2) A rank A = 2 A det A = 0 ( ) ( 1 0 A 1 01 ) A 2
xix ( ) 0.8 0.6 A = 0.4 0.3 = 0
xx A ( ) 0.8 0.6 A = 0.4 0.3 0 p A Ker A p 2 Ker A 1 = Im A 1
xxi ( 1 0.3 0.7 0.6 ) 1 2 A = ( 0.3 1 0.6 0.7 ) det A < 0 ( ) ( ) 0.3 1 1 0.3 det = det 0.6 0.7 0.7 0.6
0 1 0.1 1 0.2 2 1 5 1.1 5 1.1.1 6 1.1.2 9 1.1.3 11 1.1.4 16 1.1.5 18 1.1.6 19 1.2 19 1.2.1 19 1.2.2 1 24 1.2.3 25 1.2.4 = 28 1.2.5 31 1.2.6 = 35 1.2.7 37 1.2.8 = 44
xxiii 1.2.9 47 1.2.10 2 53 1.2.11 55 1.2.12 =??? 63 1.2.13 1 64 1.2.14 2 67 1.3 68 1.3.1 = 68 1.3.2 73 1.3.3 1 80 1.3.4 2 87 1.3.5 91 2 95 2.1 95 2.2 97 2.2.1 97 2.2.2 97 2.2.3 107 2.2.4 110 2.3 115 2.3.1 115 2.3.2 120 2.3.3 122 2.3.4 126 2.3.5 130 2.3.6 1 137 2.3.7 2 142 2.4 149 2.4.1 149 2.4.2 150 2.4.3 151 2.5 152 2.5.1 1 152 2.5.2 2 155 2.5.3 166
xxiv 2.6 167 2.6.1 167 2.6.2 170 3 1 LU 173 3.1 173 3.1.1 173 3.1.2 174 3.2 174 3.3 LU 176 3.3.1 176 3.3.2 178 3.3.3 LU 178 3.3.4 LU 180 3.4 LU 1 182 3.5 LU 186 3.6 LU 187 3.7 LU 191 3.8 LU 2 192 3.8.1 192 3.8.2 196 4 Jordan 197 4.1 197 4.2 202 4.3 203 4.4 205 4.4.1 205 4.4.2 213 4.4.3 215 4.4.4 219 4.4.5 220 4.5 220 4.5.1 220 4.5.2 225 4.5.3 232
xxv 4.5.4 240 4.6 246 4.6.1 247 4.6.2 250 4.6.3 250 4.6.4 252 4.6.5 252 4.7 255 4.7.1 255 4.7.2 Jordan 256 4.7.3 Jordan 257 4.7.4 Jordan 264 4.7.5 Jordan 271 4.7.6 Jordan 279 5 2 299 5.1 299 5.1.1 299 5.1.2 300 5.1.3 5 5 302 5.1.4 303 5.2 Jacobi 303 5.2.1 304 5.2.2 306 5.2.3 309 5.3 310 5.3.1 310 5.3.2 311 5.3.3 QR 312 5.3.4 316 5.4 QR 318 5.4.1 QR 319 5.4.2 Hessenberg 321 5.4.3 Householder 322 5.4.4 Hessenberg QR 325 5.4.5 327 5.4.6 327
xxvi 5.5 328 A 330 B 331 C 336 D 341 D.1 dx/dt = f(x) 341 D.2 dx/dt = ax + g(t) 342 E 346 E.1 346 E.1.1 346 E.1.2 347 E.1.3 347 E.1.4 349 E.1.5 351 E.1.6 351 E.2 352 E.3 353 F 354 F.1 354 F.2 354 F.3 355 357
0 0.1 3 3 2 3 2 3 n 1
2 0 1 3 0.2 2 2
0.2 3 3 3 1 2 1 2
1 0.1 1.2.13 1.1
6 1 10 10 1.1.1 1 2.9 ( ) 6 2 0.3 5 3 1/7 3 π 42 2 3 5 (2, 3, 5, 8) 2 (2, 3, 5, 8) T = 3 5 8 T Transpose 2 3 5 8 T = (2, 3, 5, 8) 1.1 2 f x f(x) A x Ax 1.2.1 x xa f(x) x.f 1 2
1.1 7 1.2 1 0, 1, 2, 3,, 2, 1, 0, 1, 2, 3.14159265 ( ) + ( ) i i (i 2 = 1) i 2 x 1 y 1 x 1 + y 1 2 7 9. +. =. : 9 + 5 = 14 (1.1) x n y n x n + y n 4 3 7 c x 1 cx 1 2 6 c. =. : 3 9 = 27 (1.2) x n cx n 4 12 1 0 0 0 7 7.000 7 3 2 j c, x j cx j x j, cx j
8 1 1.3 x 1 y 1.. = x 1 y 1., x 1. / x 1 /y 1 =. y 1. x n y n x n y n x n y n x n /y n 1.2.11 x + y = z x y z x y z x + y = z xy = z x y z xy x v e x v e 0 o = (0,, 0) T ( 1)x x x + ( 1)y x y 2x + 3y (2x) + (3y) c c x y
1.1 9 1.4 0 1.1.2 2 1.1 (3, 5) T 3 5 ( 2.2, 1.5) T 2.2 1.5 o = (0, 0) T O 3 3 O 1 1.2 3 3 1
10 1 1.1 1.2 a + b = b + a a b b a 1.5 (a) a 1.11 1.20 1 1.6 4 5 1.4
1.1 11 4 x 1.1.3 2 1.3 1.3 O 1 2 x v e x x o 1 2 1.1.1
12 1 1 1.7 x = (x 1, x 2 ) T x 2 1 + x2 2 1.1.1 1.4 1.9 E 1.8 x = (x 1, x 2 ) T y = (y 1, y 2 ) T x y = x 1 y 1 + x 2 y 2 E x y 3 2 3 3 n 2 3 4 1 E.1.3 2 3 x = (x 1, x 2, x 3 ) T y = (y 1, y 2, y 3 ) T x y = ((x 2 y 3 x 3 y 2 ), (x 3 y 1 x 1 y 3 ), (x 1 y 2 x 2 y 1 )) T
1.1 13 [6] 1 3 x y x y x y 1 1 100 10000 [6] 1.9 v 1.4 e 1 e 2 e 1 3 e 2 2 v v = 3 e 1 + 2 e 2 1 [6]
14 1 1.4 e 1 e 2 e 1 3 e 2 2 ( e 1, e 2 ) v v = (3, 2) T 1 ( e 1, e 2 ) e 1 e 2 1.5 1.5 1.10 1.1.1 v v v v 1
1.1 15 v v = (v 1, v 2 ) T v 1, v 2 v 1 2 1.2.11 1.11 v v v 1 v v 2 p p
16 1 1.12 1.6 1.6 1 1 Jacobian 2 1.1.4 ( e 1,, e n ) 1. v v = x 1 e 1 + + x n e n x 1,, x n 2. 1 1
1.1 17 2 x = (x 1,, x n ) T y = (y 1,, y n ) T x y 1.7 1.7 2 3 2 (x 1,, x n ) T (x 1,, x n) T x 1 e 1 + + x n e n x 1 e 1 + + x n e n x 1 e 1 + + x n e n = x 1 e 1 + + x n e n (x 1,, x n ) T = (x 1,, x n) T 1 u 1 e 1 + + u n e n = o u 1 = = u n = 0 2 x 1 e 1 + + x n e n = x 1 e 1 + + x n e n (x 1 x 1) e 1 + + (x n x n) e n = o 1 A B B A 2 e 1,, e n 2.3.4
18 1 u 1 = x 1 x 1 u 2 = x 2 x 2 u 1 e 1 + + u n e n e 1,, e n u 1,, u n u 1 e 1 + + u n e n e 1,, e n 1 x e 1,, e n ( e 1,, e n ) 1.1.5 n n = = C 1.13 x = (x 1, x 2, x 3, ) y = (y 1, y 2, y 3, ) c u = (x 1 + y 1, x 2 + y 2, x 3 + y 3, ) v = (cx 1, cx 2, cx 3, ) u = x + y v = cx D.1 1 u 1,, u n
1.2 19 1.1.6 3776 3776 3776 v 1.1 1.2 x y c x + y = (x 1 + y 1 ) e 1 + + (x n + y n ) e n c x = (cx 1 ) e 1 + + (cx n ) e n 1.2.11 1.2 1.2.1 1 1
20 1 ( ) 2 0 0 3 ( ) 2.2 9 1/7 7 π 42 3 1 4 1 5 9 2 6 5 3 5 8 9 7 9 2 2 2 3 5 3 1 2 2 2 3 3 3 A i j A (i, j) (2, 1) 7 (1, 3) 1/7 (i, j) a 11 a 12 a 13 a 14 A = a 21 a 22 a 23 a 24 a 31 a 32 a 33 a 34 3 4 A = (a ij ) 2 1.14 3 row column 1.8 1.8 1 2 3
1.2 21 a 11 a 1n b 11 b 1n a 11 + b 11 a 1n + b 1n.. +.. =.. (1.3) a m1 a mn : b m1 b mn a m1 + b m1 a mn + b mn ( ) ( ) ( ) 2 9 4 1 2 3 3 11 7 + = 7 5 3 4 5 6 11 10 9 c a 11 a 1n c.. = ca 11 ca 1n.. (1.4) a m1 a mn ca m1 ca mn ( ) ( ) 2 9 4 6 27 12 : 3 = 7 5 3 21 15 9 A = ( 1)A A B = A + ( B) 2A + 3B = (2A) + (3B) x x x y y y = a x + a x + a x (1.5) y = a x + a x + a x (1.6) a a ( y y ) ( ) a a a = a a a x x x (1.7)
22 1 x x x a m n n a 11 a 1n x 1 a 11 x 1 + + a 1n x n... =. (1.8) a m1 a mn x n a m1 x 1 + + a mn x n : 2 7 ( ) 2 1 + 7 2 16 9 5 1 = 2 9 1 + 5 2 = 19 4 3 4 1 + 3 2 10 1.5 1.6 x x x y y a x + a x + a x 1 A x + y = z Ax + Ay = Az cx = y c(ax) = Ay 2 3 1 0.2 2 A x y z c 3
1.2 23 1.15 f(x + y) = f(x) + f(y) f(cx) = cf(x) f x, y c f(x) A f e 1 = (1, 0, 0,, 0) T e 2 = (0, 1, 0,, 0) T f a i = f(e i ) x = (x 1,, x n ) T f(x) = x 1 a 1 + + x n a n a 1,, a n A = (a 1,, a n ) A f(x) f(x) = Ax 1.2.9 1.16 2 7 ( ) 2 1 + 7 2 16 9 5 1 2 = 9 1 + 5 2 = 19 4 3 4 1 + 3 2 10 n 1 n ( 31 4 ) 1.2.9 n 1 n m n n 1 m n n 1.2.4 1.8
24 1 1.8 1.9 1.9 j i y = Ax (i, j) j i 1.2.2 1 0.2 x x y y y = a x + a x = x + x y = a x + a x = 2x + 4x a = 1 1 1 a = 4 1 4 ( ) ( y a = y a a a ) ( ) ( ) ( x 1 1 = x 2 4 10 1 10 A B x x )
a 1.2 25 1 1 1 2 3 a 11 a 21 a 31 1 2 1 2 3 a 12 a 22 a 32 x 1 1 x 2 2 1 2 3 y 1 y 2 = a ( ) 11 a 12 a 21 a 22 x 1 x y 3 a 31 a 2 32 1 20 1000 18 000 y = Ax LCR y = Ax 1.2.3 n x m n A m y = Ax A 1 1 n m
26 1 1.10 A = ( ) 1 0.3 0.7 0.6 x Ruby 1 Ax Gnuplot 2 F 1.10 A = ` 1 0.3 0.7 0.6 x A 1 http://www.ruby-lang.org/ 2 http://www.gnuplot.info/
1.2 27 O 1 A e 1 = ( ) ( 1 0 1 ) 0.7 e2 = ( ) ( 0 1 0.3 ) 0.6 A 1 e 1 A 2 e 2 e 1 e 2 m n A n m A 1 e 1 2 e 2 1.11 1.11 e 1, e 2, 2 A B Ax = Bx x A = B 2 1 2 Ae 1 = Be 1 A 1 B 1 Ae 2 = Be 2 A 2 B 2 x A = ` ` 2 0 1 3 B = 77 0 66 3 x = (0, 1) T Ax = Bx = (0, 3) T A B
28 1 1.2.4 = k m B = (b ij ) m n A = (a jp ) b 11 b 1m a 11 a 1n.... b k1 b km a m1 a mn (b 11 a 11 + + b 1m a m1 ) (b 11 a 1n + + b 1m a mn ) =.. (1.9) (b k1 a 11 + + b km a m1 ) (b k1 a 1n + + b km a mn ) : 2 7 ( ) (2 1 + 7 2) (2 3 7 1) 16 1 9 5 1 3 = 2 1 (9 1 + 5 2) (9 3 5 1) = 19 22 4 3 (4 1 + 3 2) (4 3 3 1) 10 9 k m m n k n 1. 2. 3. a 11 a 1n B.. B a 11.,, B a 1n. a m1 a mn a m1 a mn b 11 a 11 + + b 1m a m1 b 11 a 1n + + b 1m a mn.,,. b k1 a 11 + + b km a m1 b k1 a 1n + + b km a mn (b 11 a 11 + + b 1m a m1 ) (b 11 a 1n + + b 1m a mn ).. (b k1 a 11 + + b km a m1 ) (b k1 a 1n + + b km a mn )
1.2 29 x A y = Ax y B z = B(Ax) BA x z x A BA 1.12 y B z 1.12 A B BA (BA)x = B(Ax) x BAx (BA)x B(Ax) 1.13 B A 1.13 z = (BA)x B i A j z i x j BA (i, j) B i A j 1.9