112 对称正定矩阵的Cholesky分解

1．功能 分解对称正定矩阵A=LL^T,其中L为下三角阵，L^T是L的转置.

2．算法简介

由A=LL^T，用矩阵乘法并比较两边矩阵的元素，可导出算式。对K=1,2,3…,n

L_KK=(A_KK-Σ_I=1^k-1L²_KI)^1/2

L_JK=(A_JK-Σ_I=1^k-1L_JIL_IK)/L_KK,j=k+1,k+2,…,n.

3. 使用说明

输入参数

N 整变量，方阵的阶数。

A(N,N) N×N个元素的二维数组，按行存放A的元素.

输出参数

W 标志： W=0,A 对称正定，分解为A=LL^T;W=1,说明A对称但非正定,不能作Cholesky分解。

L(N,N) 下三角阵。

10 ‘************************************** ’

20 ‘* 112 对称正定矩阵的Cholesky分解 *’

30 ‘************************************** ‘

40 INPUT "N="; N

50 DIM A(N, N)

60 PRINT TAB(3); "EXAMPLE"; TAB(8); "JUZHEN A": PRINT

70 FOR I = 1 TO N

80 FOR J = 1 TO N

90 READ A(I, J): PRINT USING "###.#"; A(I, J);

100 NEXT J

110 PRINT

120 NEXT I

130 PRINT : GOSUB 300

140 PRINT TAB(5); "W="; W: PRINT ""

150 IF W = 1 THEN 250

160 PRINT TAB(15); "JUZHEN L"

170 FOR I = 1 TO N

180 FOR J = 1 TO I

190 PRINT USING "##.######"; A(I, J); : PRINT " ";

200 NEXT J

210 PRINT

220 NEXT I

230 DATA 1,2,1,-3,2,5,0,-5,1,0,14,1,-3,-5,1,15

240 DATA

250 END

300 'ZI CHENG XU'

310 IF A(1, 1) <= 0 THEN 410

320 A(1, 1) = SQR(A(1, 1))

330 FOR I = 2 TO N

340 A(I, 1) = A(I, 1) / A(1, 1)

350 NEXT I

360 FOR K = 2 TO N

370 FOR J = 1 TO K - 1

380 A(K, K) = A(K, K) - A(K, J) ^ 2

390 NEXT J

400 IF A(K, K) > 0 THEN 430

410 W = 1

420 GOTO 530

430 A(K, K) = SQR(A(K, K))

440 IF K = N THEN 530

450 FOR I = K + 1 TO N

460 FOR J = 1 TO K - 1

470 A(I, K) = A(I, K) - A(I, J) * A(K, J)

480 NEXT J

490 A(I, K) = A(I, K) / A(K, K)

500 NEXT I

510 NEXT K

520 W = 0

530 RETURN

例题 矩阵A

1.0 2.0 1.0 -3.0

2.0 5.0 0.0 -5.0

1.0 0.0 14.0 1.0

-3.0 –5.0 1.0 15.0

W=0

矩阵l

1． 00

2． 00 1. 00

1. 00 -2. 00 3. 00

-3. 00 1. 00 2. 00 1. 00