推 joe7078 :懂了!感謝高手解答 08/16 20:09
※ 引述《joe7078 (Science & Truth)》之銘言：
: 各位好! 小弟我第一次在這裡發文,如有疏漏,懇請指教^^
: 研究中出現了一個神祕的三元二次非線性方程組,
: 百思不得其解, 懇請各位先進幫忙, 謝謝 !
: 方程組如下 :
: A * x^2 + B * y^2 + C * z^2 + D *xyz = E
: i i i i i
: i=1,2,3
: 此方程組看似無解又有解, 直觀上好像是三個橢球的交集, 不知道有沒有解
: 若把A ,B ,C 帶數值用MATLAB好像也算不太出來(MATLAB新手) @@
: i i i
: 懇請各位幫忙想想辦法 ORZ
: Please
: ※ 編輯: joe7078 來自: 1.169.163.102 (08/16 02:38)
設
A1=/=0
(A3*B1/A1-B3)=/=0
(A11*A22/A21-A12)=/=0
A1 * x^2 + B1 * y^2 + C1 * z^2 + D1 *xyz = E1 (1-1)
A2 * x^2 + B2 * y^2 + C2 * z^2 + D2 *xyz = E2 (1-2)
A3 * x^2 + B3 * y^2 + C3 * z^2 + D3 *xyz = E3 (1-3)
(1-1)*A2/A1 :
A2 * x^2 + A2*B1 * y^2/A1 + A2*C1 * z^2/A1 + A2*D1 *xyz/A1 = A2*E1/A1 (1-4)
(1-1)*A3/A1 :
A3 * x^2 + A3*B1 * y^2/A1 + A3*C1 * z^2/A1 + A3*D1 *xyz/A1 = A3*E1/A1 (1-5)
(1-4)-(1-2):
(A2*B1/A1-B2)*y^2 + (A2*C1/A1-C2)*z^2 + (A2*D1/A1-D2)*xyz = (A2*E1/A1-E2) (1-6)
取
A11=(A2*B1/A1-B2)
A12=(A2*C1/A1-C2)
A13=(A2*D1/A1-D2)
A14=(A2*E1/A1-E2)
A11*y^2 + A12*z^2 + A13*xyz = A14 (2-1)
(1-5)-(1-3):
(A3*B1/A1-B3)*y^2 + (A3*C1/A1-C3)*z^2 + (A3*D1/A1-D3)*xyz = (A3*E1/A1-E3) (1-7)
取
A21=(A3*B1/A1-B3)
A22=(A3*C1/A1-C3)
A23=(A3*D1/A1-D3)
A24=(A3*E1/A1-E3)
A21*y^2 + A22*z^2 + A23*xyz = A24 (2-2)
(2-2)*A11/A21:
A11*y^2 + A11*A22*z^2/A21 + A11*A23*xyz/A21 = A11*A24/A21 (2-3)
(2-3)-(2-1):
(A11*A22/A21-A12)*z^2 + (A11*A23/A21-A13)*xyz = (A11*A24/A21-A14) (2-4)
取
A31=(A11*A22/A21-A12)
A32=(A11*A23/A21-A13)
A33=(A11*A24/A21-A14)
A31*z^2 + A32*xyz = A33 (2-5)
取
A41=A32/A31
A42=A33/A31
z^2 + A41*xyz = A42 (2-6)
-->z^2=A42-A41*xyz (2-7)
代回(1-1)(1-2)(1-3)可得
A1 * x^2 + B1 * y^2 + (D1-c1*A41) *xyz = E1-A42*C1 (3-1)
A2 * x^2 + B2 * y^2 + (D2-c2*A41) *xyz = E2-A42*C2 (3-2)
A3 * x^2 + B3 * y^2 + (D3-C3*A41) *xyz = E3-A42*C3 (3-3)
取
Y1=x^2 (3-4)
Y2=y^2 (3-5)
Y3=xyz (3-6)
F1=(D1-c1*A41)
F2=(D2-c2*A41)
F3=(D3-C3*A41)
G1=E1-A42*C1
G2=E2-A42*C2
G3=E3-A42*C3
A1 * Y1 + B1 * Y2 + F1 *Y3 = G1 (4-1)
A2 * Y1 + B2 * Y2 + F2 *Y3 = G2 (4-2)
A3 * Y1 + B3 * Y2 + F3 *Y3 = G3 (4-3)
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到這裡Y1,Y2,Y3該就會解了
再代回(3-4)(3-5)(3-6)
Y1=x^2
Y2=y^2
Y3=xyz
就可以得到x,y,z的值
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