Numerical solution of linear algebraic systems.

Method of Gauss.

AX=B A= a11 a12... a1n b= b1

a21 a22... a2n b2

..........

an1 an2... ann bn

a₁₁x₁+a₁₂x₂+a₁₃x₃+...+a_1n-1x_n-1+a_1nxn=b1

a₂₁x₁+a₂₂x₂+a₂₃x₃+...+a_2n-1x_n-1+a_2nxn=b2

a₃₁x₁+a₃₂x₂+a₃₃x₃+...+a_3n-1x_n-1+a_3nxn=b3 (1)

.................

a_n-11x₁+a_n-12x₂+a_n-13x₃+...+a_n-1n-1x_n-1+a_n-1nxn=bn-1

a_n1x₁+a_n2x₂+a_n3x₃+...+a_nn-1x_n-1+annxn=bn

We can put system in such way, when a₁₁ < > 0;

Let’s to divide 1^st equation on a₁₁, then:

c₁+α ₁₂c₂+α ₁₃c₃+... +α _1n-1c_n-1+α _1nc_n (2)

where:

After that in succession multiply 1^st equation on coefficients with x₁ in every equation and subtract; so we except x₁ out of equation from 2 until n; system will be:

x₁+α ¢ ₁₂x₂+α ¢ ₁₃x₃+...+α ¢ _1n-1x_n-1+α ¢ _1nx_n=β ₁

a¢ ₂₂x₂+a¢ ₂₃x₃+...+a¢ _2n-1x_n-1+a¢ _2nx_n=b¢ ₂

a¢ ₃₂x₂+a¢ ₃₃x₃+...+a¢ _3n-1x_n-1+a¢ _3nx_n=b¢ ₃ (3)

................

a¢ _n-12x₂+a¢ _n-13x₃+...+a¢ _n-1n-1x_n-1+a¢ _n-1nx_n=b¢ _n-1

a¢ _n2x₂+a¢ _n3x₃+...+a¢ _nn-1x_n-1+a¢ _nnx_n=b¢ _n

a¢ _ij=a_ij-a_i1 α _1j, i=2,..., n

j=2,..., n

b¢ _i=b_i-a_i1β

Now we can consider system of n-1 equations

x2,..., xn

a¢ ₂₂< > 0, then divide 2^nd equation:

x₂+α ₂₃x₃+...+α _nn-1x_n-1+α _2nx_n=β ₂,

except x₂:

x₁+α ₁₂x₂+α ₁₃x₃+...+α _1n-1x_n-1+α _1nx_n=β ₁

x₂+α ₂₃x₃+...+α _2n-1x_n-1+α _2nx_n=β ₂

a² ₃₃x₃+...+a² _3n-1x_n-1+a² _3nx_n=b² ₃ (4)

................

a² _n-13x₃+...+a² _n-1n-1x_n-1+a² _n-1nx_n=b² _n-1

a² _n3x₃+...+a² _nn-1x_n-1+a² _nnx_n=b² _n

a ² _ij = a ¢ _ij- a ¢ _i2α _2j , i, j=2,..., n

b ² _i = b ¢ _i- a ¢ _i2β ₂, i=2,..., n

After n steps we come to such form:

(5)

This procedure is named the direct motion of the Gauss method.

During the back motion we find values of variables:

(6)

The special case: after m steps

(7)

If we have any coefficient , then system (1) is incompatible. If , then we have system with n–m equations, x_m+1, …, x_n are free and system (1) is indefinite.