Solve Equations 4x1+8x2=8;8x1+18x2+2x3=18;2x2+5x3+1.5x4=0.5;1.5x3+1.75x4=1.75 using Thomas algorithm method

Solution:
Solving tridiagonal matrix system of equations using Thomas algorithm method
Total Equations are `4`

`4x1+8x2+0x3+0x4=8 -> (1)`

`8x1+18x2+2x3+0x4=18 -> (2)`

`0x1+2x2+5x3+1.5x4=0.5 -> (3)`

`0x1+0x2+1.5x3+1.75x4=1.75 -> (4)`

Converting given equations into matrix form

	`4`	`8`	`0`	`0`		`8`
	`8`	`18`	`2`	`0`		`18`
	`0`	`2`	`5`	`1.5`		`0.5`
	`0`	`0`	`1.5`	`1.75`		`1.75`

Identify each number above with Thomas algorithm notation

	`b_1`	`c_1`	`0`	`0`		`d_1`
	`a_2`	`b_2`	`c_2`	`0`		`d_2`
	`0`	`a_3`	`b_3`	`c_3`		`d_3`
	`0`	`0`	`a_4`	`b_4`		`d_4`

Sub diagonal values are
`a_2=8,a_3=2,a_4=1.5`

Main diagonal values are
`b_1=4,b_2=18,b_3=5,b_4=1.75`

Super diagonal values are
`c_1=8,c_2=2,c_3=1.5`

Right-hand side values are
`d_1=8,d_2=18,d_3=0.5,d_4=1.75`

Step-1 :
`y_i=b_i-(a_i * c_(i-1))/(y_(i-1)), i=2,3,4`

`y_1=b_1``=4`

`y_2=b_2-(a_2 * c_1)/(y_1)`

`=18-(8 * 8)/(4)`

`=18-16`

`=2`

`y_3=b_3-(a_3 * c_2)/(y_2)`

`=5-(2 * 2)/(2)`

`=5-2`

`=3`

`y_4=b_4-(a_4 * c_3)/(y_3)`

`=1.75-(1.5 * 1.5)/(3)`

`=1.75-0.75`

`=1`

Step-2 :
`z_i=(d_i-a_i*z_(i-1))/(y_i), i=2,3,4`

`z_1=d_1/y_1``=8/4=2`

`z_2=(d_2-a_2*z_1)/(y_2)`

`=(18 - 8 * 2)/(2)`

`=(2)/(2)`

`=1`

`z_3=(d_3-a_3*z_2)/(y_3)`

`=(0.5 - 2 * 1)/(3)`

`=(-1.5)/(3)`

`=-0.5`

`z_4=(d_4-a_4*z_3)/(y_4)`

`=(1.75 - 1.5 * (-0.5))/(1)`

`=(2.5)/(1)`

`=2.5`

Step-3 :
`x_i=z_i-(c_i*x_(i+1))/(y_i), i=3,2,1`

`x_4=z_4``=2.5`

`x_3=z_3-(c_3*x_4)/(y_3)`

`=-0.5-(1.5 * 2.5)/(3)`

`=-0.5-1.25`

`=-1.75`

`x_2=z_2-(c_2*x_3)/(y_2)`

`=1-(2 * (-1.75))/(2)`

`=1+1.75`

`=2.75`

`x_1=z_1-(c_1*x_2)/(y_1)`

`=2-(8 * 2.75)/(4)`

`=2-5.5`

`=-3.5`

Solution is
`x_1=-3.5`

`x_2=2.75`

`x_3=-1.75`

`x_4=2.5`

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