Steps:
Solve by Matrix Inversion method:
x + y + z = 8
x - y + 2z = 6
3x + 5y - 7z = 14
Sol:
Writing these equation in matrix form:
+-- --+ +-- --+ +-- --+ | 2 1 1 | | x | | 8 | | 1 -1 2 | | y | = | 6 | | 3 5 -7 | | z | | 14 | +-- --+ +-- --+ +-- --+AX = B
+-- --+ | (7-10) -(-7-6) (5+3) | Cofactor = | -(-7-5) (-7-3) -(5-3) | | (2+1) -(2-1) (-1-1) | +-- --+ +-- --+ | -3 13 8 | Cofactor = | 12 -10 -2 | | 3 -1 -2 | +-- --+ Adj = Transpose of cofactor matrix +-- --+ | -3 12 3 | Adj A = | 13 -10 -1 | | 8 -2 -2 | +-- --+ Now finding |A|, |A| = 1(7-10) -1(-7-6) +1(5+3) |A| = 18 As X = A-1B +-- --+ +-- --+ | -3 12 3 | | 8 | X = | 13 -10 -1 | | 6 | | 8 -2 -2 | | 14 | +-- --+ +-- --+ +- -+ | 90 | X = | 30 | | 24 | +- -+ +-- --+ +-- --+ | x | | 5 | | y | = | 5/3 | | z | | 4/3 | +-- --+ +-- --+
Steps for Gauss Elimination
Solve the following equation by Gauss Elimination Method:
2x + y + 4z = 12
4x + 11y - z = 33
8x -3y + 2z = 20
Sol:
First we convert the above equation to augmented form (matrix)
+-- --+ | 2 1 4 : 12 | | 4 11 -1 : 33 | | 8 -3 2 : 20 | +-- --+pivot = 2
+-- --+ | 2 1 4 : 12 | | 0 9 -9 : 9 | | 8 -3 2 : 20 | +-- --+Pivot is still 2
+-- --+ | 2 1 4 : 12 | | 0 9 -9 : 9 | | 0 -7 -14 : -28 | +-- --+Now we want to set a32 to zero, and since we shifted towards the 2nd column, there will be a new pivot.
+-- --+ | 2 1 4 : 12 | | 0 9 -9 : 9 | | 0 0 -21 : -21 | +-- --+Converting this back into equation form, we get:
Understanding formula using the following example:
y(x) = \(y_0\) + u▵\(y_0\) +
\(\frac{(u-1)u}{2!}\)▵2\(y_{-1}\) +
\(\frac{(u-1)u(u+1)}{3!}\)▵3\(y_{-1}\) +
\(\frac{(u-2)(u-1)u(u+1)}{4!}\)▵4\(y_{-2}\) +
\(\frac{(u-2)(u-1)u(u+1)(u+2)}{5!}\)▵5\(y_{-2}\) + ...
u = \(\frac{x - x_o}{h}\)
Q- Use Gauss's forward formula to find the value of y when x = 3.75 from the following table:
x | 2.5 | 3.0 | 3.5 | 4.0 | 4.5 | 5.0 | y | 24.145 | 22.043 | 20.225 | 18.644 | 17.262 | 16.047 |Sol:
x | y | △y | △^2 y | △^3 y | △^4 y | △^5 y ---------+-------------+----------------------------+----------------------------+-------------+------------- 2.5 | 24.145 | | | | | | | 22.043-24.145= -2.102 (-2)| | | | 3.0 | 22.043 | | 0.284 (y-2) | | | | | -1.818 (-1)| | -0.047 (y-2) | | 3.5 (x0) | 20.225 (y0)✮| | 0.237 (y-1)✮| | 0.009 (y-2)✮| | | -1.581 (0)✮| | -0.038 (y-1)✮| | -0.003 (y-2)✮ 4.0 | 18.644 | | 0.199 (y0) | | 0.006 (y-1) | | | -1.382 (1)| | -0.032 (y0) | | 4.5 | 17.262 | | 0.167 (y1) | | | | | -1.215 (2)| | | | 5.0 | 16.047 | | | | |y(x) = 20.225 + 0.5(-1.581) + \(\frac{(0.5-1)0.5}{2!}\) (0.237) + \(\frac{(0.5-1)0.5(0.5+1)}{3!}\)(-0.038) + \(\frac{(0.5-2)(0.5-1)0.5(0.5+1)}{4!}\)(0.009) + \(\frac{(0.5-2)(0.5-1)0.5(0.5+1)(0.5+2)}{5!}\)(-0.003)
y(x) = \(y_o\) + u▵\(y_{-1}\) + \(\frac{u(u+1)}{2!}\)▵2\(y_{-1}\) +
\(\frac{(u-1)u(u+1)}{3!}\)▵3\(y_{-2}\) +
\(\frac{(u-1)u(u+1)(u+2)}{4!}\)▵4\(y_{-2}\) +
\(\frac{(u-2)(u-1)u(u+1)(u+2)}{5!}\)▵5\(y_{-3}\)
u = \(\frac{x - x_o}{h}\)
Q- Apply Gauss Backward formula to find the population of the town in 1946, given that
Year | 1931 | 1941 | 1951 | 1961 | 1971 Population | 15 | 20 | 27 | 39 | 52 (In thousands) |u = \(\frac{x - x_o}{h}\)
x | y | △y | △^2 y | △^3 y | △^4 y | ---------+---------+----------+---------+----------+-----------+ 1931 | 15 | | | | | | | 5 (y-2)| | | | 1941 | 20 | | 2 (y-2) | | | | | 7 (y-1)✮| | 3 (y-2)✮ | | 1951 (x0)| 27 (y0)✮| | 5 (y-1)✮| | -7 (y-2)✮ | | | 12 (y0) | | -4 (y-1) | | 1961 | 39 | | 1 (y0) | | | | | 13 (y1)| | | | 1971 | 52 | | | | |y(x) = 27 + (-0.5)(7) + \(\frac{(-0.5)(-0.5+1)}{2!}\)(5) + \(\frac{(-0.5-1)(-0.5)(-0.5+1)}{3!}\)(3) + \(\frac{(-0.5-1)(-0.5)(-0.5+1)(-0.5+2)}{4!}\)(-7)
Reference