Solution of a System of Linear Algebraic Equations 2. Find the unknowns by backward substitution. If the breakdown is permanent, then the system has either no solution or infinitely many solutions.
Note: at any stage of solution you can use the GCD the greatest common divisor. This will simplify the mathematical computations. Another important benefit is that when the numbers are very big then using the GCD 3. Solution of a System of Linear Algebraic Equations will prevent the rounding of big numbers and thus we will obtain exact results. This definition is also used for other rows. Rows with blank definition mean that there are no changes occur to these rows in these stages.
Backward substitution Check the solution by substituting x values in the original system of equations. Solution Multiply Eq. Solution of a System of Linear Algebraic Equations Check the solution by substituting x values in the original system of equations.
This is a contradiction and therefore no solution exists. Thus, there are 3 unknowns and 2 equations. Thus 3. Solution of a System of Linear Algebraic Equations Backward substitution Check the solution by substituting x values in the original system of equations. Numerical Methods 4. Matrix Inversion 4. Matrix Inversion Integer or Rational: A. If it is not possible to reduce matrix A to I then matrix A is not an invertible matrix.
Note: at any stage of solution you can use the GCD the greatest common divisor to simplify the computations. Matrix Inversion Since it is not possible to reduce matrix A to I we have obtained a row of zeros on the left side , A is not invertible. Numerical Methods 5. Interpolation 5. Interpolation Substitute the given points into the polynomial equation Solve this linear system of equations to get Substitute these values into the polynomial equation to get 5.
Then 5. Solution 5. Then differentiate or integrate this polynomial. The subintervals must be even while there are 5 subintervals. To solve this problem, there are two choices:. Therefore, a polynomial is fitted to the given data. This means that x is a function of y. This subject is called Inverse Interpolation. Numerical Methods 6. Curve Fitting 6. Curve Fitting Given: the following experimental data xi x0 x1 …… Required: Finding the constants of the best fit equation 6.
Solve the normal equations to find the constants of the best fit equation a, b, c Linear relationship Given: the following experimental data xi x0 x1 …… These are 2 linear algebraic equations in a and b.
Solve these equations to obtain the a and b values. Then substitute the a and b values into Eq. Now, the y value for any x in the closed interval [x0, xn] can be found by substituting the x value in the best fit equation and finding the corresponding y value.
Note: Substituting the tabulated x values in the best fit equation will not give the corresponding tabulated y values except when the best fit curve passes through the given points. This is the major difference between interpolation and best fit. Conclusion 6. Curve Fitting From the above derivation, we conclude that the normal equations of the linear equation can be obtained by multiplying the linear equation by the coefficients of the constants a and b and sum the results.
This is true for any polynomial of degree n. Here a, b We can find the normal equations of the given best fit equation by multiplying the given best fit equation by the coefficients of the constants a, b, c However, for simplicity, the nonlinear x or y or both can by replaced by linear terms to obtain the shape of a linear equation in the new terms. Then, we find the normal equations of the new linear equation. Note that sometimes we need some manipulations before replacing the nonlinear x or y or both by linear terms.
Curve Fitting. Set 2 using ln or log Here a or b or both are nonlinear. Use ln or log to convert the best fit equation to a linear equation. Here we cannot find the normal equations directly by the least squares of errors method. So we convert the given best fit equation into a linear equation and then finding the normal equations for the new linear equation. Curve Fitting x 0 1 4 9 16 y 0. Curve Fitting Example 4: Fit to the following data, work to 4dp x 1.
Numerical Methods 7. Numerical Solutions of Ordinary Differential Eqs 7. Numerical Solutions of Ordinary Differential Eqs and so on. Solution and so on to finally get y 0. Use Euler Method for y and z. Solution [ ] [ ] 7. To browse Academia. Log in with Facebook Log in with Google. Remember me on this computer. Enter the email address you signed up with and we'll email you a reset link. Need an account? Click here to sign up. Download Free PDF.
Numerical Methods - Lecture Notes — Najm Alghazali. A short summary of this paper. Download Download PDF. Translate PDF. Najm Alghazali Civil Eng. Numerical Integration Solution of a System of Linear Algebraic Equations Matrix Inversion Curve Fitting Numerical Solution of Ordinary Differential Equations Numerical Solution of Partial Differential Equations Wylie, C. Finney, R.
B References 1. Gerald C. Burden R. Numerical Methods 1. That is, the zero of f x is approximated by the zero of the tangent line of f x. Let x0 be the initial estimate of r Solve for x1 1. Open Methods: Sometimes we need to find the root of an equation near a point x.
Bracketing methods: Sometimes we need to find the root s of an equation in an interval [a, b]. Examine the sign of f x at the ends of the interval [a, b].
The best initial point is the point that makes the value of f x closer to zero. Determine the root position x 0 0. Numerical Methods 2. Numerical Integration 2. Therefore, numerical integration is used.
When the function only exists as a table of values i. Here n must be even. Then H. Work to 4dp. Solution 2. Numerical Methods 3. Solution of a System of Linear Algebraic Equations 3. Steps of solution Alghazali Technique 1. Solution of a System of Linear Algebraic Equations 2. Find the unknowns by backward substitution. If the breakdown is permanent, then the system has either no solution or infinitely many solutions.
Note: at any stage of solution you can use the GCD the greatest common divisor. This will simplify the mathematical computations. Another important benefit is that when the numbers are very big then using the GCD 3. Solution of a System of Linear Algebraic Equations will prevent the rounding of big numbers and thus we will obtain exact results. This definition is also used for other rows.
Rows with blank definition mean that there are no changes occur to these rows in these stages. Backward substitution Check the solution by substituting x values in the original system of equations. Solution Multiply Eq. Solution of a System of Linear Algebraic Equations Check the solution by substituting x values in the original system of equations. This is a contradiction and therefore no solution exists. Thus, there are 3 unknowns and 2 equations.
Thus 3. Solution of a System of Linear Algebraic Equations Backward substitution Check the solution by substituting x values in the original system of equations. Numerical Methods 4. Matrix Inversion 4. Matrix Inversion Integer or Rational: A.
If it is not possible to reduce matrix A to I then matrix A is not an invertible matrix. Note: at any stage of solution you can use the GCD the greatest common divisor to simplify the computations.
Matrix Inversion Since it is not possible to reduce matrix A to I we have obtained a row of zeros on the left side , A is not invertible. Numerical Methods 5.
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