Numerical Analysis Lecture 45 Summing up NonLinear Equations
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Numerical Analysis Lecture 45
Summing up
Non-Linear Equations
Bisection Method (Bolzano) Regula-Falsi Method of iteration Newton - Raphson Method Muller’s Method Graeffe’s Root Squaring Method
In the method of False Position, the first approximation to the root of f (x) = 0 is given by (2. 2) Here f (xn-1) and f (xn+1) are of opposite sign. Successive approximations to the root of f (x) = 0 is given by Eq. (2. 2).
METHOD OF ITERATION can be applied to find a real root of the equation f (x) = 0 by rewriting the same in the form,
In Newton – Raphson Method successive approximations x 2, x 3, …, xn to the root are obtained from N-R Formula
Secant Method This sequence converges to the root ‘b’ of f (x) = 0 i. e. f( b ) = 0.
The Secant method converges faster than linear and slower than Newton’s quadratic.
In Muller’s Method we can get a better approximation to the root, by using
Where we defined
Systems of Linear Equations
Gaussian Elimination Gauss-Jordon Elimination Crout’s Reduction Jacobi’s Gauss- Seidal Iteration Relaxation Matrix Inversion
In Gaussian Elimination method, the solution to the system of equations is obtained in two stages. i)the given system of equations is reduced to an equivalent upper triangular form using elementary transformations ii)the upper triangular system is solved using back substitution procedure
Gauss-Jordon method is a variation of Gaussian method. In this method, the elements above and below the diagonal are simultaneously made zero
In Crout’s Reduction Method the coefficient matrix [A] of the system of equations is decomposed into the product of two matrices [L] and [U], where [L] is a lower-triangular matrix and [U] is an upper-triangular matrix with 1’s on its main diagonal.
For the purpose of illustration, consider a general matrix in the form
Jacobi’s Method is an iterative method, where initial approximate solution to a given system of equations is assumed and is improved towards the exact solution in an iterative way.
In Jacobi’s method, the (r + 1)th approximation to the above system is given by Equations
Here we can observe that no element of replaces entirely for the next cycle of computation.
In Gauss-Seidel method, the corresponding elements of replaces those of as soon as they become available. It is also called method of Successive Displacement.
The Relaxation Method is also an iterative method and is due to Southwell.
Eigen Value Problems Power Method Jcobi’s Method
In Power Method the result looks like Here, is the desired largest eigen value and is the corresponding eigenvector.
Interpolation
Finite Difference Operators Newton’s Forward Difference Interpolation Formula Newton’s Backward Difference Interpolation Formula Lagrange’s Interpolation Formula Divided Differences Interpolation in Two Dimensions Cubic Spline Interpolation
Finite Difference Operators Forward Differences Backward Differences Central Difference
Thus Similarly
Shift operator, E
The inverse operator is defined as Similarly, -1 E
Average Operator,
Differential Operator, D
Important Results
The Newton’s forward difference formula for interpolation, which gives the value of f (x 0 + ph) in terms of f (x 0) and its leading differences.
An alternate expression is
Newton’s Backward difference formula is,
Alternatively, this formula can also be written as Here
The Lagrange’s formula for interpolation
Newton’s divided difference interpolation formula can be written as
Where the first order divided difference is defined as
Numerical Differentiation and Integration
We expressed D in terms of ∆ :
Using backward difference operator , we have On expansion, we have
Using Central difference Operator Differentiation Using Interpolation Richardson’s Extrapolation
Thus, is approximated by which is given by
Basic Issues in Integration What does an integral represent? = AREA = VOLUME
Y y = f(x) (x 1, y 1) (x 2, y 2) (x 0, y 0) y 0 y 1 y 2 y 3 yn-1 yn O X x 0 = a x 1 x 2 x 3 xn-1 xn = b
Y y = f(x) (x 2, y 2) (x 0, y 0) y 0 y 1 y 2 O X x 0 = a x 1 x 2 x 3 xn-1 xn = b
TRAPEZOIDAL RULE
DOUBLE INTEGRATION We described procedure to evaluate numerically a double integral of the form
Differential Equations
Taylor Series Euler Method Runge-Kutta Method Predictor Corrector Method
In Taylor’s series we expanded y (t ) by Taylor’s series about the point t = t 0 and obtain
In Euler Method we obtained the solution of the differential equation in the form of a recurrence relation
We derived the recurrence relation Which is the modified Euler’s method.
The fourth-order R-K method was described as
where
In general, Milne’s predictorcorrector pair can be written as
This is known as Adam’s predictor formula. Alternatively, it can be written as
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