Analytical Numerical Analysis

S .R aja R ama mana nath tha an

APPLIED NUMERICAL ANALYSIS NEWTON RAPHSON METHOD:

ORDER OF CONVERGENCE: 2 ADVANTAGES: 1. The advantage of the method is its order of convergence c onvergence is quadratic. quadratic. 2. Convergence rate is one of the fastest when it does converges 3. Linear convergence near multiple roots.

DISADVANTAGES: 1. It needs the function evaluation and then the derivative ev aluation. 2. If the tangent is parallel or nearly parallel to the x-axis, then the method does not converge. 3. Usually Newton method is expected to converge only near the solution.

REGULA FALSI METHOD:

ORDER OF CONVERGENCE: 1.618 ADVANTAGES: 1. Better-than-linea Better-than-linearr convergence near simple root 2. Linear convergence near multiple root 3. No derivative needed DISADVANTAGES 1. Iterates may diverge 2. No practical & rigorous error bound

GRAEFFE’S METHOD:

ADVANTAGES: 1. It is used to find real, distinct, complex and double roots. 2. It is the direct method of finding f inding solution. DISADVANTAGES: 1. It has numerically complicated procedure. LIN-BAIRSTOW METHOD

ADVANTAGES: 1. It is used to find all a ll roots of the equation.

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S .R aja R ama mana nath tha an

APPLIED NUMERICAL ANALYSIS

DISADVANTAGES: 1. Long division of the polynomial to be solved GAUSS ELIMINATION METHOD:

ADVANTAGES: 1. It is the direct method of solving s olving linear simultaneous equations. 2. It uses back substitution. substitution. 3. It is reduced to equivalent upper triangular matrix. DISADVANTAGES: 1. It requires right vectors to be known. GAUSS JORDAN:

ADVANTAGES: 1. It is direct method. 2. The roots of the equation are found immediately without using back substitution. 3. It is reduced to equivalent identity matrix. DISADVANTAGES: 1. The additional steps increase round off errors. 2. It requires right vectors to be known.

LU DECOMPOSITION:

ADVANTAGES: 1. Suppose we want want to solve a m×n system AX = b. If we can find a LU-decomposition for A , then to solve AX =b, it is enough to solve the systems

Thus the system LY = b can be solved by the method of forward substitution and the system UX = Y can be solved by the method of backward substitution. 2. The advantage is that the solution of triangular set of equations is t rivial to solve. 3. 4. 5. 6. 7.

It is direct method. Applicable for any matrix Finds all solutions Easy to program Fast Page 2 of 3

S .R aja R ama mana nath tha an

APPLIED NUMERICAL ANALYSIS

DISADVANTAGES; 1. Does not Find approximate solutions (least square) 2. Could easily be unstable

CROUT’S METHOD:

ADVANTAGES: 1. The advantage is that the solution of triangular set of equations is t rivial to solve. 2. 3. 4. 5. 6.

It is direct method. Applicable for any matrix Finds all solutions Easy to program Fast

DISADVANTAGES: 1. Complete pivoting is almost impossible and only the partial column wise pivoting is possible

GAUSS JACOBI METHOD:

It is iterative method. 2. The system of equations must be diagonally dominant. 3. It suits better for large numbers of unknown unknownss 4. It is self correcting method. 1.

GAUSS SEIDEL METHOD:

1. 2. 3. 4. 5.

It is iterative method. The system of equations must be diagonally dominant. It suits better for large numbers of unknown unknownss It is self correcting method. The number of iterations is less than Jacobi method.

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