MTH603 Final Term Giga File

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Table of Content

TABLE OF CONTENT ..................................................................................................................................................................... 1

ERROR .......................................................................................................................................................................................... 6

Method of Iterations ................................................................................................................................................................. 13

Graeffe’s Root Square Method .................................................................................................................................................. 20 LINEAR EQUATIONS ............................................................................................................................................................................. 20 Gaussian Elimination Method ................................................................................................................................................... 20 Guass-Jordan Elimination Method ............................................................................................................................................ 23 Jacobi’s Iterative Method .......................................................................................................................................................... 23 Gauss-Seidel Iteration Method .................................................................................................................................................. 25

Matrix Inversion ........................................................................................................................................................................ 28 EIGEN VALUE PROBLEMS ...................................................................................................................................................................... 28 Power Method ........................................................................................................................................................................... 28 Jacobi’s Method ......................................................................................................................................................................... 31 INTERPOLATION .................................................................................................................................................................................. 35 Forward Differences .................................................................................................................................................................. 37 Backward Differences ................................................................................................................................................................ 38 Divided Differences .................................................................................................................................................................... 40 By Newton’s Forward Difference Formula ................................................................................................................................. 47 b) By Lagrange’s Formula ......................................................................................................................................................... 48 Langrange’s Interpolation ......................................................................................................................................................... 49 Differentiation Using Difference Operators ............................................................................................................................... 50 NUMERICAL INTEGRATION .................................................................................................................................................................... 51 Trapezoidal Rule ........................................................................................................................................................................ 51 Simpson’s 1/3 and 3/8 rules ...................................................................................................................................................... 55

Taylor Series Method ................................................................................................................................................................. 57 Euler Method ............................................................................................................................................................................. 59 Euler Modified Method:............................................................................................................................................................. 60 Runge-Kutta Method ................................................................................................................................................................. 61 Milne’s Predictor Corrector Method .......................................................................................................................................... 64 Adam Moultan’s Predictor Corrector Method ........................................................................................................................... 65 FAQ UPDATED VERSION. ................................................................................................................................................ 66

QUESTION:

WHAT IS BRACKETING METHOD? ................................................................................................................. 66

QUESTION:

WHAT IS AN OPEN METHOD? ......................................................................................................................... 67

QUESTION:

EXPLAIN MULLER’S METHOD BRIEFLY. ........................................................................................................ 67

QUESTION:

EXPLAIN THE DIFFERENCE BETWEEN THE LINEAR AND NON-LINEAR EQUATIONS. ................................ 67 QUESTION: EXPLAIN WHICH VALUE IS TO BE CHOOSED AS X0 IN N-R METHOD. ...................................................... 67

QUESTION:

DEFINE ITERATIVE METHOD OF SOLVING LINEAR EQUATIONS WITH TWO EXAMPLES. .......................... 67 QUESTION: DEFINE PIVOTING. ........................................................................................................................................... 67

QUESTION:

WRITE THE TWO STEPS OF SOLVING THE LINEAR EQUATIONS USING GAUSSIAN ELIMINATION METHOD.

QUESTION:

DESCRIBE GAUSS-JORDAN ELIMINATION METHOD BRIEFLY ..................................................................... 67

QUESTION:

DESCRIBE BRIEFLY CROUT’S REDUCTION METHOD. .................................................................................. 68

QUESTION:

DESCRIBE BRIEFLY THE JACOBI’S METHOD OF SOLVING LINEAR EQUATIONS. ...................................... 68 QUESTION: WHAT IS THE DIFFERENCE BETWEEN JACOBI’S METHOD AND GAUSS SEIDAL METHOD? ................... 68 QUESTION: WHAT IS THE BASIC IDEA OF RELAXATION METHOD? ............................................................................... 68

QUESTION:

HOW THE FAST CONVERGENCE IN THE RELAXATION METHOD IS ACHIEVED?........................................ 68 QUESTION: WHICH MATRIX WILL HAVE AN INVERSE? .................................................................................................... 68

QUESTION:

WHAT ARE THE POPULAR METHODS AVAILABLE FOR FINDING THE INVERSE OF A MATRIX? ............... 68 QUESTION: EXPLAIN GAUSSIAN ELIMINATION METHOD FOR FINDING THE INVERSE OF A MATRIX. ....................... 68 QUESTION: WHAT ARE THE STEPS FOR FINDING THE LARGEST EIGEN VALUE BY POWER METHOD. ....................... 68 QUESTION:

WHAT IS THE METHOD FOR FINDING THE EIGEN VALUE OF THE LEAST MAGNITUDE OF THE MATRIX

[A]?

QUESTION:

WHAT IS INTERPOLATION? ............................................................................................................................. 69

QUESTION:

WHAT IS EXTRAPOLATION? ............................................................................................................................ 69

QUESTION:

WHAT HAPPENS WHEN SHIFT OPERATOR E OPERATES ON THE FUNCTION. ............................................ 69 QUESTION: WHAT IS THE BASIC CONDITION FOR THE DATA TO APPLY NEWTON’S INTERPOLATION METHODS? .. 69 QUESTION: WHEN IS THE NEWTON’S FORWARD DIFFERENCE INTERPOLATION FORMULA USED? ........................... 69 QUESTION: WHEN IS THE NEWTON’S FORWARD DIFFERENCE INTERPOLATION FORMULA USED? ........................... 69 QUESTION: WHEN THE NEWTON’S BACKWARD DIFFERENCE INTERPOLATION FORMULA IS USED? ......................... 69 QUESTION:

IF THE VALUES OF THE INDEPENDENT VARIABLE ARE NOT EQUALLY SPACED THEN WHICH FORMULA

SHOULD BE USED FOR INTERPOLATION? ................................................................................................................................. 69

QUESTION:

TO USE NEWTON’S DIVIDED DIFFERENCE INTERPOLATION FORMULA, WHAT SHOULD THE VALUES OF

INDEPENDENT VARIABLES BE? .................................................................................................................................................. 69

QUESTION:

WHICH DIFFERENCE FORMULA IS SYMMETRIC FUNCTION OF ITS ARGUMENTS? ................................... 69 QUESTION:

IS THE INTERPOLATING POLYNOMIAL FOUND BY LAGRANGE’S AND NEWTON’S DIVIDED DIFFERENCE

FORMULAE IS SAME? .................................................................................................................................................................. 70

QUESTION:

WHICH FORMULA INVOLVES LESS NUMBER OF ARITHMETIC OPERATIONS? NEWTON OR LAGRANGE’S? 70

QUESTION:

WHEN DO WE NEED NUMERICAL METHODS FOR DIFFERENTIATION AND INTEGRATION? .................... 70 QUESTION: IF THE VALUE OF THE INDEPENDENT VARIABLE AT WHICH THE DERIVATIVE IS TO BE FOUND APPEARS AT THE BEGINNING OF THE TABLE OF VALUES, THEN WHICH FORMULA SHOULD BE USED? ........................................... 70 QUESTION: WHY WE NEED TO USE RICHARDSON’S EXTRAPOLATION METHOD? ....................................... 70

QUESTION:

TO APPLY SIMPSON’S 1/3 RULE, WHAT SHOULD THE NUMBER OF INTERVALS BE? ............................. 70 QUESTION: TO APPLY SIMPSON’S 3/8 RULE, WHAT SHOULD THE NUMBER OF INTERVALS BE? ............................. 70 QUESTION:

QUESTION:

WHAT IS THE ORDER OF GLOBAL ERROR IN TRAPEZOIDAL RULE? .......................................................... 70

QUESTION:

WHAT IS THE FORMULA FOR FINDING THE WIDTH OF THE INTERVAL? ................................................... 70