# NUMERICAL METHODS INTRODUCTION

Views:

Category: Education

## Presentation Description

No description available.

## Comments

By: orbit7oz (81 month(s) ago)

hi could you please allow me to download the slides... the slides are very precisely describe the main idea in learning numerical method...thanks

## Presentation Transcript

### NUMERICAL METHODS :

NUMERICAL METHODS INTRODUCTION

### NUMERICAL METHODS/ANALYSIS :

NUMERICAL METHODS/ANALYSIS IT IS THE DEVELOPMENT & STUDY OF PROCEDURES FOR SOLVING PROBLEMS WITH A COMPUTING INSTRUMENT OR COMPUTER.

### ALGORITHM :

ALGORITHM IT IS USED FOR A SYSTEMATIC PROCEDURE THAT SOLVES A PROBLEM OR A NUMBER OF PROBLEMS. ITS EFFICIENCY MAY BE MEASURED BY THE NUMBER OF STEPS IN THE ALGORITHM, THE COMPUTER TIME, AND THE AMOUNT OF MEMORY (OF THE COMPUTING INSTRUMENT) THAT IS REQUIRED.

### NOTE AND UNDERSTAND :

NOTE AND UNDERSTAND THE MAJOR ADVANTAGE OF NUMERICAL ANALYSIS IS THAT A NUMERICAL VALUE CAN BE OBTAINED EVEN WHEN THE PROBLEM HAS NO “ANALYTICAL” SOLUTION. THE MATHEMATICAL OPERATIONS REQUIRED (GENERALLY) ARE ESSENTIALLY ADDITION, SUBTRACTION, MULTIPLICATION, AND DIVISION PLUS MAKING COMPARISONS.

### NOTE AND UNDERSTAND :

NOTE AND UNDERSTAND IT IS IMPORTANT TO REALIZE THAT A NUMERICAL ANALYSIS SOLUTION IS ALWAYS NUMERICAL. ANALYTICAL METHODS, ON THE OTHER HAND, USUALLY GIVE A RESULT IN TERMS OF MATHEMATICAL FUNCTIONS THAT CAN THEN BE EVALUATED FOR SPECIFIC INSTANCES.

### NOTE AND UNDERSTAND :

NOTE AND UNDERSTAND NUMERICAL ANALYSIS IS AN APPROXIMATION, BUT RESULTS CAN BE MADE AS ACCURATELY AS DESIRED. TO ACHIEVE HIGH ACCURACY, NUMEROUS SEPARATE OPERATIONS MUST BE CARRIED OUT, BUT CURRENT COMPUTERS DO THEM SO RAPIDLY WITHOUT EVER MAKING MISTAKES.

### SOME OPERATIONS THAT NUMERICAL ANALYSIS CAN DO :

SOME OPERATIONS THAT NUMERICAL ANALYSIS CAN DO SOLVE FOR THE ROOTS OF A NON-LINEAR EQUATION. SOLVE FOR LARGE SYSTEMS OF EQUATIONS. GET THE SOLUTIONS OF A SET OF NON-LINEAR EQUATIONS. INTERPOLATE TO FIND THE INTERMEDIATE VALUES WITHIN A TABLE OF DATA.

### SOME OPERATIONS THAT NUMERICAL ANALYSIS CAN DO :

SOME OPERATIONS THAT NUMERICAL ANALYSIS CAN DO FIND EFFICIENT & EFFECTIVE APPROXIMATIONS OF FUNCTIONS. APPROXIMATE DERIVATIVES OF ANY ORDER FOR FUNCTIONS EVEN WHEN THE FUNCTION IS KNOWN ONLY AS A TABLE OF VALUES. INTEGRATE ANY FUNCTION EVEN WHEN IT IS KNOWN ONLY AS A TABLE OF VALUES.

### SOME OPERATIONS THAT NUMERICAL ANALYSIS CAN DO :

SOME OPERATIONS THAT NUMERICAL ANALYSIS CAN DO SOLVE ORDINARY DIFFERENTIAL EQUATIONS WHEN GIVEN INITIAL VALUES OR CONDITIONS FOR THE VARIABLES. THESE CAN BE OF ANY ORDER &/OR COMPLEXITY. SOLVE BOUNDARY-VALUE PROBLEMS & DETERMINE EIGENVALUES & EIGENVECTORS.

### SOME OPERATIONS THAT NUMERICAL ANALYSIS CAN DO :

SOME OPERATIONS THAT NUMERICAL ANALYSIS CAN DO OBTAIN NUMERICAL SOLUTIONS OF ALL TYPES OF PARTIAL DIFFERENTIAL EQUATIONS. FIT CURVES TO DATA BY A VARIETY OF METHODS.

### GENERAL STEPS IN SOLVING MATHEMATICAL OR ENGINEERING PROBLEMS :

GENERAL STEPS IN SOLVING MATHEMATICAL OR ENGINEERING PROBLEMS STATE THE PROBLEM CLEARLY, INCLUDING ANY SIMPLIFYING ASSUMPTIONS. DEVELOP A MATHEMATICAL STATEMENT OF THE PROBLEM IN A FORM THAT CAN BE SOLVED FOR A NUMERICAL ANSWER.

### GENERAL STEPS IN SOLVING MATHEMATICAL OR ENGINEERING PROBLEMS :

GENERAL STEPS IN SOLVING MATHEMATICAL OR ENGINEERING PROBLEMS THIS PROCESS MAY INVOLVE, AS IN THE PRESENT CASE, THE USE OF CALCULUS. IN OTHER SITUATIONS, OTHER MATHEMATICAL PROCEDURES MAY BE EMPLOYED. WHEN THIS STATEMENT IS A DIFFERENTIAL EQUATION, APPROPRIATE INITIAL CONDITIONS AND/OR BOUNDARY CONDITIONS MUST BE SPECIFIED.

### GENERAL STEPS IN SOLVING MATHEMATICAL OR ENGINEERING PROBLEMS :

GENERAL STEPS IN SOLVING MATHEMATICAL OR ENGINEERING PROBLEMS SOLVE THE EQUATIONS THAT RESULT FROM STEP #2. SOMETIMES THE METHOD WILL BE ALGEBRAIC, BUT FREQUENTLY MORE ADVANCED METHODS WILL BE NEEDED. THE RESULT OF THIS STEP IS A NUMERICAL ANSWER OR SET OF ANSWERS.

### GENERAL STEPS IN SOLVING MATHEMATICAL OR ENGINEERING PROBLEMS :

GENERAL STEPS IN SOLVING MATHEMATICAL OR ENGINEERING PROBLEMS INTERPRET THE NUMERICAL RESULT TO ARRIVE AT A DECISION. THIS WILL REQUIRE EXPERIENCE & AN UNDERSTANDING OF THE SITUATION IN WHICH THE PROBLEM IS EMBEDDED.

### THEORETICAL MATTERS :

THEORETICAL MATTERS ANY USER OF MATHEMATICAL PROCEDURES SHOULD ALWAYS BE CONCERNED WITH ITS THEORETICAL UNDERPINNINGS BECAUSE THESE EXPLAIN THE LIMITATIONS OF THE PROCEDURES TO PRODUCE THE RELIABLE RESULTS.

### WHEN THEORY IS TO BE DISCUSSED, TWO QUESTIONS ARISE: :

WHEN THEORY IS TO BE DISCUSSED, TWO QUESTIONS ARISE: WHERE DO WE START? WHAT BACKGROUND IS ASSUMED? DOES EVERY DEFINITION &/OR POSTULATE HAVE TO BE STATED OR CITED BEFORE PERTINENT THEOREMS ARE DEVELOPED?

### WHEN THEORY IS TO BE DISCUSSED, TWO QUESTIONS ARISE: :

WHEN THEORY IS TO BE DISCUSSED, TWO QUESTIONS ARISE: 2. HOW ARE THE THEOREMS PRESENTED? IS IT BETTER TO USE THE RATHER CONDENSED NOTATIONS OF MATHEMATICIANS & THEIR SPECIAL SYMBOLS, OR TO USE LANGUAGE & STYLE THAT IS MORE ACCESSIBLE TO THE AVERAGE PERSON?

### MOST OF THE METHODS OF NUMERICAL ANALYSIS ARE ITERATIVE; APPROXIMATE ANSWERS ARE OBTAINED THROUGH A SEQUENCE OF IMPROVED ESTIMATES. :

MOST OF THE METHODS OF NUMERICAL ANALYSIS ARE ITERATIVE; APPROXIMATE ANSWERS ARE OBTAINED THROUGH A SEQUENCE OF IMPROVED ESTIMATES. UNDER WHAT CONDITIONS DOES THE METHOD APPLY? FOR WHAT KINDS OF FUNCTIONS DO THE METHOD WORK AND HOW CAN WE KNOW THAT THE CONDITIONS ARE SATISFIED?

### MOST OF THE METHODS OF NUMERICAL ANALYSIS ARE ITERATIVE; APPROXIMATE ANSWERS ARE OBTAINED THROUGH A SEQUENCE OF IMPROVED ESTIMATES. :

MOST OF THE METHODS OF NUMERICAL ANALYSIS ARE ITERATIVE; APPROXIMATE ANSWERS ARE OBTAINED THROUGH A SEQUENCE OF IMPROVED ESTIMATES. 2. DOES THE METHOD CONVERGE? DO THE SUCCESSIVE APPROXIMATIONS REACH THE TRUE ANSWER TO A GIVEN ACCURACY?

### MOST OF THE METHODS OF NUMERICAL ANALYSIS ARE ITERATIVE; APPROXIMATE ANSWERS ARE OBTAINED THROUGH A SEQUENCE OF IMPROVED ESTIMATES. :

MOST OF THE METHODS OF NUMERICAL ANALYSIS ARE ITERATIVE; APPROXIMATE ANSWERS ARE OBTAINED THROUGH A SEQUENCE OF IMPROVED ESTIMATES. WHAT BOUNDS CAN BE PLACED ON THE ERROR OF EACH ESTIMATE? CAN WE KNOW IN ADVANCE THE MAXIMUM SIZE OF THE ERROR AFTER A CERTAIN NUMBER OF ITERATIONS?

### MOST OF THE METHODS OF NUMERICAL ANALYSIS ARE ITERATIVE; APPROXIMATE ANSWERS ARE OBTAINED THROUGH A SEQUENCE OF IMPROVED ESTIMATES. :

MOST OF THE METHODS OF NUMERICAL ANALYSIS ARE ITERATIVE; APPROXIMATE ANSWERS ARE OBTAINED THROUGH A SEQUENCE OF IMPROVED ESTIMATES. HOW RAPIDLY DO THE ERRORS OF THE SUCCESSIVE ESTIMATES DECREASE? DO ERRORS DECREASE PROPORTIONALLY TO THE NUMBER OF ITERATIONS?

### MOST OF THE METHODS OF NUMERICAL ANALYSIS ARE ITERATIVE; APPROXIMATE ANSWERS ARE OBTAINED THROUGH A SEQUENCE OF IMPROVED ESTIMATES. :

MOST OF THE METHODS OF NUMERICAL ANALYSIS ARE ITERATIVE; APPROXIMATE ANSWERS ARE OBTAINED THROUGH A SEQUENCE OF IMPROVED ESTIMATES. IS THE ACCURACY IMPROVED MORE RAPIDLY (EXPONENTIALLY) RATHER THAN LINEARLY? (MOST DESIRABLE SITUATION) HOW ACCURATELY DO WE KNOW THE ERROR?