Introductory to Numerical Analysis 01417343

Introductory to Numerical Analysis01417343by

Suriya Na nhongkaiType of Errors (Inherent error) (Round-off error) (Truncation error) (continuous system) (discrete system)Definition of Error

Error: Example

Accuracy Identification

General Rounding off n

General Rounding off 5000... n 5000... 5000... n General Rounding off: Example

Propagated Error

9Propagated Error: Addition and Subtraction

Propagated Error Addition and Subtraction: Example11Propagated Error: Multiplication

Propagated ErrorMultiplication: Example 1

Propagated ErrorMultiplication: Example 2Propagated Error: Division

Propagated ErrorDivision: Example

Fundamental Theorem in Calculus

Fundamental Theorem in Calculus

Fundamental Theorem in Calculus

Fundamental Theorem in Calculus

Fundamental Theorem in Calculus

Fundamental Theorem in Calculus

Fundamental Theorem in Calculus

Fundamental Theorem in Calculus

Fundamental Theorem in Calculus

Fundamental Theorem in Calculus

Fundamental Theorem in Calculus

Fundamental Theorem in Calculus

Fundamental Theorem in Calculus

Fundamental Theorem in Calculus

Fundamental Theorem in Calculus: Example

Fundamental Theorem in Calculus: Taylors Theorem

Fundamental Theorem in Calculus: Taylors TheoremFundamental Theorem in Calculus Taylors Theorem: Example

Fundamental Theorem in Calculus Taylors Theorem: Example

Fundamental Theorem in Calculus Taylors Theorem: Example

Fundamental Theorem in Calculus Taylors Theorem: Example

Fundamental Theorem in Calculus Taylors Theorem: Example

Fundamental Theorem in Calculus Taylors Theorem: Example

Fundamental Theorem in Calculus Taylors Theorem: Example

Fundamental Theorem in Calculus Taylors Theorem: Example

Fundamental Theorem in Calculus Taylors Theorem: Example

Fundamental Theorem in Calculus Taylors Theorem: Example

Fundamental Theorem in Calculus Taylors Theorem: Example

Rounding off and Computer Arithmetic

Rounding off and Computer Arithmetic

Rounding off and Computer Arithmetic: Example 32 IBM 3000 IBM 4300 1 7 ( 16) 24

7 0 127 64 -64 63Rounding off and Computer Arithmetic: Example

Sign bit 0

Rounding off and Computer Arithmetic: Example

179.015625

Rounding off and Computer Arithmetic: IEEE-754 Single Precision

Rounding off and Computer Arithmetic: IEEE-754 Double PrecisionRounding off in Calculator

Chopping and Rounding

Rounding off in Calculator: ExampleRounding off in Calculator: Example

Rounding off in Calculator: Example

Rounding off in Calculator: Example

Rounding off in Calculator: Example

1

(Absolute error)

= 0.5225 =0.5237

= 0.5225-0.5237 = -0.0012

= = 0.0023 (Decimal Place, D.P.) x = 3.14725 3 D.P. x = 3.147 (Significant Digit, S.D.)

0.012041 0.31470 5 4 (Error Bound)

0.0012 123.456789012345623456

n 123.456789012345623000 5.4565725 2 D.P. 5.46 3 D.P. 5.457 6 D.P. 5.456572 5 S.D. 5.4566 n D.P. 3 S.D.

3 D.P. 8 D.P.

3 D.P. 2 D.P.

1 D.P. 2 D.P. 3 D.P.

0 D.P. -151

()

(2 D.P.) (3 D.P.) 1 D.P.

1. 2.

(Rolls Theorem)

(Mean Value Theorem)

Slope slope (Extreme Value Theory) , (Riemann Integral)

,

(Weighted Mean Value Theorem for Integral) ,

(Intermediate Value Theorem) . (Taylors Theorem)

(Taylor Polynomial) (Remainder Term or Truncation Error) () (Maclaurin Series) (1) (2)

0

8 D.P.

9 D.P. 0.99995000042 9(2)

0.099833417 6 D.P.

( )

,

0.04

3 (6 D.P.)

5 D.P.

5 D.P.

2

3

3

(Significant Number) (Mantissa) (Characteristic) () 123.4567 0.00021378

179.0156097412109375 179.0156402587890625Single Precision 32 1 () 8 () 23 ()

Double Precision 64-bit 1 () 11 () 52 ()

()

(Chopping) (Rounding)Chopping Rounding 1 33 Single Precision IEEE-754 33 24 Normalize1. 1 ( 1 )2. 0 ()

1 1

Sign (s)Characteristic (c)Mantissa (f)

+128

Single Precision IEEE-754 0.1

Normalize

Rounding

4 Sign (s)Characteristic (c)Mantissa (f)

+123