Numerical Methods Formula Sheet

FORMULA SHEET FOR NUMERICAL METHODS FOR ENGINEERS COEB223 / MATB324

Part 1: Modeling, Computers and Error Analysis Error Definitions True error: True percent relative error:

Approximate percent relative error:

Stopping criterion: Terminate computation when εa < εs where εs is the desired percent relative error

Taylor Series Taylor series expansion (

)

( )

( )

( )

( )(

( )

)

where remainder (

)(

(

)

)

(

or

)

Error propagation For n independent variables x1, x2, …, xn having errors ̃ ,

̃ , …,

function f can be estimated via |

| ̃

|

| ̃

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|

| ̃

̃ , the error in the

COEB223 / MATB324 Formula Sheet

Part 2: Roots of Equations Method

Formulation

Bisection

( )( ( )

False Position

(

) )

If ( )

( )

set

If ( )

( )

set

If ( )

( )

set

If ( )

( )

set

( ) ( )

Newton Raphson

Secant

( )( ( )

) ( )

Part 3: Linear Algebraic Equations Gauss Elimination

[

]

[

]

LU decomposition Back Substitution

Decomposition

[

]

[

]{

}

{ }

[

]{ }

Forward Substitution

Page 2 of 8

{

}

{ }

COEB223 / MATB324 Formula Sheet

Gauss-Seidel method

|

|

} With relaxation, (

)

Part 4: Curve Fitting Method Linear Regression

Formulation

Errors √

where )

∑( ∑ ∑

∑ ∑ (∑ )

where: ∑(

Polynomial Regression

̅)

For a 2nd order polynomial fit, √

(

)

where )

∑(

by differentiating Sr with respect to each coefficients and setting the partial derivatives equal to zero, we have: ∑

∑

∑

∑

∑

[∑

∑

∑

∑ { ]

}

∑ {∑

Page 3 of 8

}

where: ∑(

̅)

COEB223 / MATB324 Formula Sheet

Multiple Linear Regression

√

(

)

For a two-variable linear fit,

where where:

)

∑(

∑( by differentiating Sr with respect to each coefficients and setting the partial derivatives equal to zero, we have: ∑ ∑ [∑

Newton’s divided difference interpolating polynomial

∑

∑

∑

∑

∑

{

∑

}

{∑

]

}

For third order: ( )

(

) (

( )(

)( )(

where ( ) [

]

[ [ Lagrange interpolating polynomial

∑

( )

∑

] ]

( ) ( )

where, ( )

∏

Page 4 of 8

) )

̅)

COEB223 / MATB324 Formula Sheet

Part 6: Numerical Differentiation and Integration A. Numerical Differentiation Method Forward finitedivided difference

Formulation First Derivative: (

( )

( )

( ) )

(

)

( ) (

)

First Derivative: ( )

( )

(

( )

( )

Centred finitedivided difference

) (

( )

Backward finitedivided difference

Errors

( )

) (

)

(

) (

)

(

)

(

)

First Derivative (

( )

)

(

(

( )

)

) (

)

(

)

(

B. Numerical Integration Method

Formulation

Trapezoidal rule

(

)

Multiple-application trapezoidal rule

(

)

( )

( )

( ( )

Page 5 of 8

∑ ( )

(

))

)

COEB223 / MATB324 Formula Sheet

Simpson’s 1/3 rule

Multiple-application Simpson’s 1/3 rule

Simpson’s 3/8 rule

Gauss Quadrature

(

)

(

)

(

)

( )

( )

( )

( ( )

( )

( )

( )

∑

( )

( )

( )

(

∑

( )

)

Gauss-Legendre For two-point Gauss-Legendre:

For three-point Gauss-Legendre:

Change of variables: (

Page 6 of 8

)

(

)

( )

(

))

COEB223 / MATB324 Formula Sheet

Part 7: Ordinary Differential Equations Method

Formulation

Euler’s First-Order RK (

Heun’s Second Order RK

)

(

)

(

)

(

)

Midpoint Second Order RK (

)

(

Ralston’s Second Order RK

)

(

)

(

)

(

Classical Fourth Order RK

)

( (

) )

(

)

(

)

(

Page 7 of 8

)

COEB223 / MATB324 Formula Sheet

Part 8: Partial Differential Equations Method

Formulation

Elliptic PDEs Liebmann’s Method

Parabolic PDEs (one dimensional)

(

)

Explicit Method (

)

(

Simple Implicit Method (

)

)

(

Crank-Nicolson Method

) (

(

)

Page 8 of 8

)