CIS 541 – Numerical Methods

Mathematical Preliminaries

April 22, 2023 OSU/CIS 541 2

Derivatives

• Recall the limit definition of the first derivative.

0

( ) ( )'( ) limh

dy f x h f xf xdx h

April 22, 2023 OSU/CIS 541 3

Partial Derivatives

• Same as derivatives, keep each other dimension constant.

0

( , , ) ( , , )limxh

f f x h y z f x y zfx h

April 22, 2023 OSU/CIS 541 4

Tangents and Gradients

• Recall that the slope of a curve (defined as a 1D function) at any point x, is the first derivative of the function.

• That is, the linear approximation to the curve in the neighborhood of t is l(x) = b + f’(t)x

f(x)

t

(1,f’(t))

x

y

April 22, 2023 OSU/CIS 541 5

Tangents and Gradients

• Since we also want this linear approximation to intersect the curve at the point t.

l(t) = f(t) = b + f’(t)t• Or, b = f(t) - f’(t)t• We say that the line l(x) interpolates the

curve f(x) at the point t.

April 22, 2023 OSU/CIS 541 6

Functions as curves

• We can think of the curve shown in the previous slide as the set of all points (x,f(x)).

• Then, the tangent vector at any point along the curve is

, ( ) 1, '( )d x f x f xdx

April 22, 2023 OSU/CIS 541 7

Side note on Curves

• There are other ways to represent curves, rather than explicitly.

• Functions are a subset of curves (x,y(x)).• Parametric equations represent the curve by the

distance walked along the curve (x(t),y(t)).Circle: (cos, sin)

• Implicit representations define a contour or level-set of the function: f(x,y) = c.

April 22, 2023 OSU/CIS 541 8

Tangent Planes and Gradients

• In higher-dimensions, we have the same thing:

• A surface is a 2D function in 3D:Surface = (x, y, f(x,y) )

• A volume or hyper-surface is a 3D function in 4D:

Volume = (x, y, z, f(x,y,z) )

April 22, 2023 OSU/CIS 541 9

Tangent Planes and Gradients

• The linear approximation to the higher-dimensional function at a point (s,t), has the form: ax+by+cz+d=0, or z(x,y) = …

• What is this plane?

April 22, 2023 OSU/CIS 541 10

Construction of Tangent Planes

Images courtesy of TJ Murphy: http://www.math.ou.edu/~tjmurphy/Teaching/2443/TangentPlane/TangentPlane.html

April 22, 2023 OSU/CIS 541 11

Construction of Tangent Planes

April 22, 2023 OSU/CIS 541 12

Construction of Tangent Planes

April 22, 2023 OSU/CIS 541 13

Tangent Planes and Gradients

• The formula for the plane is rather simple:

• z(s,t) = f(s,t) - interpolates• z(s+dx,t) = f(s,t) + fx(s,t)dx = b + adx

– Linear in dx

• Of course, the plane does not stay close to the surface as you move away from the point (s,t).

( , ) ( , ) ( , )( ) ( , )( )x yz x y f s t f s t x s f s t y t

April 22, 2023 OSU/CIS 541 14

Tangent Planes and Gradients

• The normal to the plane is thus:

• The 2D vector:is called the gradient of the function.

• It represents the direction of maximal change.

( , ), ( , ),1x yN f s t f s t

( , ), ( , )T

x yf s t f s t

April 22, 2023 OSU/CIS 541 15

Gradients

• The gradient thus indicates the direction to walk to get down the hill the fastest.

• Also used in graphics to determine illumination.

( , ) ( , ), ( , )T

f ff x y x y x yx y

April 22, 2023 OSU/CIS 541 16

Review of Functions

• Extrema of a function occur where f’(x)=0.• The second derivative determines whether

the point is a minimum or maximum.• The second derivative also gives us an

indication of the curvature of the curve. That is, how fast it is oscillating or turning.

April 22, 2023 OSU/CIS 541 17

The Class of Polynomials

• Specific functions of the form:

2 3 4 50 1 2 3 4 5

0

0 1 2 3 4 5

( )

( ( ( ( ( ) )))))

ii

i

p x a a x a x a x a x a x

a x

a x a x a x a x a x a

April 22, 2023 OSU/CIS 541 18

The Class of Polynomials

• For many polynomials, the latter coefficients are zero. For example:

p(x) = 3+x2+5x3

April 22, 2023 OSU/CIS 541 19

Taylor’s Series

• For a function, f(x), about a point c.

• I.E. A polynomial

2 3

( )

0

( ) ( )( ) ( ) ( )( ) ( ) ( )2! 3!

( ) ( )!

kk

k

f c f cf x f c f c x c x c x c

f x x ck

April 22, 2023 OSU/CIS 541 20

Taylor’s Theorem

• Taylor’s Theorem allows us to truncate this infinite series:

( )

10

( 1)1

1

( )( ) ( )!

( ) ( )( 1)!

( ) ( , )

knk

nk

nn

n

f cf x x c Ek

fE x cn

where x c x

April 22, 2023 OSU/CIS 541 21

Taylor’s Theorem

• Some things to note:1. (x-c)(n+1) quickly approaches zero if |x-c|<<12. (x-c)(n+1) increases quickly if |x-c|>>13. Higher-order derivatives may get smaller (for

smooth functions).

April 22, 2023 OSU/CIS 541 22

Higher Derivatives

• What is the 100th derivative of sin(x)?

• What is the 100th derivative of sin(3x)?– Compare 3100 to 100!

• What is the 100th derivative of sin(1000x)?

April 22, 2023 OSU/CIS 541 23

Taylor’s Theorem

• Hence, for points near c we can just drop the error term and we have a good polynomial approximation to the function (again, for points near c).

• Consider the case where (x-c)=0.5

• For n=4, this leads to an error term around 2.6*10-4 f()• Do this for other values of n.• Do this for the case (x-c) = 0.1

( 1)

1 1

( ) 1( 1)! 2

n

n n

fEn

April 22, 2023 OSU/CIS 541 24

Some Common Derivatives

1( )

(sin ) cos

(cos ) sin

( )

1(ln )

n n

x x

d ax naxdxd x xdxd x xdxd e edxd xdx x

1( )

(sin ) cos

(cos ) sin

( )

1(ln )

n n

u u

d duau naudx dxd duu udx dxd duu u chain ruledx dxd due edx dxd duudx u dx

April 22, 2023 OSU/CIS 541 25

Some Resulting Series

• About c=02 3 4

1 2 2 3 3

1 2 2 3 3

2 3 4

12! 3! 4!

( 1) ( 1)( 2)( )2! 3!

1 2 31 1 1 1

1

x

n n n n n

n n n n

x x xe x

n n n n na x a na x a x a x

n n na a x a x a x

x x x x xx

April 22, 2023 OSU/CIS 541 26

Some Resulting Series

• About c=03 5 7

2 4 6

2 3 4 5

3 5 7

sin3! 5! 7!

cos 12! 4! 6!

ln(1 ) 1 12 3 4 5

1ln 2 1 11 3 5 7

x x xx x

x x xx

x x x xx x x

x x x xx xx

April 22, 2023 OSU/CIS 541 27

Book’s Introduction Example

• Eight terms for first series not even yielding a single significant digit.

• Only four for second serieswith foursignificantdigits.

3 5 7

ln 2 0.6931471801 1 1 1 1 1 1ln(1 1) 12 3 4 5 6 7 8

0.63452

1 1 111 1 3 3 33ln 21 3 3 5 713

0.69313

April 22, 2023 OSU/CIS 541 28

Mean-Value Theorem

• Special case of Taylor’s Theorem, where n=0, x=b.

• Assumes f(x) is continuous and its first derivative exists everywhere within (a,b).

1( ) ( )( ) ( ) ( ) ( , )

f b f a Ef a b a f a b

April 22, 2023 OSU/CIS 541 29

Mean-Value Theorem

• So what!?!?! What does this mean?• Function can not jump away from current value

faster than the derivative will allow.

f(x)

a b secant

April 22, 2023 OSU/CIS 541 30

Rolles Theorem

• If a and b are roots (f(a)=f(b)=0) of a continuous function f(x), which is not everywhere equal to zero, then f’(t)=0 for some point t in (a,b).

• I.e., What goes up, must come down.

f(x)

a b

f’(t)=0

t

April 22, 2023 OSU/CIS 541 31

Caveat

• For Taylor’s Series and Taylor’s Theorem to hold, the function and its derivatives must exist within the range you are trying to use it.

• That is, the function does not go to infinity, or have a discontinuity (implies f’(x) does not exist), …

April 22, 2023 OSU/CIS 541 32

Implementing a Fast sin()

const int Max_Iters = 100,000,000;float x = -0.1;float delta = 0.2 / Max_Iters;float Reimann_sum = 0.0;for (int i=0; i<Max_Iters; i++){

Reimann_Sum += sinf(x);x+=delta;

}Printf(“Integral of sin(x) from –0.1->0.1 equals:

%f\n”, Reimann_Sum*delta );

April 22, 2023 OSU/CIS 541 33

Implementing a Fast sin()

const int Max_Iters = 100,000,000;float x = -0.1;float delta = 0.2 / Max_Iters;float Reimann_sum = 0.0;for (int i=0; i<Max_Iters; i++){

Reimann_Sum += my_sin(x); //my own sine funcx+=delta;

}Printf(“Integral of sin(x) from –0.1->0.1 equals:

%f\n”, Reimann_Sum*delta );

April 22, 2023 OSU/CIS 541 34

Version 1.0

my_sin( const float x ){

float x2 = x*x;float x3 = x*x2;

return (x – x3/6.0 + x2*x3/120.0 );}

April 22, 2023 OSU/CIS 541 35

Version 2.0 – Horner’s Rule

Static const float fac3inv = 1.0 / 6.0f;Static const float fac5inv = 1.0 / 120.0f;

my_sin( const float x ){

float x2 = x*x;

return x*(1.0 – x2*(fac3inv - x2*fac5inv));}

April 22, 2023 OSU/CIS 541 36

Version 3.0 – Inline code

const int Max_Iters = 100,000,000;float x = -0.1;float delta = 0.2 / Max_Iters;float Reimann_sum = 0.0;for (i=0; i<Max_Iters; i++){

x2 = x*x;Reimann_Sum += x*(1.0–x2*(fac3inv-x2*fac5inv);x+=delta;

}Printf(“Integral of sin(x) from –0.1->0.1 equals:

%f\n”, Reimann_Sum*delta );

April 22, 2023 OSU/CIS 541 37

Timings

• Pentium III, 600MHz machineTime in seconds Result Max(

|sin(x)-my_sin(x)| )

Using sinf 27 -0.0041943

Version 1.0 20 -0.0041943 1.93765*10-11

Version 2.0 13 -0.0041943 8.0495*10-12

Version 3.0 2 -0.0041943 8.0495*10-12

April 22, 2023 OSU/CIS 541 38

Observations

• Is the result correct?• Why did we gain some accuracy with

version 2.0?• Is (–0.1,0.1) a fair range to consider?• Is the original sinf() function optimized?• How did we achieve our speed-ups?• We will re-examine this after Lab1.

April 22, 2023 OSU/CIS 541 39

Homework

• Read Chapters 1 and 2 for next class.• Start working on Lab 1 and Homework 1.