© 2005 Microchip Technology Incorporated. All Rights Reserved. Slide 1 NUM 1019 Numerical Methods...

© 2005 Microchip Technology Incorporated. All Rights Reserved. Slide 1

NUM 1019

Numerical Methods on the PIC Micro

© 2006 Microchip Technology Incorporated. All Rights Reserved. Class Slide 2

Class Objective

When you finish this class you will: Know the basics of numerical methods

as applied to embedded systems Know what the PIC micro can really do See a complex mechatronics demo Know where to find additional design

resources


Agenda

What is numerical methods? The basics Trigonometry Finding the square root Dynamic systems

Introduction Introduction The BasicsThe BasicsTrigonometryTrigonometryFinding the square rootFinding the square rootDynamic systemsDynamic systems


What is Numerical Methods?


Introduction

Numerical methods use numbers to simulate mathematical processes (The world around us)



Introduction

“The Purpose of Computing Is Insight, Not Numbers” –R.W. Hamming

Computers are for Computing



The Basics


Numbers

The three number systems Counting numbers

0,1,2,…,32,767

Fixed point 3.1415, 0.01234, -123.4

Floating point -.12345678 x 104



Fixed Point Fixed point numbers represent whole

numbers and fractions

10.4 can also be represented as 104 when scaled by 10 (decimal shifted once right)

Computations can be done on whole numbers as long as we remember where to put the decimal in the end (called the radix point)


10

4104.10


Fixed Point

The same applies to binary numbers often referred to as the Qx.x format or Qi.j

i and j are the number of bits to the left and right of the decimal point respectively.

Ex) 11011010.1101 Q8.4

1

0

1

0

2*)(22*)(n

k

n

k

krrk kakaA



Fixed Point

Signed-magnitude format for negative numbers First bit is used as a sign bit ex) 10010011 = -19

2’s complement is most often used for fixed point negative numbers Complement +1 ex) 9 = 00001001 11110111 = -9



Floating Point Notation: Floating Point Notation: Sign * (Mantissa) * 2Sign * (Mantissa) * 2exponentexponent = =

Fractional component stored in signed-magnitude format and is in binary form

Floating PointIntroduction Introduction The BasicsThe BasicsTrigonometryTrigonometryFinding the square rootFinding the square rootDynamic systemsDynamic systems

eMS 2

1

0

2*n

k

kkmMbitsignS


Floating Point

Exponent stored in biased format

This makes number comparisons easier Floating point number stored as:


Fxeeb 70

eb M2 M1 M0

IEEE 32bit S e7 … e2 e1 e0 m22 … m17 m16 m15 m14 … m9 m8 m7 m6 … m1 m0

Mchp 32bit e7 e6 … e1 e0 S m22 … m17 m16 m15 m14 … m9 m8 m7 m6 … m1 m0

Mchp 24bit e7 e6 … e1 e0 S m14 … m9 m8 m7 m6 … m1 m0


Good for large and small numbers (1038,10-38)

Number system is not linear, distance between numbers jumps

Using Floating point numbers for counting and/or logical decision making is generally a bad idea

Floating PointIntroduction Introduction The BasicsThe BasicsTrigonometryTrigonometryFinding the square rootFinding the square rootDynamic systemsDynamic systems

4444333 104103102101109108107


Fixed Float

Example:

1) Find exponent:

2) Find fractional component:


3)int(6780719.2)2ln(

)15625.0ln(15625.02

zez

z

)1bealwayswillx(25.12

15625.03

x


Fixed Float

3) Find binary number:


3

212

221

0

225.115625.02)1(Float)bitsign(0

00000000000000000000010.125.1

0;1yes?,225.025.0;0no?,225.0

25.0125.1;1yes?,225.1

eS

g

MS

bindecM

xmxm

xm


Fixed Float

4) Convert to Microchip float format eb= Biased Exponent

Microchip Float Format:


CxebFxebFxeeb

7070370

000020715625.0012

CMMMExp


Bases

We typically do calculations in decimal (base 10)

130ten = (1 * 102) + (3 * 101) + (0 * 100)

Machines only understand 1’s and 0’s so they use binary (base 2)

130ten = (1*27)+(0*26)+(0*25)+(0*24)+(0*23)+(0*22)+(1*21)+(0*20)

= 1000 0010two

Hexadecimal (base 16) serves as a shorthand for binary numbers

1000 0010two = (8*161) + (2 * 160) = 82sixteen (0x82)



Bases

Dec Bin


0110000122

102

012

122

152

0112

222

4

3

2

1

0

a

a

a

a

a

11010000.8125.000.1

2

500.0

2

250.1

2

625.1

2

8125.

Fraction Bin


Bases

twoten 00110.1.0

21

2

61

2

80

2

40

repeating

2

20

2

1.0

Watch out for non-terminating numbers

Never enough precision

Try it:


11.010


Bases

Making the Dec Bin Dec conversion doesn’t always return the same result

Example:

numbersdistinct64

.1001

.1001

.1001

000000.1001

99.9

98.9

01.9

00.9

numbers100

xxxxxx

xxxxxx

xxxxxx



Bases

Do not confuse the kind of number system used (Counting, Fixed Point, Float) with the form of the number representation (Dec, Bin, Hex)

(they are the same number)

twoten 113 81000000021100.4

2

0

x



BCD

Binary Coded Decimal (BCD) Counting to ten in binary

Solves 0.1 approximation problem

BCD arithmetic is sometimes required by law for financial applications


1300

001100011101

xxD

0001.00001.0


Four Functions

Addition:


ten

ten

ten

380010011012000011002610101000

11

ten

ten

ten

200001010017000100013701010100

20

Subtraction:


Four FunctionsIntroduction Introduction The BasicsThe BasicsTrigonometryTrigonometryFinding the square rootFinding the square rootDynamic systemsDynamic systems

Division:

1011

10111

10111

1001111101111

1111101010111

101

Multiplication:


Multiplication A trick that can speed up multiplication in software

Round by inspection to allow shift and add instead of multiply

Use two or three shift and add routines instead of a multiply routine Need to multiply x(n) by 10010101 (dec 149) 10010000 (dec 144) => error = -5 10100000 (dec 160) => error = +11 x x(n)*10010101 x(n)<<7 + x(n)<<4 = x(n)*27+x(n)*24

To multiply y(n) by 00001011(dec 11) 00001100 (dec 12) => error = +1 00001010 (dec 10) => error = -1 In general round down for y(n)



Evaluating

Horner’s method for a polynomial

Evaluated:

Used to convert thermocouple voltage to temperature (oC) over a wide range of temperatures.

0123455

012

21

1

))))(((()(

5

)(

axaxaxaxaxaxP

n

axaxaxaxaxP nn

nn


end

kbckakb

nkfor

nanb

);1(*)()(

0:1:1

);()(


Evaluating Using unsigned coefficients is sometimes beneficial Subtract values with negative coefficients in algorithm For scaled coefficients between -128 and 127 convert

these to 0 to 255 IIR digital filter example:

Start with difference equation:

)2()1()2()1()()( 21210 nyanyanxbnxbnxbny

)2()128()1()128(

)2()128()1()128()()128()(

21

2101

nyanya

nxbnxbnxbny

128)2(128)1(128)()( nxnxnxnX

)()()( 1 nXnyny



Error analysis

Things to watch out for: Loss of significance when subtracting two

nearly equal numbers Example with algebraically equivalent

expressions:

1748.111

1500.11)1(500

xx

x

xxxx



Error analysis

Two kinds of error, Round-Off and Truncation

Round-Off is a result of a fixed precision (machine dependant)


999.0333.333.333.

000.131

31

31

46.223.123.14645.223.12345.1


Error analysis

Round-off example:


Decimal Equivalents for a set of binary numbers

-3 -2 -1 0 1 2 3 40.0625000 0.125000 0.25000 0.5000 1.000 2.00 4.0 8.00.0703125 0.140625 0.28125 0.5625 1.125 2.25 4.5 9.00.0781250 0.156250 0.31250 0.6250 1.250 2.50 5.0 10.00.0859375 0.171875 0.34375 0.6875 1.375 2.75 5.5 11.00.0937500 0.187500 0.37500 0.7500 1.500 3.00 6.0 12.00.1015625 0.203125 0.40625 0.8125 1.625 3.25 6.5 13.00.1093750 0.218750 0.43750 0.8750 1.750 3.50 7.0 14.00.1171875 0.234375 0.46875 0.9375 1.875 3.75 7.5 15.0

4-bit mantissa Exponent e0.xxxx1000

110111101111

1001101010111100


Error analysis

Round-off example: 61

51

101 )(

2103

2251

23101

20011.1

21101.021101.0

201101.021101.0

two

twotwo

twotwo

01157

1261

11103

210000.0211111.0

201011.021011.0

21010.021010.0

twotwo

twotwo

twotwo

%)14.7(033333.010000.0157 twooferror



Error analysis

What is the difference between Chop-off and Round-off?

Chop-off will introduce unnecessary bias errors

Round-off minimizes bias


457.1456.1456852.1RoundorChop


Error analysis

Truncation error is the result of implementing a finite number of steps of an infinite step sequence (algorithm dependant)

For example, using a fixed number of terms in a Taylor expansion See hand out for further information on

Taylor approximations


10

)1(

00

0)(

)()!1(

)()(

)(!

)()()(

NN

N

N

k

kk

N

xxN

cfxE

xxk

xfxPxf


Trigonometry


Trigonometry

Taylor approximation for sin(x) of degree 9, P9(x) about x0 = 0


!9!7!5!3!

)0()(

97539

0

)(

9

xxxxxx

k

fxP

k

kk

109 !10

)sin()( x

cxE

-1.5

-1

-0.5

0

0.5

1

1.5

0.E+00

2.E+06

4.E+06

6.E+06

8.E+06

1.E+07

1.E+07

P9(X)

SIN(X)

Error


CORDIC Transform

Coordinate Rotation Digital Computer An iterative algorithm used for calculating

trig functions The CORDIC Transform only works

for angles between ±90o


cossin

sincos'' yxyx

])}[tan(*][][{*])[cos(]1[])}[tan(*][][{*])[cos(]1[iiXiYiiYiiYiXiiX


Trigonometry

How does it work? Rotates the phase of a complex number,

by multiplying it by a succession of constant values

The multiplier used is in 2, so in binary arithmetic we will use the shift command

See handout for more detail on the derivation



Trigonometry

Range finders have been around for a long time

They used right triangles to find distances


distance)tan( L


Trigonometry

Lets make a laser range finder!

Our setup:

Use law of Sines to find L:


Line up the Line up the laser pointers!laser pointers!

AA BB

cc

LLbb aa

)sin(*)sin(

)sin(*1800 AbL

C

BcbBAC

CC


Trigonometry

What is the error using CORDIC SIN(X)?

What is the error using Taylor approximation to SIN(X)?



Finding the square root


Square Root

Fixed point Newton-Raphson method (AN526)

2)(5.0 0

1

01

Nx

x

xxx

xN

nnn

n



Square Root

MINIMAX floating point approximation

1) Limit your input range



Square Root

MINIMAX floating point approximation

2) Find the best linear solution


Sqr(2)Sqr(2)

11 22

Same maximum error at mid-point and both end-pointsSame maximum error at mid-point and both end-points


Square Root

Fast integer square root (TB040)



Dynamic Systems


Differentiation

Backward difference “three-point formula”

M = maximum value for the 4th derivative Find a step size h to minimize error E3

24

)(

2

43)(

)4(4210

0

cfh

h

fffxf

242

43)( 4

4210

3

Mh

h

eeexE



Integration

Simpson’s 3/8th rule

)(80

3)33(

8

3)( )4(

5

3210

3

0

cfh

ffffh

dxxfx

x

)(xfy



Dynamic simulation

World of differential equations



Summary

What is numerical methods? The basics Trigonometry Finding the square root Dynamic systems


References

AN670 “Floating Point to ASCII conversion”

TB040 “Fast Integer Square root” AN526 “PIC16 Math Utility Routines” AN544 “Math Utility Routines” AN617 “Fixed Point Routines” AN660 “Floating Point Math

Functions”


References “German Optical Rangefinders” by Peter

Lienau “Fixed-Point Trigonometry With CORDIC

Iterations” By Ken Turkowski “A survey of CORDIC algorithms for

FPGA based computers” By Ray Andraka “Math Toolkit for Real-Time

programming” by Jack Crenshaw “Numerical Methods Using Matlab” by

Kurtis D. Fink “Numerical Methods for Scientists and

Engineers” by Richard Hamming

© 2005 Microchip Technology Incorporated. All Rights Reserved. Slide 1 NUM 1019 Numerical Methods...

Documents

Transcript of © 2005 Microchip Technology Incorporated. All Rights Reserved. Slide 1 NUM 1019 Numerical Methods...