Squeezing the most out of casio Fx 570es calculator for matrix comptation in numerical methods

SQUEEZING THE MOST OUT OF CASIO

FX-570ES CALCULATOR FOR MATRIX

COMPTATION IN NUMERICAL METHODS

TAY KIM GAIK

ROSMILA ABDUL KAHAR

3RD INTERNATIONALCONFERENCE ON

SCIENCE AND MATHEMATICS EDUCATION

SQUEEZING THE MOST OUT OF CASIO FX-570ES CALCULATOR FOR

MATRIX COMPUTATION IN NUMERICAL METHODS

T a y K i m G a i k R o s m i l a A b d u l - K a h a r C e n t r e for S c i e n c e S t u d i e s , C e n t r e for S c i e n c e S t u d i e s , U n i v e r s i t i T u n H u s s e i n O n n M a l a y s i a U n i v e r s i t i T u n H u s s e i n O n n M a l a y s i a

< t a y @ u t h m . e d u . m y > < r o s m i l a @ u t h m . e d u . m y >

Abstract

In most examinations, scientific calculator is used to do calculation. As compared to Casio 570MS, Casio fx-570ES calculator is more users friendly. The display of the n x m matrix is more natural which is exactly in their mathematical form. This natural textbook display will avoid confusion among students as Casio fx-570MS only shows one line expression. In this paper, a collection of keystroke sequences is illustrated to highlight its features of matrix computation. Once the students can utilize their calculator throughout matrix computation in numerical methods, they can expand their ability to any other problems involving matrix in a similar pattern for any subjects in undergraduate level.

I n t r o d u c t i o n

W h e n u s i n g c a l c u l a t o r in a c l a s s r o o m , a l m o s t i m m e d i a t e l y in a p i c t u r e o f a m i n d is a g r a p h i n g calculator w i t h all t h e a c c e s s o r i e s a t t a c h e d t o t h e b i g screen d i s p l a y . H o w e v e r , it is n o t practical if student c a n n o t b r i n g a n d u s e t h e g r a p h i n g c a l c u l a t o r a n y w h e r e b e c a u s e it is c o s t l y , b u l k y a n d h e a v y . In an e x a m i n a t i o n hall w i t h a l a r g e n u m b e r o f s t u d e n t s t a k i n g n u m e r i c a l m e t h o d , o n l y a h a n d h e l d c a l c u l a t o r is a l l o w e d . A s f x - 5 7 0 M S c a l c u l a t o r ( G u e r r e r o - G a r c i a & S a n t o s - P a l o m o , 2 0 0 8 ; T a y , 2 0 0 6 ) p r o m o t e s o n l y o n e line display a n d s o o n b e c o m e o u t d a t e d , t h e C a s i o f x - 5 7 0 E S is a n e v e r y d a y c a l c u l a t o r w h i c h s t u d e n t c a n afford t o b u y . S t u d e n t s n e e d t o g e t t h e m a x i m u m benefit out o f this c a l c u l a t o r . N o t t o m a k e t h e full u s e of this c a l c u l a t o r is s u c h a w a s t e . It c a n b e u s e d in a l m o s t all s u b j e c t s in a u n i v e r s i t y w h i c h n e e d calculator. W i t h t h e h e l p o f C a s i o f x - 5 7 0 E S e m u l a t o r , lecturers c a n easily d i s p l a y their k e y s t r o k e s e q u e n c e s from their laptop in t h e c l a s s r o o m .

In n u m e r i c a l m e t h o d , m a t r i x is u s e d in t o p i c s such as s y s t e m of linear e q u a t i o n s , e i g e n v a l u e -e i g -e n v -e c t o r , b o u n d a r y v a l u -e p r o b l -e m ( B V P ) in o r d i n a r y diff-er-ential -e q u a t i o n ( O D E ) , partial diff-er-ential e q u a t i o n ( P D E ) a n d finite e l e m e n t m e t h o d ( F E M ) . N o t t o forget t h e m a n u a l c a l c u l a t i o n o f m a t r i x w h i c h w a s i n t r o d u c e d in l i n e a r a l g e b r a , s t u d e n t w i l l still n e e d a step further u s i n g t h e i r c a l c u l a t o r in s o l v i n g matrix. W i t h C a s i o f x - 5 7 0 E S c a l c u l a t o r display o f t h e n x m m a t r i x e x a c t l y in t h e i r m a t h e m a t i c a l form, students w i l l r e d u c e t h e a m o u n t of t i m e s p e n d i n g in s o l v i n g r e l a t e d p r o b l e m s .

M a t r i x

K e y s t r o k e s e q u e n c e s o f c a s i o f x - 5 7 0 E S for t h e b a s i c o p e r a t i o n of t w o m a t r i c e s , A a n d B .

Find A + B, A - B, AxB, A 1 , AT and BT. "3 2 0" "3 2 l"

Let A = 2 3 1 and B = 4 1 2

0 1 3 2 3 1

Activity K e y s t r o k e C a l c u l a t o r Display Input 3x3 matrix A

raisfTjLiJllJlsltsllEllsltllli)

W(D m[=)(3r

=](Ti(=)[3](=)

'[ 1

Press |AC] button to clear screen

Input 3x3 matrix B » l l [ 4 J ( T J ( T ) ( T J [ 3 J ( ^ ( j j @ ( T j @

iju [=) [ I ] ( = ] [ I ] ( D [ S( = ) L I ] (D

B

r 3 e 1 " 1 a 1 3 M l

1 Press (ACI button to clear screen

Adding matrix A and B

e ®

& m m

B

I 2 U II

Press (ACI button to clear screen Subtracting matrix A

a n d B

ISHIFTl 14J13J [-1 |SHIFT|[4] | 4 | MatA-MatH

0

H

- 2 2 -1

I - 2 - 2 2

1

Press (ACI button to clear screen Multiplying matrix A

a n d B

|SHIFT| Lfl 13J IXJ lSH|FT| [4J {_4J HatfixMatB

0

s

en 10 a

I 10 10 s

1 " Press [AC] button to clear screen

Checking if A"1 exist,

determinant is nonzero.

b b e

m

scaLH[=]

12 A"1 ISHIFTl 14 1 1 3 1 | g 1 1=1 ""SiBH^r - 0 . 5 0. IBCG

- D . 5 0. 1 9 - 0. 2 5 O.IEEE - 0 . 3 5 D.WEE

?.r The transpose of matrix

A,

A

1m f t| | 4| | 8 | [ s h i r| [ 4 ] 1 3 1 1 ) 1 Trn(MatA)

0

H

2 3 1

I 0 1 3

B

AnS _{II 2} 2 1 3

L 1 2 1

System of linear equations

Gauss Elimination Method

Suppose the task is to solve the linear system below by Gauss Elimination method. The keystrokes sequences in Casio fx-570ES is the same as Casio fx-570MS (Tay, 2006), therefore, the keystroke will not be discussed:

2*; + 3J C2 + 1 0* 3= 9

5Xj -

x2

+

4x, + 7x, +

x,

( 0.660 ")

- 0 . 2 1 0

v 0.831

j

Student can use direct solution of 3x3 matrix using Casio fx-570ES for comparing their answer in solving system of linear equations in numerical methods.

Activity Keystroke Calculator Display

Entering the linear system

IM0DE1 L5J 12 :: a n X + b n Y = C n a n X + b n Y + C n Z = d n

3 :a x z+ b x+ c= a

ru

0

First row

_rj]

(3

rjj on

, b C *

1 3 ID

mmm

2 D 0 0

3l D D 0

Second row ( s j E i s e c i J i E i r a i s i r a i s ] If 2 3. \ b c 10

2 5 " I 3

j l m M r o 0 a

Third row

_{rs m m in ru i=j}

i

21

O d ) x=

242

EJ

77 "367

( 3

305

*r>7

T h e a n s w e r is

V c 2 4 2 ^

367 ' 0.660 " ' 0.660 "

=

=D-367 = - 0 . 2 1 0 w h i c h is t h e s a m e as x = - 0 . 2 1 0 305

V 367

J

Cholesky Method

T o s o l v e t h e s y s t e m o f l i n e a r e q u a t i o n s b e l o w b y C h o l e s k y m e t h o d , t h e m a t r i x s h o u l d be s h o w n first as p o s i t i v e definite.

3 1 0

1 3 - 1

0 - 1 3 _v4y

In s o l v i n g t h e a b o v e e q u a t i o n s u s i n g C h o l e s k y M e t h o d , m a t r i x

A-r3 1 0 ^

1 3 - 1

0 - 1 3

m u s t b e s y m m e t r i c

positive-definite. T h e o r e m : A s y m m e t r i c m a t r i x A is p o s i t i v e definite if a n d o n l y if e a c h of its l e a d i n g p r i n c i p a l s u b m a t r i c e s h a s a p o s i t i v e d e t e r m i n a n t .

Activity K e y s t r o k e C a l c u l a t o r Display

det (3)=3>0

d e t

f

det

f3 1 0

1 3 - 1

- 1 3 =8>0

=21>0

*m e ] @] ru (d

It is s h o w n t h a t t h e 3 x 3 m a t r i x a b o v e is s y m m e t r i c p o s i t i v e - d e f i n i t e .

F r o m A = LLT, s o l v e for L a n d LT.

' 3 1 0 ^ f a 0 oN 'a 6 o 6 ad

1 3 - 1 = c 0 0 c e = / 32+ c2 bd + ce

y0 - 1 3 , e /,

,°

Activity K e y s t r o k e C a l c u l a t o r Display

Compare element by

element a2 = 3,a = ?

[MODE) LU |SH|FT| ^tODE) ( j j (ALPHA) [_)J 1*3 (ALPHA) [CALCJ

_x

_=a

a2 = 3,a = X 1AIPHA1 ICALCl Solve for X

0 0

X= 1.732850808 L-R= R

ab = \,b = ? IALPHAI1) ] |ALPHA| 1(-)| |ALPHA| ICALCl [ 1 I XA=1I

a = X,b = A (ALPHA] [CALCJ ft?

ad = 0,d = 0 Manually e

0

b2+c2=3,c = ? c = B

[ALPHAJ 1^11™) [CALCJ [ViJ |_3J | - | |AIPHA| | ( - ) | |CALC| [=J _LW3-A2

3

bd + ce = -\,d = 0, e = - 1

e = D

I ALPHA) [sinl |ALPHA| ICALCI | ( - ) | 11 I | - r | IALPHAJ M 1=1 D=-l-B

4

d2+e2+ f2 = 3 , / = ?

f = M

g A | [MjJ [ALPHAJ [CALCJ [Vi) |_3J | - | [ALPHA] Isinl lac'l ICALCI _M=J3-D2

4

F r o m LY = b, s o l v e Y b y f o r w a r d s u b s t i t u t i o n :

U s i n g direct s o l u t i o n f r o m C a s i o f x - 5 7 0 E S ,

0

3 0

=

f 0 . 5 7 7A

1.021

2.855

F r o m £rx = Y, s o l v e x b y b a c k w a r d substitution:

'1.732 0.577 0 ^ '0.577

0 2V6 _{3 4} x2

=

v 0 0

•1*2

4 / \X-I) ^2.855

U s i n g direct s o l u t i o n f r o m C a s i o f x - 5 7 0 E S ,

^ - 0 . 0 9 6 ^

1.286

1.762

Advantages and Shortcomes of Casio fx-570ES calculator

C a s i o f x - 5 7 0 E S c a l c u l a t o r h e l p s e d u c a t o r s a n d students in s o l v i n g n u m e r i c a l m e t h o d s . Its h e l p s t o r e d u c e t h e a m o u n t of t i m e s p e n d in s o l v i n g t h e t a s k s g i v e n in t h e c l a s s . E v e n t h o u g h it is d e s i g n e d for a s y s t e m o f m a x i m u m 3 x 3 m a t r i x , it h e l p s s t u d e n t a lot in t h e e x a m i n a t i o n hall. F o r m o r e t h a n 3 x 3 m a t r i x , student s h o u l d u s e o t h e r r e s o u r c e s s u c h as E X C E L , M A T H C A D , M A P L E , M A T L A B , M A T H E M A T I C A or C P r o g r a m m i n g . H e n c e u s e r s c a n e x t e n d t h e c o n c e p t p r e s e n t e d h e r e t o a s y s t e m o f n linear e q u a t i o n s .

Conclusion

S e v e r a l e x a m p l e s of s o l v i n g m a t r i x in n u m e r i c a l m e t h o d s a r e s h o w n in this p a p e r . A s s o m e of t h e k e y s t r o k e s s e q u e n c e s for C a s i o f x - 5 7 0 E S are t h e s a m e as C a s i o f x - 5 7 0 M S in t h e b o o k w r i t t e n b y Tay, ( 2 0 0 6 ) , t h e steps t o s o l v e specific t o p i c are n o t d e s c r i b e d . F o r future r e s e a r c h on s q u e e z i n g t h e m o s t out of C a s i o f x - 5 7 0 E S c a l c u l a t o r , m o r e t o p i c s o n n u m e r i c a l m e t h o d will b e s t u d i e d .

R e f e r e n c e s

Tay, K . G . ( 2 0 0 6 ) . How To Use Calculator Casio FX-570MS In Numerical Methods. B a t u P a h a t : P e n e r b i t K U i T T H O .

G u e r r e r o - G a r c i a , P . & S a n t o s - P a l o m o . A . ( 2 0 0 8 , J u n e ) . Squeezing the most out of the casio fx-570ms for

electrical/electronics engineers. P a p e r p r e s e n t e d at t h e P r o c e e d i n g s o f t h e I n t e r n a t i o n a l C o n f e r e n c e on

C o m p u t a t i o n a l a n d M a t h e m a t i c a l M e t h o d s in S c i e n c e a n d E n g i n e e r i n g , S p a i n .

C o p y r i g h t S t a t e m e n t

C o p y r i g h t © 2 0 0 9 < T a y K i m G a i k > & < R o s m i l a A b d u l - K a h a r > . T h e a u t h o r s g r a n t a n o n - e x c l u s i v e license t o t h e o r g a n i s e r s of t h e 3r d C o s M E D I n t e r n a t i o n a l C o n f e r e n c e 2 0 0 9 , S E A M E O R E C S A M t o p u b l i s h this