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CS 101: Computer Programming

and Utilization

11-MatricesMultidimensional Arrays

Instructor: Sridhar IyerIIT om!ay

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Acti"ity # $oad !alancing

There are t%o truc&s' A and ' each ha"ing

pac&ages %ith di((erent %eights) *e %ish to !alancethe load in !oth the truc&s !y s%apping e+actly onepair o( pac&ages !et%een them)

*rite a program to determine i( such load !alancingis possi!le (or a gi"en set o( input %eights' and i(yes' identi(y %hich pac&ages to s%ap) Assume that

all %eights are integers ,e)g) . /g' /g' 2 /g'etc)3)

4iscuss: pseudo-code5 6un: demo11-!alance)cpp

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$oad !alancing truc&s # !asic idea

I( + is element o( A' and y element o( ' %e need to

(ind + and y such that' i( + and y are e+changed' theresulting sums match)

sumA 7y -+ 8 sum -y 7 + ,a(ter e+changing + and y3

or y 8 ,sum-sumA39 7 +

or some i' i( %e are loo&ing at A;i

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inary Search

Can %e do (aster than se>uentially searching (or y in

Array ?● @o% long does it ta&e any%ay?

● proportional to num!er o( elements - ,n3

nce the Array is sorted

● *e can use an idea similar to (inding a root !y

!isection method● @o% much is the time reduced?

● ,log n3

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inding a root !y !isection method

Use this idea to de"elop the code (or inary Search

● 4o Thin&-Pair-Share

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B+ecution o( inary Search - e+ample

;0< 10012

;1< 100.

;< 100D1

;< 100E

;< 100E

;.< 100.

;E< 100DE.;2< 1011D.

;< 1012

;D< 101E2D

or the array gi"en' sho% the "alueso( lo' hi and mid (or each iteration'

●  %hen num is gi"en as 100.

●  %hen num is gi"en as 101D.

Indi"idually' %or& it out in %ritingF

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inary Search - code

int n8100' A;n

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Multidimensional Arrays

*e can use arrays %ith more than one dimension

int A;.0

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Storage: ro% ma=or

● Internally' no distinction !et%een ;

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Matri+ manipulations

4 arrays are commonly used to represent Matrices

and are (ound in programs such as:

● Matri+ multiplication

● Oi"en: m + n matri+ A5 n + p matri+ ' the matri+product C 8 A %ill !e m p matri+)

● C[i][j] =k=0

Qn-1A;i

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asic operations

● Initialize s>uare matri+ to identity

for (int rx = 0; rx < rows; ++rx) {  for (int cx = 0; cx < cols; ++cx) {

  dmat[rx][cx] = (rx == cx)? 1 : 0; } }

● etter code uses t%o loops ,%hy?3● In (irst r+'c+ loop set all elements to zero

for (int rx=0; rx

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Transpose a s>uare matri+ - e+ample

1 2 3

4 5 6

7 8 9

1 4 3

2 5 6

7 8 9

1 4 3

2 5 6

7 8 9

1 4 7

2 5 6

3 8 9

1 4 7

2 5 6

3 8 9

1 4 7

2 5 8

3 6 9

4o Thin&-Pair-Share (or coding

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Transpose a s>uare matri+ - code

● 4onKt dou!le transpose !ac& to s>uare oneF

for (int rx = 0; rx < rows; ++rx) {

  for (int cx =rx+1

; cx < cols; ++cx) {  float tmp = fmat[rx][cx];

  fmat[rx][cx] = fmat[cx][rx];

  fmat[cx][rx] = tmp;

  }

}

No need to transpose diagonal or below

Swap [rx][cx]with [cx][rx]

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Matri+ "ector multiplication - e+ample

a00 a01 a02

a10 a11 a12

x0

x1

x2

y0

y1=

a00*x0 +a01*x1 +a02*x2

a10*x0 +a11*x1 +a12*x2

4o Thin&-Pair-Share (or coding

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Matri+-"ector multiplication - code

float A[rows][cols];

float x[cols] ![rows];"" fill #p A and x

for (int rx=0; rx < rows; ++rx) {

  ![rx] = 0;  for (int cx=0; cx < cols; ++cx) {

  ![rx] += A[rx][cx] $ x[cx];

  }}

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Matri+-matri+ multiplication

float amat[lsi%&][msi%&] 'mat[msi%&][nsi%&] cmat[lsi%&][nsi%&];

for (int crx=0; crx

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Oaussian elimination method

Consider t%o e>uations: + 7 y 8 and + 7 .y 8 1

They can !e represented as:

 ; < ;+< 8 ;<

 ; .< ;y< 8 ;1<

Oauss elimination method in"ol"es trans(ormations,!y using multiplication and linear com!inations3 toreduce the coe((icient matri+ to upper triangular (orm:

;1 < ;+< 8 ;<

;0 1< ;y< 8 ;<

Then sol"ing !y !ac&-su!stitution to get y 8 ' + 8 -)

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Oaussian elimination representation

The pre"ious coe((icient matri+ can !e stored as:

A;0

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Oaussian elimination pseudo-code

● 6ead matrices A;

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Oaussian elimination program

● *or& %ith your neigh!our to %rite the C77 code

● Compare %ith: demo11-gauss)cpp

dli l i

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@andling large inputs

● Tedious to type the input "alues one at a timeF

● Solution: I9 6edirection R

● *hen %e e+ecute our program' S assigns it

three standard (iles - stdin' stdout' stderr

● y de(ault' S JconnectsK these to de"ices:

● stdin to &ey!oard' stdout and stderr to monitor

● )9a)out input)t+t 6edirects stdin (rom a (ile

● )9a)out G output)t+t 6edirects stdout to a (ile