Computer Applications LabComputer Applications LabLab 3Lab 3

Arithmetic Operations on Arithmetic Operations on Arrays Arrays

Chapter 2Chapter 2Sections 3,4,5Sections 3,4,5

Dr. Iyad Jafar

Adapted from the publisher slides

OutlineOutline

� Element-by-element operations

� Matrix operations

� Special Matrices

� Polynomial operations using arrays� Polynomial operations using arrays

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ElementElement--byby--Element OperationsElement Operations

� It is used to perform the desired mathematical operation between corresponding elements of arrays or by corresponding elements of arrays or by applying the operation between a scalar and each element of the array

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ElementElement--byby--Element OperationsElement Operations

� scalar-array operations◦ Addition B = A + 4

◦ Subtraction B = A – 4

◦ Multiplication B = A * 4

◦ Division B = A / 4

◦ The operation is repeated for each element in the array separately. ◦ The operation is repeated for each element in the array separately.

� Example

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ElementElement--byby--Element OperationsElement Operations

� Array-array operations

◦ The mathematical operation is repeated between corresponding elements in the arrays

◦ Arrays should be of equal dimensions

� Example� Example

5

NoteSubtraction is

performed in similar way

ElementElement--byby--Element OperationsElement Operations

� Array-array operations

◦ For multiplication (division) use .* (./) to multiply (divide) corresponding elements of the two arrays

� Example

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NoteDivision is performed

in similar way

ElementElement--byby--Element OperationsElement Operations

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ElementElement--byby--Element OperationsElement Operations

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ElementElement--byby--Element OperationsElement Operations

� Array-array operations

◦ For exponentiation, use .^ to raise each element in the array to specified base

� Example

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ElementElement--byby--Element Element OperationsOperations

� Can be used to evaluate functions at a set of points

� Example

evaluate f(x,y) = xy^2 + 8x at (1,2), (3,1), (5,9)evaluate f(x,y) = xy^2 + 8x at (1,2), (3,1), (5,9)

>> x = [1,3,5];

>> y = [2,1,9];

>> f = x.*(y.^2) + 8*x

ans =

12 27 445

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Matrix OperationsMatrix Operations

� We have discussed addition and subtraction earlier

� Multiplication

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Matrix OperationsMatrix Operations• In MATLAB, use * to perform multiplication of matrices

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Matrix OperationsMatrix Operations� Matrix division is more challenging topic than

matrix multiplication.

� Matrix division uses both right and left division operators, / and \, for various applications

� One application is solving linear algebraic equations� One application is solving linear algebraic equations6x+ 12y+ 4z= 70

7x–2y+ 3z= 5

2x+ 8y–9z= 64

>> A = [6,12,4;7,-2,3;2,8,-9];

>> B = [70;5;64];

>> Solution = A\B

Solution = 3 5 -2

The solution is x= 3, y= 5,and z= –2.13

Matrix OperationsMatrix Operations

� Solving the previous set of linear equations can be done by matrix inversion.

>> A = [6,12,4;7,-2,3;2,8,-9];

>> B = [70;5;64];

>> solution = inv(A) * B

solution =

3.0000

5.0000

-2.0000

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Matrix OperationsMatrix Operations

� Matrix exponentiation

◦ An is equivalent to multiplying the matrix by itself n times

◦ A should be a square matrix

◦ AB , where B is a matrix, is not defined

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Special MatricesSpecial Matrices

� Identity matrix I◦ eye(n) , eye(m,n) , eye(m,n)

� All-ones matrix � All-ones matrix ◦ ones(n) , ones(m,n) , ones (size(A))

� All-zeros matrix◦ zeros(n), zeros(m,n), zeros(size(A))

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Polynomial Operations Using ArraysPolynomial Operations Using Arrays� The polynomial

can be represented in MATLAB by

� Use the function roots(A) to find the polynomial roots

>> c = roots ([1,5,6])

n n 1 n 21 2 3 n 1f (x) = a x + a x + a x + ... + a− −

+

1 2 n 1A = [a , a , ... , a ]+

>> c = roots ([1,5,6])

c = -2 -3

� Use the function poly(c) to find the coefficients of the polynomial from its roots.

>> c = [-1,2]

>> ploy(c)

ans = 1 -1 -2

This means that the answer is x2 – x – 2 17

Polynomial Operations Using ArraysPolynomial Operations Using Arrays

� Multiplication

Use the function conv(a,b) to compute the product of the two polynomials described by the coefficient arrays a and b. The two polynomials need not be the same degree. The result is the coefficient array of the product polynomial.

� DivisionUse the function [q,r] = deconv(num,den) to compute the result of dividing a numerator polynomial, whose coefficient array is num, by a denominator polynomial represented by the coefficient array den. The quotient polynomial is given by the coefficient array q, and the remainder polynomial is given by the coefficient array r.

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Polynomial Operations Using ArraysPolynomial Operations Using Arrays

� Example: if a = 9x3 – 5x2 + 3x + 7 and b = 6x2 + 2, then find a x b and a / b.

>> a = [9,-5,3,7] ; b = [6,0,2] ; >> product = conv(a,b)

product = product = 54 -30 36 32 6 14

(This is equivalent to 54x 5 – 30x 4 + 36x 3 + 32x 2 + 6x + 14)

>> [Q,R] = deconv(a,b) Q =

1.5000 -0.8333R =

0 0 0 8.6667

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Polynomial Operations Using ArraysPolynomial Operations Using Arrays

� Evaluation of polynomials

The function polyval(a,x) evaluates a polynomial at specified values of its independent variable x, which can be a matrix or a vector. The polynomial’s coefficients of descending powers are stored in the can be a matrix or a vector. The polynomial’s coefficients of descending powers are stored in the array a. The result is the same size as x.

>> a = [9,-5,3,7];

>> x = [-2,3,4,5];

>> f = polyval(a,x)

f = -91 214 515 1022

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