RvsZ3

Comparison of R vs Z3

Simple manipulations; numbers and vectors

Vectors and assignment

Z3 operates on named data structures. The simplest such structure is the numeric vector, which is a single entity consisting of an ordered collection of numbers. 

To set up a vector named x, say, consisting of five numbers, namely 10.4, 5.6, 3.1, 6.4 and 21.7, use the R command

> x <- c(10.4, 5.6, 3.1, 6.4, 21.7)

Z3 command to set up a vector is:

x<==[10.4, 5.6, 3.1, 6.4, 21.7];

Alternatively we can use the simple "=" also.

 x=[10.4, 5.6, 3.1, 6.4, 21.7]

Assignment can also be made using the function ASSIGN(). An equivalent way of making the same assignment as above is with: In R,

> assign("x", c(10.4, 5.6, 3.1, 6.4, 21.7))

In Z3, use the "ASSIGN" function as:

ASSIGN("x", [10.4, 5.6, 3.1, 6.4, 21.7])

Assignments can also be made in the other direction, using the obvious change in the assignment operator. So the same assignment could be made using

[10.4, 5.6, 3.1, 6.4, 21.7]==>x 

The reciprocals of the above five values for x in R,

> 1/x

In Z3, We can use the function called Reciprocal,

RECIPROCAL(x) (the value of x is [10.4, 5.6, 3.1, 6.4, 21.7] 

Also we can use directly,

([10.4,5.6,3.1,6.4,21.7]<>d40)@(x=>1/x)

The further assignment

> y <- c(x, 0, x)

would create a vector y with 11 entries consisting of two copies of x with a zero in the middle place.

Vector arithmetic

Vectors can be used in arithmetic expressions, in which case the operations are performed element by element. Vectors occurring in the same expression need not all be of the same length. If they are not, the value of the expression is a vector with the same length as the longest vector which occurs in the expression. So with the above assignments the R command

> v <- 2*x + y + 1

generates a new vector v of length 11 constructed by adding together, element by element, 2*x repeated 2.2 times, y repeated just once, and 1 repeated 11 times.

With the same Assignment Z3 command is.

v=2*x+y+1

In Z3 the elementary arithmetic operators are the usual +, -, *, / and ^ for raising to a power. Also we can use the functions SUM,SUB,PRODUCT,DIVIDE and POWER instead of using arithmetic operators.

In addition all of the common arithmetic functions are available. LOG, EXP, SQRT, SIN, COS, TAN, SEC,COSEC,COTAN, Hyperbolic functions and so on. For trignometric functions we can find the values in Deg and Radians also.

MAX and MIN select the largest and smallest elements of a vector respectively.

In R,two statistical functions are mean(x) which calculates the sample mean, which is the same as sum(x)/length(x), and var(x) which gives

sum((x-mean(x))^2)/(length(x)-1)

or sample variance.

In Z3,to find the mean value we can use function called MEAN(x), AVG(x) or AVERAGE(x).

In R, sort(x) returns a vector of the same size as x with the elements arranged in increasing order.

SORTING(x) returns the vector in increasing order in Z3.

• • • Is there any other functions like order or list.

To work with complex numbers, supply an explicit complex part. Thus sqrt(-17) will give NaN and a warning, but sqrt(-17+0i) will do the computations as complex numbers in R.

In Z3, while computing complex numbers simply we can use as

SQRT(-17) or SQRT(-17+0i).

• • • Need to know any functions equivalent to The parallel maximum and minimum functions pmax and pmin return a vector.

Generating regular sequences

The function seq() is a more general facility for generating sequences. To get the sequence of values from a particular range with step value with R as > seq(-5, 5, by=.2) -> s3 generates in s3 the vector c(-5.0, -4.8, -4.6, ..., 4.6, 4.8, 5.0). Similarly > s4 <- seq(length=51, from=-5, by=.2) generates the same vector in s4.

Z3 command to get a sequence value as:

s3=-5..5..0.2

generates in s3 vector.

s4=Array(51).seq(-5,0.2)

generates s4 vector with same as s3.

To print the five copies of x ene-to-end in s5 the R command is > s5 <- rep(x, times=5) Alternatively > s6 <- rep(x, each=5) which repeats each element of x five times before moving on to the next.

The z3 command of replicating the array value is: s5=x.replicate(5)

The another way is: s6=RECURSIVEARRAY(5,x)

Arrays and Matrices

Arrays

A 3 by 5 by 100 dimension vector z of 1500 elements is defined with R command as:

> dim(z) <- c(3,5,100)

Z3 command to define an array is:

DIM(3,5,100)

Alternatively it can also be represented in array form as:

|3,5,100|

Array indexing. Subsections of an array

A 4 x 2 array with array elements is represented by R command as:

c(a[2,1,1], a[2,2,1], a[2,3,1], a[2,4,1],
a[2,1,2], a[2,2,2], a[2,3,2], a[2,4,2])

In Z3, the above 4 X 2 array with array elements is defined using square brackets as:

[[2,1,1], [2,2,1], [2,3,1], [2,4,1], [2,1,2], [2,2,2], [2,3,2], [2,4,2]]

The above array can be stored with a variable name 'Z' as:

z = [[2,1,1], [2,2,1], [2,3,1], [2,4,1], [2,1,2], [2,2,2], [2,3,2], [2,4,2]]

The contents of variable Z can be obtained using Z3 command:

DIM(z)

Also, to identify the size of 'z', use the Z3 command:

DIMENSIONS(z) 

which gives the result as: 8 3 (8 rows, 3 columns)

Index matrices

A matrix 'x' with 4 rows and 5 colums containing values from 1 to 20, is defined using R command as:

> x <- array(1:20, dim=c(4,5))

This command displays the result as:

    [,1] [,2] [,3] [,4] [,5]
   
[1,] 1    5    9    13   17

[2,] 2    6   10    14   18

[3,] 3    7   11    15   19

[4,] 4    8   12    16   20

The Z3 command can be used as:

x= |4,5,1..20|

In Z3, the array elements are stroed row wise. Hence using a TRANSPOSE command will result into the same output as above.

TRANSPOSE(x) 

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