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)

Logical vectors

The elements of a logical vector can have the values TRUE, FALSE, and NA. Logical vectors are generated by conditions.

For example

> temp <- x > 13 sets temp as a vector of the same length as x with values FALSE corresponding to elements of x where the condition is not met and TRUE where it is.

Z3 command to generate the logical vector is:

[10.4,5.6,3.1,6.4,21.7]|[x,x>13]|; 

Logical vectors may be used in ordinary arithmetic, in which case they are coerced into numeric vectors, FALSE becoming 0 and TRUE becoming 1.

Missing values

The function is.na(x) gives a logical vector of the same size as x with value TRUE if and only if the corresponding element in x is NA.

> z <- c(1:3,NA); ind <- is.na(z)

Z3 command for ISNA is:

z=([1,2,3,"NA"]);ISNA(z)

There is a second kind of “missing” values which are produced by numerical computation, the so-called Not a Number, NaN, values. In R, examples are

> 0/0

or

> Inf - Inf which both give NaN since the result cannot be defined sensibly.

In z3,

0/0 will give the result as NaN.

∞-∞; \\Symbol of Infinity

will give the result as Null.

Character vectors

Character quantities and character vectors are used frequently in R, for example as plot labels.The paste() function takes an arbitrary number of arguments and concatenates them one by one into character strings.

The arguments are by default separated in the result by a single blank character, but this can be changed by the named parameter, sep=string, which changes it to string, possibly empty.

For example,

> labs <- paste(c("X","Y"), 1:10, sep="")

makes labs into the character vector

c("X1", "Y2", "X3", "Y4", "X5", "Y6", "X7", "Y8", "X9", "Y10")

Z3 command to execute the above is:

|10|.fillwith("x","y").joincolumnswith(1..10)  //need to add "=" symbol

Index vectors; selecting and modifying subsets of a data set

Subsets of the elements of a vector may be selected by appending to the name of the vector an index vector in square brackets. Such index vectors can be any of four distinct types.

• A logical vector:

Values corresponding to TRUE in the index vector are selected and those corresponding to FALSE are omitted. For example > y <- x[!is.na(x)] creates (or re-creates) an object y which will contain the non-missing values of x, in the same order. Note that if x has missing values, y will be shorter than x. Also > (x+1)[(!is.na(x)) & x>0] -> z

Corresponding z3 command is:

y = x(!=ISNA(x))

To create z is:

z=(x+1)[(!ISNA(x)) & x>0]          //!= is not working(1..10[!=]1..130)

• A vector of positive integral quantities:

The corresponding elements of the vector are selected and concatenated, in that order, in the result. In R, x[6] is the sixth component of x and > x[1:10] The same in Z3 command is:

 x.any(10)  //Need to check

selects the first 10 elements of x (assuming length(x) is not less than 10). Also > c("x","y")[rep(c(1,2,2,1), times=4)]

• A vector of negative integral quantities:

Such an index vector specifies the values to be excluded rather than included. Thus > y <- x[-(1:5)] gives y all but the first five elements of x. \\it is not showing the first five elements of x. It is just showing the result as numeric(0)

Z3 command to index vector specification is:

                                                      // To include

• A vector of character strings:

In this case a sub-vector of the names vector may be used in the same way as the positive integral labels in item 2 further above.

> fruit <- c(5, 10, 1, 20) > names(fruit) <- c("orange", "banana", "apple", "peach") > lunch <- fruit[c("apple","orange")]

The advantage is that alphanumeric names are often easier to remember than numeric indices.

The same in Z3 command is:

fruit=[5, 10, 1, 20]
["names"]<<<"orange", "banana", "apple", "peach"    //Here names(fruit) is not giving result
["lunch"] <<<fruit[("apple","orange")]   // Giving the result as null

The vector assigned must match the length of the index vector, and in the case of a logical index vector it must again be the same length as the vector it is indexing. For example > x[is.na(x)] <- 0

replaces any missing values in x by zeros and > y[y < 0] <- -y[y < 0]

has the same effect as > y <- abs(y)

We can write in Z3,

 x[ISNA(x)] = 0

Replacing values in x is:

y[y < 0] = -y[y < 0]

Also we can write as,

y = ABS(y)

Objects, their modes and attributes

Intrinsic attributes: mode and length

R consists of a number of data objects to perform various functions. There are 6 types of objects in R Programming. They include vector, list, matrix, array, factor, and data frame.

Vectors in R programming data objects: logical, integer, character, raw, double, and complex.

Z3 language also supports data objects: logical, integer, character, raw, double, and complex.

Lists in R contain various types of elements including strings, numbers, vectors, and a nested list inside it. It can also consist of matrices or functions as elements. It can be created with the help of the list() function.

Z3 stores all the data in array format. The data can be strings, numbers, vectors, matrices or functions as elements.

List of elements can be displayed using Z3 command LISTALL.

Matrices in R Programming are used to arrange elements in the two-dimensional layout to perform mathematical operations.

Matrices in Z3 can be of any dimensions. A matrix can be defined in many ways such as:

MATRIX(3)     //Displays 3x3 matrix

or

MATRIX("anti-diagonal",4,200..204)  //Displays 4x4 anti-diagonal matrix with values inbetween 200 and 204

or

|5|   //Displays 5x5 matrix

or

|2,3,4|   //Displays 2x3x4 matrix

An array in R is used to store data in multi-dimensional format. It can be created with the help of an array() function.

Z3 has n number of commands for using array functions such as:

ARRAY(3,4)    //Defines a 3-dimensional array with each element value of 4

a=[[1,3,4],[2,3,4]]		//Defines an array 'a'
a.add(45)			//Adds 45 in each array element

Row, Column, Diagonal, concatination etc operations are possible using Z3 commands. (Refer list of Array Manipulation Functions here: https://wiki.zcubes.com/Z%5E3_Array_Manipulation_Member_Functions)

Factors are data objects that are used in order to categorize and store data as levels. They can be strings or integers. They are extremely useful in data analytics for statistical modeling. They can be created using factor() function.

Factors can be identified or retrieved in Z3 by giving variable name as a command.

a=[[11,3,4],[21,3,4]]		//Defines array 'a'
a				//Displays elements of array 'a'

Dataframe is a 2-dimensional data structure wherein each column consists of the value of one variable and each row consists of a value set from each column.

.................Need to add explanation for this..............................

Properties of an object are provided by attributes such as mode, length. Change of mode in R is represented as:

> z <- 0:9	//z is defined with elements 0 to 9
> digits <- as.character(z)   // digits is the character vector c("0", "1", "2", ..., "9")
> d <- as.integer(digits)	//Now d and z are the same

The above mode change can be represented in Z3 as:

z=[0..9]
digits=CHAR(z)
d= INT(digits)

Changing the length of an object

An “empty” object can be defined in R language as:

> e <- numeric()	 //makes e an empty vector structure of mode numeric. 

> e <- character()	 //makes e an empty vector structure of mode character.

Using below Z3 command, an empty object can be defined as:

e=NUM()
e=CHAR()

Once an object of any size has been created, new components may be added to it simply by giving it an index value outside its previous range. Thus

> e[3] <- 17	//makes e a vector of length 3

Z3 command is:

e[3]=17	//length of e vector is 3 

The length of a vector can be retrieved by R command:

>length(e)

Z3 command used is:

LEN(e)     //displays output as 3

Getting an setting attributes

R command: attr(z, "dim") <- c(10,10)

Z3 command: ' to do *** '

The class of an object

Object in R with class "data.frame", plot() and other functions such as summary() will display the output values in certain ways. Using Z3, the data output values can be displayed in list format, spreadsheet format, graphical format etc.

In R, unclass() removes temporarily the effects of class. For example if winter has the class "data.frame" then

> winter

will print it in data frame form, which is rather like a matrix, whereas

> unclass(winter)

will print it as an ordinary list.

Z3 command: ' to do *** '

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 stored row wise.

1	2	3	4	5
6	7	8	9	10
11	12	13	14	15
16	17	18	19	20

' to do *** ' : Z3 command to be added to obtain same output as in R

Array function

Array function in R:

> Z <- array(data_vector, dim_vector)

For example

> Z <- array(h, dim=c(3,4,2))
same as
> Z <- h ; dim(Z) <- c(3,4,2)

> Z <- array(0, c(3,4,2))  //makes Z an array of all zeros.

Z3 Commands used are:

z=[3,4,2]   //Defines an array z with elements specified

REPLACE(z,0)  //Replaces elements in z with '0'

So if A, B and C are all similar arrays, then D is a similar array with its data vector being the result of the given element-by-element operations. R command:

> D <- 2*A*B + C + 1

Z3 command:

d= 2*a*b+c+1 

The outer product of two arrays

In R language, if a and b are two numeric arrays, their outer product is formed by the special operator %o%:

> ab <- a %o% b
or also alternatively,
> ab <- outer(a, b, "*")

In Z3, array can be multiplied using inbuilt '.multiply' function. e.g

[[1,3,4],[2,3,4]].multiply(45)

Multiplication of two or more matrices can be carried using Z3 commands:

MATRIXMULTIPLY([2,-3,4;-5,6,7],9)
or
MATRIXPRODUCT([2,3,4;5,6,7],5)
or
MATRIXPRODUCT([[6,7,8],[10,12,-22],[7,17,23]],[[20,12,16],[7,8,13],[4,8,9]])

In R, the multiplication function can be replaced by an arbitrary function of two variables as:

> f <- function(x, y) cos(y)/(1 + x^2)
> z <- outer(x, y, f)

In Z3, the above function can be defined as:

f(x,y)=COS(y)/(1+x^2)
...' to do *** '............

Refer all array manipulation functions here: Listing of Z3 Array Manipulation Member Functions

Generalized transpose of an array

Transpose of an array can be calculated in R language using aperm() or t() functions as:

> B <- aperm(A, c(2,1))
or
B <- t(A)

In Z3 language, transpose can be calculated using commands array.flip(),t(), MATRIXTRANSPOSE() functions as:

[[1,8,3],[7,4,5],[9,13,45]].flip()
or
MAGICSQUARE(3).t()
or
MATRIXTRANSPOSE([[12,17,18],[6,15,36],[13,19,25]])

Matrix facilities

R has matrix functions such as:

t(X) is the matrix transpose
nrow(A) gives number of rows in the matrix A
ncol(A) give the number of columns in the matrix A

Z3 has number of matrix functions such as:

MATRIXTRANSPOSE()  returns transpose of a matrix
MATRIXROW          returns specified row elements
MATRIXCOLUMN       returns specified column elements

To know more matrix functions, read here: [| Matrix Functions 1] [| Matrix Functions 2]

Matrix multiplication

In R, the operator %*% is used for matrix multiplication. If, for example, A and B are square matrices of the same size, then

> A * B is the matrix of element by element products and
> A %*% B is the matrix product.
 If x is a vector, then > x %*% A %*% x is a quadratic form

In Z3, functions such as MATRIXMULTIPLY() or MMULT() are used for matrix multiplication

e.g 1 MATRIXMULTIPLY([4,7.2,6;9,-8,12],[2,3;6,5;9,8])

e.g 2 MMULT([[2.5,4,3,7],[1,3,5,4]],[[2,5,6],[7.3,4,9],[10,4,1],[6,2,8]])

Function crossprod() in R language forms crossproducts of two matrices. Functions such as CROSSPRODUCT() and VECTORPRODUCT() are used.

CROSSPRODUCT([2,7,8],[3,9,5]) =-37 14 -3
or
VECTORPRODUCT([2,3,5],[8,6,4]) = -18 32 -12

IN R, diag(v) displays diagonal elements of vector v. In Z3, DIAG() function is used to display diagonal elements as:

DIAG([[21,43,-56],[1,-6,-15],[2,3.2,8]]) displays result as 21, -6, 8

Lists and data frames

Lists

An R list is an object consisting of an ordered collection of objects known as its components.

An Example to make a list is: > Lst <- list(name="Fred", wife="Mary", no.children=3,child.ages=c(4,7,9))

An Example in Z3 is:

Lst=[name="Fred",wife="Mary",NoofChildren=3,Childages=[4,7,9]];

To know the number of component it has in R, length(Lst)

Lst.length gives the number of component in Z3.

Components of lists may also be named, > name$component_name

**To do for Z3

Simple Example to get the right component in R:

Lst$name is the same as Lst1 and is the string "Fred"

Lst$wife is the same as Lst2 and is the string "Mary"

Lst$child.ages[1] is the same as Lst4[1] and is the number 4

Also Lst"name" is the same as Lst$name.

The same result will get in Z3,

name or Lst0 
wife or Lst1
Childages or Lst3 and the result as 4,7,9

Also we can use,

name@Lst           //To check this format result

When the name of the component to be extracted is stored in another variable in R as, > x <- "name"; Lstx

In Z3,

x=["name"];x@Lst;

The names of components may be abbreviated down to the minimum number of letters needed to identify them uniquely. Thus Lst$coefficients may be minimally specified as Lst$coe and Lst$covariance as Lst$cov.

In Z3, we can use the COVAR@Lst instead of COVARIANCE@Lst.

Constructing and modifying lists

New lists may be formed from existing objects by the function list(). An assignment of the form > Lst <- list(name_1=object_1, ..., name_m=object_m) \\not giving any result in R

• • To do the same in Z3.

Lists, like any subscripted object, can be extended by specifying additional components.

For example > Lst[5] <- list(matrix=Mat) \\Not giving any result

• • To do in Z3.

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