Example Sheet 8 – Doing more with arrays and using
Range Variables
The aim of this worksheet is to introduce you to more things you can do with vectors and
matrices in MathCAD.
By the end of the sheet you will know how to:
 Create vectors and matrices
 Perform simple matrix and vector mathematics
 Access individual elements of arrays
 Access columns and submatrices within arrays
 Create and use range variables
 Use range variables to create vectors and matrices
 Read in data to analyse from a file.
This worksheet takes the form of a number of examples which illustrate and introduce
these principles. Work through them all.
Exercise 1 – Creating vectors and matrices and doing simple
maths with them
In this exercise we will create some vectors and matrices, which will then form the basis
for our later investigations.
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

VA
MA
Start MathCAD and create a new, blank worksheet
Display the Matrix Toolbar by selecting View|Toolbars|Matrix from the menu
Use the [:::] button on the matrix toolbar to create the following matrices:
o The vectors VA and VB are defined with 2 rows and 1 column
o The matrices MA and MB are defined with 2 rows and 2 columns
1
2
1 2
3 4
VB
MB
3
4
5 6
7 8
Simple operators and matrix math
Having created these arrays, we can use them to perform simple matrix maths.
Vectors and matrices may be added or subtracted, if they are of the same size.
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VA
MathCAD Example Sheet 8
4
VB 
VA
6
6
MA MB 
8
10 12
VB 
MB MA 
2
Addition and subtraction of vectors
2
4 4
Addition and subtraction of matrices
4 4
MathCAD enforces rules of matrix maths and will not allow addition of matrices or
vectors of different sizes.
MA VA 
The number of rows and/or columns
in these arrays do not match
MathCAD will allow addition or multiplication of an array by a scalar. In this case the
scalar is added or multiplied by each element of the matrix in turn:
VA
1
VA  3 
2
MA 1 
3
3
MA 3 
6
2 3
4 5
3 6
9 12
Addition of constant
Multiplication by scalar
MathCAD will also calculate the dot products of vectors and matrices, providing the
inner dimensions are compatible…
VA  VB  11
MA MB 
19 22
MA VA 
5
Dot product of 2 vectors
Dot product of 2 matrices
43 50
11
Dot product of matrix and vector
The ‘dot’ operator for matrix multiplication is the same ‘*’ as is used for conventional
multiplication, alternatively you can select the ‘dot product’ operator from the matrix
toolbar.
For practice, create the following arrays and matrices:
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MathCAD Example Sheet 8
3
Vec3DA
4
5
Vec3DB
7
7
Vec3DC
11
1.2
3.4
5.6
7.8
9.0
1.2
MC
1
MD
2
3
0 1
3
5 0
5
ME
9 11 13
1 1
1
2 4
8
3 9 27
4 16 64
Exercise 2 – Accessing parts of Arrays
In the last exercise we applied operators (such as addition and multiplication) to entire
arrays. In this exercise we will see how to access individual elements or parts of arrays by
using indices.
MathCAD, by default, starts counting with indices at 0, so the first element of an array is
at index 0. This can be changed but I would strongly advise against doing so.
MathCAD uses array subscripts to address individual elements of an array. Array
subscripts may be created either by selecting the subscript operator from the matrix
toolbar or by using the <[> key.
We will use the arrays you defined in the last exercise to practice on.
VA
MA
MC
1
VB
2
1 2
MB
3 4
1.2
3.4
5.6
7.8
9.0
1.2
MD
3
4
5 6
7 8
0 1
3
5 0
5
9 11 13
ME
1 1
1
2 4
8
3 9 27
4 16 64
Using these arrays, we will first use subscripts to extract values from the arrays and then
use subscripts to set individual elements of the array.
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MathCAD Example Sheet 8
On the worksheet you have been using so far, position the cursor below the last
matrix definition.
Type VA[0=
MathCAD responds by giving the value of the ‘zeroth’ element of the vector VA
In the same way, query the value of VA1 by typing VA[1=
The two values should be displayed as shown.




VA0  1
VA1  2
For two dimensional arrays, (matrices), mathCAD uses 2 indices separated by a comma,
in the order Row,Column
MD0 1  1
MD1 0  5
ME1 2  8
ME2 1  9
We can use the same subscript operator to set individual elements of arrays. To do this,
simply follow the expression which references the array with the assignment operator
<:>
For example, to set value of the zeroth element of VB to 7, type VB[0:7
VB0
VB 
7
7
4
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Exercise 3 – Accessing parts of arrays
As well as using the matrix indices to access individual elements of an array, we can use
functions to access a subset of the matrix containing several elements. We met this
operator earlier when we used it to separate out different parts of experimental data.
MathCAD has an operator to access an entire column of a matrix at one time. It is shown
on the worksheet as shown below
Column
Operator
Column
Index
1
Matrix to
operate
on
 1
ME

4
9
16
To insert the row operator into your worksheet, you can either select it from the matrix
toolbar or type <ctrl-6> from the keyboard. The ability to extract a column from a matrix
is particularly useful when reading experimental data from a file into mathCAD, as we
will see later.
MathCAD also has a function, submatrix(), which can extract a number of rows and
columns from an array
First Column
Matrix to
operate on
Last Column
1 1
submatrix( ME  0  2  1  2 )  4 8
9 27
First Row
Last Row
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As an exercise, use the functions and operators to print out the following parts of the
arrays:
The zeroth column of array MA
The first column of array MB
The zeroth and first row of matrix MC
The 2x2 submatrix at the bottom right hand corner of matrix MD
The submatrix with 2 columns and 3 rows at the bottom right hand corner of ME
Transposing Rows and Columns
Another operator which works with arrays is the transpose operator (MT). This can be
found on the matrix toolbar and swaps the rows and columns of a matrix.
You can use the transpose operator in conjunction with the column select operator to
extract a single row from a matrix. The examples below show the transpose operator in
action.
1 1
1
2 4
8
3 9 27
1 2 3 4
T
ME  1 4 9 16
4 16 64
1 8 27 64
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ME
T
ME
 2 T
 ( 3 9 27 )
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MathCAD Example Sheet 8
Exercise 4 – Other matrix functions and operators
There are many other functions which operate on vectors and matrices. Some of these
may be found on the matrix toolbar and others can be found on the functions dialog box
(from the f(x) toolbar button) under the heading “Vector and Matrix”.
I have given brief notes on some of the important ones below. In the summary I have
followed the MathCAD convention of using V to refer to a vector function, M to refer to
a matrix function and A to refer to an array, which can be either a vector or a matrix.
rows(M), cols(M)
min(A), max(A)
last(V)
length(V)
matrix(r,c,F)
stack(A,B)
augment(A,B)
identity(n)
V
sort(V)
reverse(A)
rsort(M,n)
csort(M,n)
Return the number of rows and columns in matrix M.
Return the minimum or maximum value in array A.
Returns the index of the last element of V
Returns the number of elements in V
Returns a matrix with r rows and c columns using the function F.
Returns the result of putting matrix A on top of B. A & B must
have the same number of columns
Returns the result of placing matrix B to the right of matrix A
Returns the identity matrix with n rows and n columns
Sums the elements in the vector V – from toolbar
The following sorting functions work on arrays and vectors and
may be found under the “sorting” heading of function dialog box
Sorts a vector V into ascending order
Reverses the elements of a vector or the rows of an array
Arranges the columns of M until row n is in ascending order
Arranges the rows of M until column n is in ascending order
Experiment with the using these functions on your example vectors and matrices to solve
the following problems:





What is the largest element in matrix ME ?
What is the smallest element in matrix MC ?
What is the smallest element in column 1 of matrix MD ?
What is the largest element in row 2 of matrix MD ?
Combine matrices MD & ME and sort the result to produce a matrix with 7 rows
and 3 columns with column 0 in ascending order.
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Element by element evaluation with the Vectorise operator
The determinant operator (|M|) can be used to return the determinant of a matrix. We can
also use the same operator to return the absolute value of a number.
2 2
7 7
Suppose that instead of wanting to find the determinant of a matrix we wish to find the
absolute value of all the elements in the matrix. In order to do this we use the vectorise
operator from the matrix toolbar. This is shown as an arrow over f(M).
1
2
3
4
2
1
2
3
4
1 2

3 4
In the example on the left, the determinant operator is applied to the 2x2 matrix and
calculates the determinant as normal. In the example on the right, the vectorise operator
has been applied to the determinant operator, causing it to be applied to each element of
the matrix in turn.
The vectorise operator also works with matrix multiplication, as shown below:
1
4
4
4
2  5  32
2  5
 10
3
3
6
1
6
18
In the left hand case, two vectors are multiplied together using the normal rules of matrix
multiplication, giving a scalar product. In the right hand case the vectorise operator forces
the multiplication to take place on an element by element basis.
The vectorise operator works for all operations that normally have a special meaning for
matrices.
Exercise 5 – Range Variables and filling matrices
Up until now, all the variables you have used in mathCAD have taken a single value (or
set of values in the case of arrays). A range variable is a special type of variable which
takes on a whole series of values, any expression involving the range variable is
evaluated a number of times, once for each value in the series.
Because range variables take on a series of values, mathCAD places restrictions on where
they may be used, if you attempt to use a range variable in the wrong place, mathCAD
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will highlight the offending expression in red and the error message will refer to ‘Illegal
Context’.
Range values correspond to FOR loops in conventional programming languages, and are
very useful when used with arrays to identify elements of the array.
To start with, we will create a range variable, i, and do some experiments with it.
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Start with a fresh mathCAD sheet.
Define the range variable by typing [i][:][0][;][3]
Notice how you define the range variable by typing the semi-colon character [;]
and mathCAD displays this as a series of dots.
Once you have defined the range variable, you an ask mathCAD to print out the
value of it, like you would any other variable, by typing [i][=]
In this case, however, mathCAD prints a table of values.
Your sheet should now look like this:
0  3
i
i
0
1
2
3
Notice how the display of the values of I is subtly different from the way in which
mathCAD displays a matrix or a vector.
You can create an expression involving the range variable and mathCAD will evaluate
the expression for each of the values taken by the range variable. In the example below I
have asked mathCAD to do evaluate a number of expressions involving the range
variable.
0  3
i
i
0
1
2
3
i 1
1
2
3
4
2
i
i 
2 
0
1
4
9
1
2
4
8
Again, you should note that mathCAD is simply displaying the result of a calculation
repeated a number of times, changing the value of i each time.
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Range variables find their greatest use when filling an array with a set of values.
Suppose that we were performing an experiment where we took a reading every 10
minutes. We could create an array and type the times into it by hand, like this:
0
10
time
20
 min
30
But this is tedious and prone to mistakes, especially if we had 100 times to type in instead
of the 4 in this case.
Looking at the series of numbers in the array, we can say that the i'th element is i times
10 minutes, which we express to mathCAD like this:
i
0  3
timei
i 10 min
0
0
600
time 
1.2  10
3
1.8  10
3
s
change the units to give:
time 
10
20
min
30
Note how the range variable on the left hand side of the assignment symbol is used to
index through the array elements while the same variable on the right hand side is used to
create their values. For practice, after you have created the ‘time’ array, have a go at
creating the following arrays using range variables:



The array ‘evens’, which contains the first 5 even numbers.
The array ‘odds’ which contains the first 5 odd numbers
The array ‘squares’ which contains the squares of the first 6 integers.
When you have finished, your arrays should look like this:
0
1
0
2
3
1
evens  4
odds  5
6
7
8
9
squares
4
9
16
25
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MathCAD Example Sheet 8
Exercise 6 – Statistical functions and arrays
MathCAD has a range of statistical and related functions which operate on arrays.
As a simple example, we are going to perform some statistical analysis on a series of
results. These can be placed into a vector and the mean and standard deviation calculated:
2.55
2.5
2.6
2.42
2.31
Diameters
2.53
2.56
2.62
2.28
2.44
2.41
2.49
mean( Diameters)  2.476
stdev( Diameters)  0.103
The mean and stdev functions may be selected from the f(x) dialog box, under the
function category ‘statistics’.
Note that you can add units to a vector. If all elements of a vector have the same units,
the easiest way to do this is to select the entire matrix and multiply it by the appropriate
unit. The units will then propagate through to the answers as shown below:
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2.55
2.5
2.6
2.42
2.31
Diameters
2.53
2.56
 mm
2.62
2.28
2.44
2.41
2.49
mean( Diameters)  2.476  10
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Exercise 7 – Getting data from a file.
Although for simple experiments it is possible to type values directly into MathCAD
using an input table, frequently we have to process sets of results that come from
computer controlled test and measurement equipment, or that have been tabulated by
other researchers. MathCAD can read data files stored on the computer and bring those
into its calculations.
As an exercise, I have prepared two files for you to bring into MathCAD and perform
some simple analysis upon. The files are data taken from one of my detector experiments.
The first column shows the detector number, and the second column the number of
counts received on each detector over a 10 second period. The two files are nominally
from the same spectrum but shifted along the array.
Use a web browser to bring up the page
http://users.aber.ac.uk/dpl/ph15720/handouts/handouts.htm and at the bottom of the page
you should see a link to datafiles for exercise 8. Right click on the link and save the first
file on your M: drive with your other MathCAD files, use the default filename of
detectordata.txt. Repeat to save the second file in the same place, with the default
filename of detectordata2.txt. Once the files have finished downloading, close
the browser and return to MathCAD.
On a blank MathCAD worksheet, select “Insert|Component…” from the menu. This will
bring up a wizard where you should first indicate that you want to read data from a file.
On the next page of the dialog box you should leave the file type as ‘Text File’ and use
the browse button to locate the datafile you have stored on your M: drive in the previous
step. Once the wizard closes, your worksheet should show an icon representing a file read
component, with a little disk on it. The cursor will be positioned in a placeholder ready to
hold the name you wish to give the dataset. Once you have given it a name your
worksheet should look like this:
Now, using the techniques we started to explore with the resistor experiment and the
photoelectric effect, carry out the following analyses on the data. I have put some hints at
the bottom of the page which may help if you get stuck.


From the matrix of data, create a vector to hold the Detector number and another
to hold the counts.
Plot the counts against detector number.
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MathCAD Example Sheet 8
Create a plot of count rate vs detector, given that the sample period was 10
seconds.
Each detector is 20um x 2mm. Produce another graph with the plot rate scaled in
terms of counts per second per cm2.
There are 2 clear peaks in the data. Find the height of the largest peak in terms of
counts per second.
Create an expression which will give the number of the detector with the highest
counts.
Use the submatrix function to create a matrix containing the data from the highest
counting channel together with the 6 channels on either side.
Calculate the total number of counts in the highest peak and its adjacent channels
that you have just created above.
Create a function called PeakArea which takes as its arguments the channel
number and the width of the peak in channels. It is to return the total number of
counts of the given channel together with those up to 1 width away. For example,
PeakArea(42,3) will give the total counts from detectors 39,40,41,42,43,44 and
45.
Use the PeakArea function to re-calculate the area of the highest peak and the 6
channels either side.
Create a graph which will show the largest peak and a specified number of
adjacent channels on either side.
Go back to the file read component and right click on it, select “Choose File…”
from the pop-up menu and pick the file “DetectorData2.txt”. Click out of the file
read component and check that all of your answers and graphs change to reflect
the new data file.
Hints
The following hints may be useful to you. Use the help system and conduct some
experiments with MathCAD to see how.
You may want to create a unit for the micron (1 um is 10-6 m)
The max() function will tell you the largest number in a vector
The csort() function can be used to sort a matrix so that a particular column is in
ascending order.
You can reverse() the results of a sort.
You can use variables in the placeholders which determine the axis scaling of a graph.
The submatrix() function can be used to pick off part of a matrix.
The matrix toolbar has one way of summing the elements of a vector.
The calculus toolbar has two more ways of doing summations.
END
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