PH15010
Laboratory Techniques An Introduction to MATHCAD
Introduction
•
•
•
•
Review of last week
Vector and Matrix Maths
Statistics
Solving simultaneous equations
with matrices
Review of Last Week
• Processing experimental data
with arrays
• Adding Error bars to graphs
• Pre-Processing Data
– Convert l to f
– Logarithms for power law
• Extract slope & intercept from
data
Functions Reviewed
• Take one or more inputs
(parameters) and return a value
or values
• From f(x) toolbar if you don’t
know the name
• Afunction(parameters)=
• Avariable:=Afunction(p1,p2,p3…)
• Use the help system/f(x) dialog
Matrix & Vector Maths
• Used for:
–
–
–
–
–
–
Processing experimental data
Statistics
Resolving forces etc in 2D, 3D
Co-ordinate transformations
Simultaneous Equations
Applying transforms to data
Creating Arrays in
MathCAD
• 2 ways of entering arrays
– Input Table (already encountered)
• Spreadsheet type entry
• Expand to fit data available
– Insert Matrix Dialog box
• Array of placeholders
• Good if you know how many elements
in matrix/vector
Creating an array with
Insert|Matrix…
• Same for defining ordinary
variable
• Follow : with
– [:::] from toolbar
– <ctrl-m>
– Insert|Matrix
• Dialog box for rows & columns
Creating arrays with
Insert|Matrix…
Empty 3 – element
vector
MyVector
1
MyVector
2
Placeholders to fill in
3
1
MyVector
2  mm
3
Can use units
Matrix Maths #1
VA
1
VA  3 
2
MA 1 
3
3
MA 3 
6
VA  VB  11
MA MB 
19 22
MA VA 
5
2 3
4 5
3 6
9 12
Addition of constant
Multiplication by scalar
Dot product of 2 vectors
Dot product of 2 matrices
43 50
11
Dot product of matrix and vector
Matrix Maths #2
(Symbolically)
a b
c d

y
a b
a d
c d
a b
c d
a x
c x
x
K
• Matrix
Multiplication
b y
d y
b c
• Determinant
a
K b
K
c
K d
K
• Addition of
constant
Matrix & Vector Maths
• Add, Multiply & Divide
matrices/vectors
• Must have correct numbers of
rows & columns
Addressing Arrays with
Indices
• Index addressing allows access
to individual elements of array
• Examine & Set elements
• Indices start at 0
• Indices use [ key
• 2D matrices use two indices for
row, column selection
Indices #1
Vector Example
VA
12
666
VA0  12
VA0
VA 
42
42
666
VA1  666
• Simple 2
element vector
• Examine
elements
• Set element 0
• Result of setting
Indices #2
Matrix Example
1
MA
2
3
999 42 666
MA0  0  1
MA0  1  2
MA0  3 
MA1  0
MA 
27
1
2
3
27 42 666
MA0  2  3
• Sample matrix
• Examine
elements
• Subscript too big
• Set element
• Result
Accessing Parts Of
Arrays
• Index addressing – single
element 
• Column Extract operator M<> 
• Submatrix() function
• Transposing matrices
The submatrix() function
1

2

MyMatrix
3

4
1
1 
4
8 
27 
9

Define a matrix

16 64 
Matrix to operate on
 1 1 
submatrixMyMatrix0
(
  2  1  2)   4 8 
 9 27 


Rows
Columns
Use submatrix()
to extract part
of the array
Transpose Rows &
Columns
1

2

MyMatrix
3

4
1
1 

4 8 
9 27 

16 64 
 1 2 3 4 
T
MyMatrix   1 4 9 16 
 1 8 27 64 


• Uses MT
operator
• From matrix
toolbar
Other matrix functions
1. Information
• Return information about:
– Size of matrix
• rows(M), cols(M), last(V), length(V)
– Contents of matrix
• min(A), max(A)
• V
Other matrix functions
2. Manipulation
• Create new arrays
– Put 2 arrays together
• stack(A1,A2)
• augment(A1,A2)
– Sorting
• sort(V), reverse(A)
• rsort(M,r), csort(M,c)
Statistical Functions
on Vectors #1
• From f(x) dialog box under
“Statistics”
• Averaging
– mean() ‘Average value’
– median() ‘Half way item’
– mode() ‘Most common value’
Statistical Functions
on Vectors #2
• Variance & Standard Deviation
• 2 forms of each
• Analysis of sample
– Stdev(), Var()
• Analysis of whole population
– stdev(), var()
• Note capitalisation
Vectorise Operator #1

f (x)
• From matrix toolbar
• Forces evaluation on element by
element basis
• Overrides normal matrix maths

f (x)
– ‘dot’ product
– Determinant |M|
– Powers Mx
Vectorise Operator #2
1
• Dot product
4
2  5  32
3
1 4
6
2 5
3  6  32
4
4
2  5
 10
1
3
6
18
• With vectorise
operator
Range Variables
• Take on a series of values
• Define using semicolon [;]
i
0  10
Start
i
5
Finis h 15
Start Finis h
Range Variables #2
• Restrictions on where used
• ‘illegal context’ error message
• Useful for filling arrays
Filling Arrays using RVs
• Define RV
• Use RV in array index &
expression
i
0  3
timei
time 
i 10 min
0
0
600
10
3
1.2  10
1.8  10
3
s
change the units to give:
time 
20
30
min
Processing Arrays using
RVs
• Use RV in expressions to create
output array from input array
Range Variables with
different step sizes
• Can create RVs with different
step sizes.
• Enter first 2 values separated by
comma [,] before typing [;]
RV
First Second Last
Summation #1
• Summation operator on matrix
palette
• Sum of all elements in a vector
V
Summation #2
• Operators on Calculus palette
• Do summation on any expression
• 2 forms
– Plain => defines local RV
– Range Variable => uses existing RV
m
 Expression
n l
 Expression
n
Summation #3
5
i  15
• Sum of first 5
integers
Countsi  9
• Sum of elements
20-30 of vector
Counts
i= 1
30
i = 20
Summation #4
Reciprocals of factorials
• 2 terms
1
1
i
 2
i= 0
• 4 terms
3
1
i
 2.667
i= 0
• 9 terms
8
1
i
i= 0
 2.718
Data Files
• Used Input table to put data into
MathCAD by hand.
• Can use Import on input table to
get from a file (makes a copy)
– Select table & right-click
• Also can link to data file with
File Read Component (preferred)
Reading Data Files #1
• File Read Component
• Reads from datafile  array
• Text files
– Numbers separated by comma or
tab
• Other formats supported
Reading Data Files #2
• Insert|Component…|File Read or
Write
• Wizard:
– (Select file type)
– Browse for file
– Finish
• Give name for array
Reading Data Files #3
• Example
This weeks worksheet
•
•
•
•
•
Arrays & Indices
Columns (Revision)
Submatrix() function
Other array functions
Reading data from file