TKK2144 1314 Semester 1 Instructor Rama Oktavian Email ramaoktavianubacid Office Hr M1315 W 1315 Th 1315 F 1315 Outlines 1 Interpolation Basic Theory 2 ID: 321091
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Slide1
Computer Programming(TKK-2144)
13/14 Semester 1
Instructor: Rama
Oktavian
Email: rama.oktavian@ub.ac.id
Office Hr.: M.13-15, W. 13-15 Th. 13-15,
F. 13-15Slide2
Outlines
1.
Interpolation: Basic Theory
2. Methods of interpolation
3. Interpolation in Excel
4. Interpolation in chem.engSlide3
Interpolation (basic theory)
Interpolation function
a function that passes exactly through a set of data points.
Interpolating functions to interpolate values in tables
find functional values at other values of the independent
variable, e.g. sin(0.63253)Slide4
Interpolation (basic theory)
Interpolation
Linear Interpolation
Polynomial Interpolation
Cubic
Spline Interpolation Lagrange Polynomial Interpolation Slide5
Interpolation methods
Linear Interpolation
Linear interpolation is obtained by passing a straight line between2 data points
the exact function for which values are known only at a discrete set of data points
the interpolated approximation to
the data points(also referred to as interpolation points or nodes)Slide6
Interpolation methods
Linear Interpolation
In tabular form:Slide7
Interpolation methods
Linear Interpolation
If is a linear function then
To pass through points and Slide8
Interpolation methods
Linear Interpolation
Advantages:
• Easy to use (homework, exams)
• Exact for linear functions
Disadvantages:• Not very accurate for nonlinear functionsSlide9
Interpolation methods
Linear Interpolation
Sin(
0.63
) = ??Slide10
Interpolation methods
Error in Linear Interpolation
Error is defined as:Slide11
Interpolation methods
Polynomial Interpolation
Two steps:
• obtain polynomial coefficients
• evaluate the value of the polynomial at the desired location (x
i
)For N data points, there is a unique polynomial (usually of order n-1) that goes through each pointThis is an interpolating polynomial, because it goes exactly through each data pointSlide12
Interpolation methods
Polynomial Interpolation
First step:
• obtain polynomial coefficients
For general
n, the goal is to determine the coefficients a1, a2,…,an so that for all i
= 1:nPn-1(x) = a1 + a2x + a
3
x
2
+…+ a
n
x
n-1
Writing these equations in matrix-vector form, we obtain
Vandermonde
matrixSlide13
Interpolation methods
Polynomial Interpolation
Second step:
• evaluate the value of the polynomial at the desired location (xi)
P
n-1(xi) = a1 + a2
xi + a3xi2 +…+ anxin-1Sin(
0.63
) = ??Slide14
Interpolation in excel
Lookup Tables
Often, we need to retrieve data that is stored in a table
For example, consider these tables:Slide15
Interpolation in excel
LOOKUP(Lookup_value,Lookup_vector,Result_vector)
search down
the lookup
vector
find either
a matching
value, or
a value
greater than
the lookup
value, whichever
occurs first
go to the same position in
the result vector, and extract
the value thereSlide16
Interpolation in excel
Metal
Modulus of Elasticity, psi
Density, lb/in
3
Yield
Strength, psi
Aluminum 2014-T6
10,600,000
0.101
60,000
Aluminum 6061-T6
10,000,000
0.098
37,000
Stainless Steel 304
28,000,000
0.284
30,000
Structural Steel A36
29,000,000
0.284
36,000
Since the properties to be retrieved are in columns, the table is called a vertical lookup table
The vertical lookup table command in Excel is:
VLOOKUP(lookup value, table range, column number, true/false)Slide17
Interpolation in excel
VLOOKUP(Lookup_value,Table_array,col_index_num,Range_lookup)
search
down
left
column
of
table
array
to
find
match
Which column number for
extraction of result? 1, 2, 3 or 4?
result
optional argument:
if FALSE, requires
an exact matchSlide18
Interpolation in excel
ExamplesSlide19
Interpolation in excel
Examples
Using VLOOKUP to find the freezing point of a 33% solution
The formula
=VLOOKUP(F3,$A$3:$D$54,2,0)Slide20
Interpolation in excel
Other ways to Lookup Tables
Match(
Lookup_value,
lookup_array, match type) Returns the relative position in the array that matches the lookup value. Index(range, position) Returns the value from the range in the position specifiedSlide21
Interpolation in excel
Example
=MAX($B$5:$B$l6)
83119Slide22
Interpolation in excel
Example
use the MATCH function to find the position of the maximum value in the range
MATCH(
lookup_value,lookup_array,match_type_num
)=MATCH(83119,$B$5:$B$16,0)Slide23
Interpolation in excel
Example
use the INDEX function to return the value in the same position in the array of months:
=INDEX( $A$5:$A$16,4)
=INDEX( $A$5:$A$16, MATCH( MAX( $B$5: $B$16), $B$5:$B$16,0))Slide24
Interpolation in excel
Lookup value for two-way tablesSlide25
Interpolation in excel
Lookup value for two-way tables
=VLOOKUP(
Temp,Table
, MATCH( Percent, P-Row, 1 )+ 1,1)
=VLOOKUP( M2, $A$4:$1$32, MATCH( N2, $B$3:$K$3,1 )+1, 1) Slide26
Interpolation in excel
A Note About Lookup Tables
Lookup tables will
not
interpolate values!When looking up a numerical value, if an exact match is not found (and the “TRUE” option allows for an approximate match), then the value searched for is rounded down to the next tabulated value.
We will illustrate this in the following exampleSlide27
Interpolation in excel
ExampleThe population data of a town is given in the table
We want to use a lookup table to report the population for any year enteredSlide28
Interpolation in excel
ExampleNote that using the VLOOKUP function (with the TRUE/FALSE option left off or set to TRUE) returns the population for the next year in the table
lower
than the input value:Slide29
Interpolation in excel
Linear interpolation in excel
For each population interval, construct an IF statement to see if the input year falls within the interval range…
If the input year is within the interval, then the interpolation formula should be used…
If the input year is not within the interval, then a space is entered into the cell to make it blank.Slide30
Interpolation in excel
Linear interpolation in excelIF statements add a great deal of flexibility to Excel spreadsheets – allow control over how calculations are performed
Lookup tables are convenient for accessing data that is stored in table form. Important to remember that lookup tables do not interpolate between values
Linear interpolation formulas can be added to tables in order to provide a better estimate of values between tabulated pointsSlide31
Interpolation in excel
Application of VLOOKUP: Linear interpolation from a table of physical properties
For any %HNO
3
from 0 to 90, we
want an automaticestimate of theheat capacitySlide32
Interpolation in excel
Application of VLOOKUP: Linear interpolation from a table of physical properties
Create index columns to the right and left of the tableSlide33
Interpolation in excel
Application of VLOOKUP: Linear interpolation from a table of physical properties
Enter VLOOKUP functions based on “NitricTable2” and using the low and high row indices to extract the bracketing values of %HNO
3
and Heat Capacity.Slide34
Interpolation in excel
LowPct
HighPct
LowHC
HighHC
linear approximation
between to locations
in the table
input
%HNO
3
HeatCap
estimateSlide35
Interpolation in excel
Automatic linear interpolationformula and resultsSlide36
Interpolation in excelChange values of the input to get different results
[including bad results when input outside of acceptable range]
the good
the bad
and the ugly!Slide37
Interpolation in excel
Linear Interpolation using TREND functionTREND can be used to perform linear interpolation between two adjacent data points.
TREND(
known_y's
, known_x's
, new_x 's, const)=TREND( B20: B21 , A20:A21, F18,1)Slide38
Interpolation in excel
Linear Interpolation in a Two-way TableFind the viscosity value in the table for x = 76°F, y = 56.3 wt% ethylene glycolSlide39
Interpolation in excel
Linear Interpolation in a Two-way TableFirst step, interpolate the value at y = 56.3
Second step, interpolate the value at x = 76Slide40
Extrapolation
DefinitionThe process of taking data values at points x1, ..., xn
, and approximating a value outside the range of the given points.
The process of estimating and forecasting
Sin (2.6) = ??Slide41
Extrapolation
MethodsLinear extrapolation
If the two data points nearest the point
to be extrapolated are
and
, linear extrapolation gives the function:
Polynomial extrapolationSlide42
Extrapolation
MethodsLinear extrapolation
Given the following data which is known to be linear, extrapolate the
y
value when
x = 2.3.
(0.3 0.80), (0.7, 1.3), (1.2, 2.0), (1.8, 2.7) Slide43
Interpolation in Chem. EngExample - Vapor Pressure of Water
Determine vapor pressure of water at:• 25 °C
• 92 °C
• 105 °CSlide44
Interpolation in Chem. EngExample - “Real” Gases
p
- pressure
z
- compressibility factor
R- gas constantT- temperature - Molar volumeFind z at T
= 725K and p = 8bar.Slide45
Thank You !