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Computer Programming (TKK-2144)

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13/14 Semester 1

Computer Programming(TKK-2144)

Instructor: Rama Oktavian

Email: [email protected]

Office Hr.: M.13-15, W. 13-15 Th. 13-15, F. 13-15

1. Interpolation: Basic Theory

2. Methods of interpolation

3. Interpolation in Excel

4. Interpolation in chem.eng

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)

Interpolation

- Linear Interpolation
- Polynomial Interpolation
- Cubic Spline Interpolation
- Lagrange Polynomial Interpolation

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)

Linear Interpolation

In tabular form:

Linear Interpolation

If is a linear function then

To pass through points and

Linear Interpolation

Advantages:

• Easy to use (homework, exams)

• Exact for linear functions

Disadvantages:

• Not very accurate for

nonlinear functions

Linear Interpolation

Sin(0.63) = ??

Error in Linear Interpolation

Error is defined as:

Polynomial Interpolation

For N data points, there is a unique polynomial (usually of order n-1) that goes through each point

This is an interpolating polynomial, because it goes exactly through each data point

Two steps:

• obtain polynomial coefficients

• evaluate the value of the polynomial at the desired location (xi)

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:n

Pn-1(x) = a1 + a2x + a3x2 +…+ anxn-1

Writing these equations in matrix-vector form, we obtain

Vandermonde matrix

Polynomial Interpolation

Second step:

• evaluate the value of the polynomial at the desired location (xi)

Pn-1(xi) = a1 + a2xi + a3xi2 +…+ anxin-1

Sin(0.63) = ??

Lookup Tables

- Often, we need to retrieve data that is stored in a table
- For example, consider these tables:

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 there

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)

VLOOKUP(Lookup_value,Table_array,col_index_num,Range_lookup)

Which column number for

extraction of result? 1, 2, 3 or 4?

search

down

left

column

of

table

array

to

find

match

optional argument:

if FALSE, requires

an exact match

result

Examples

Examples

Using VLOOKUP to find the freezing point of a 33% solution

The formula

=VLOOKUP(F3,$A$3:$D$54,2,0)

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 specified

Example

=MAX($B$5:$B$l6)

83119

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)

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))

Lookup value for two-way tables

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)

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 example

A Note About Lookup Tables

The population data of a town is given in the table

We want to use a lookup table to report the population for any year entered

Example

Note 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:

Example

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.

IF 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 points

Linear interpolation in excel

Application of VLOOKUP: Linear interpolation from a table of physical properties

For any %HNO3

from 0 to 90, we

want an automatic

estimate of the

heat capacity

Application of VLOOKUP: Linear interpolation from a table of physical properties

Create index columns to the right and left of the table

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 %HNO3 and Heat Capacity.

linear approximation

between to locations

in the table

LowHC

HeatCap

estimate

HighHC

input

%HNO3

HighPct

LowPct

Automatic linear interpolation

formula and results

Change values of the input to get different results

[including bad results when input outside of

acceptable range]

the good

the bad

and the ugly!

Linear Interpolation using TREND function

TREND 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)

Linear Interpolation in a Two-way Table

Find the viscosity value in the table for x = 76°F, y = 56.3 wt% ethylene glycol

Linear Interpolation in a Two-way Table

First step, interpolate the value at y = 56.3

Second step, interpolate the value at x = 76

Definition

The 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) = ??

Methods

Linear extrapolation

If the two data points nearest the point to be extrapolated are and , linear extrapolation gives the function:

Polynomial extrapolation

Methods

Linear 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)

Example - Vapor Pressure of Water

Determine vapor pressure of water at:

• 25 °C

• 92 °C

• 105 °C

Example - “Real” Gases

Find z at T = 725K and p = 8bar.

p- pressure

z- compressibility factor

R- gas constant

T- temperature

- Molar volume

Thank You !