Computer Programming(TKK-2144)

13/14 Semester 1

Instructor: Rama OktavianEmail: rama.oktavian@ub.ac.idOffice Hr.: M.13-15, W. 13-15 Th. 13-15, F. 13-15

Outlines

1. Interpolation: Basic Theory

2. Methods of interpolation

3. Interpolation in Excel

4. Interpolation in chem.eng

Interpolation (basic theory)Interpolation functiona 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 independentvariable, e.g. sin(0.63253)

Interpolation (basic theory)Interpolation

1. Linear Interpolation

2. Polynomial Interpolation

3. Cubic Spline Interpolation

4. Lagrange Polynomial Interpolation

Interpolation methodsLinear Interpolation Linear interpolation is obtained by passing a straight line between2 data points

)(xf the exact function for which values are known only at a discrete set of data points

)(xg the interpolated approximation to )(xf

10 , xx the data points(also referred to as interpolation points or nodes)

Interpolation methodsLinear Interpolation

In tabular form:

Interpolation methodsLinear Interpolation

If is a linear function then)(xg bAxxg )(

To pass through points and ))(,( 00 xfx ))(,( 11 xfx

)()()()( 0101

00 xfxf

xxxxxfxg

Interpolation methodsLinear Interpolation

Advantages:• Easy to use (homework, exams)• Exact for linear functions

Disadvantages:• Not very accurate for nonlinear functions

Interpolation methodsLinear Interpolation

Sin(0.63) = ??

Interpolation methodsError in Linear Interpolation

Error is defined as:

Interpolation methodsPolynomial Interpolation

Two steps:

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

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 point

Interpolation methodsPolynomial Interpolation First step:• obtain polynomial coefficientsFor general n, the goal is to determine the coefficients a1, a2,…,an so that for all i = 1:n

iin yxp )(1

Pn-1(x) = a1 + a2x + a3x2 +…+ anxn-1

Writing these equations in matrix-vector form, we obtain

Vandermonde matrix

Interpolation methodsPolynomial Interpolation Second step:• evaluate the value of the polynomial at the desired location (xi)

iin yxp )(1

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

Sin(0.63) = ??

Interpolation in excelLookup Tables • Often, we need to retrieve data that is stored in a table• For example, consider these tables:

Interpolation in excelLOOKUP(Lookup_value,Lookup_vector,Result_vector)

search downthe lookupvector

find eithera matchingvalue, ora valuegreater thanthe lookupvalue, whicheveroccurs first go to the same position in

the result vector, and extractthe value there

Interpolation in excelMetal Modulus of

Elasticity, psi Density, lb/in3 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)

Interpolation in excelVLOOKUP(Lookup_value,Table_array,col_index_num,Range_lookup)

searchdownleftcolumnoftablearraytofindmatch

Which column number forextraction of result? 1, 2, 3 or 4?

result

optional argument:if FALSE, requiresan exact match

Interpolation in excelExamples

Interpolation in excelExamplesUsing VLOOKUP to find the freezing point of a 33% solution

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

Interpolation in excelOther 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

Interpolation in excelExample

=MAX($B$5:$B$l6) 83119

Interpolation in excelExample

use the MATCH function to find the position of the maximum value in the rangeMATCH(lookup_value,lookup_array,match_type_num)=MATCH(83119,$B$5:$B$16,0)

Interpolation in excelExample

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))

Interpolation in excelLookup value for two-way tables

Interpolation in excelLookup 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)

Interpolation in excelA 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 example

Interpolation in excelExample

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

Interpolation in excelExample

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:

Interpolation in excelLinear interpolation in excelFor 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.

Interpolation in excelLinear interpolation in excel

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

Interpolation in excelApplication of VLOOKUP: Linear interpolation from a table of physical properties

For any %HNO3from 0 to 90, wewant an automaticestimate of theheat capacity

Interpolation in excelApplication of VLOOKUP: Linear interpolation from a table of physical properties

Create index columns to the right and left of the table

Interpolation in excelApplication of VLOOKUP: Linear interpolation from a table of physical propertiesEnter VLOOKUP functions based on “NitricTable2” and using the low and high row indices to extract the bracketing values of %HNO3 and Heat Capacity.

Interpolation in excel

LowPct HighPct

LowHC

HighHC

linear approximationbetween to locationsin the table

input%HNO3

HeatCapestimate

input LowPct

HeatCap HighHC LowHC LowHCHighPct LowPct

Interpolation in excelAutomatic linear interpolationformula and results

Interpolation in excelChange values of the input to get different results

[including bad results when input outside ofacceptable range]

the good the bad and the ugly!

Interpolation in excelLinear 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)

Interpolation in excelLinear Interpolation in a Two-way Table

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

Interpolation in excelLinear Interpolation in a Two-way Table

First step, interpolate the value at y = 56.3

Second step, interpolate the value at x = 76

ExtrapolationDefinition

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) = ??

ExtrapolationMethods

Linear extrapolation

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

Polynomial extrapolation

ExtrapolationMethods

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)

Interpolation in Chem. EngExample - Vapor Pressure of Water

Determine vapor pressure of water at:• 25 °C• 92 °C• 105 °C

Interpolation in Chem. EngExample - “Real” Gases

zRTVp ˆp- pressurez- compressibility factorR- gas constantT- temperature - Molar volumeV̂

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