Tag: numerical integration

Gaussian quadrature and weights listed as scrapeable data

This file consists of Gauss quadrature points and weights from n=1 to n=30. The blank spaces separate out the value of n, quadrature points, and weights. The points and weights are for the integration limits of [-1,1]. To convert to any finite integration limits [a,b], multiply the weights by (b-a)/2 and use (b-a)/2*QuadPoint+(b+a)/2 for the quadrature points. The data is also given in a scrapeable textfile at https://mathforcollege.com/nm/blog/QuadPointsWeightsUpTo30.txt

 1

0

2.0

 2

-0.57735026918962576450914878050196
0.57735026918962576450914878050196

1.0
1.0

 3

-0.77459666924148337703585307995648
0
0.77459666924148337703585307995648

0.55555555555555555555555555555556
0.88888888888888888888888888888889
0.55555555555555555555555555555556

 4

-0.86113631159405257522394648889281
-0.33998104358485626480266575910324
0.33998104358485626480266575910324
0.86113631159405257522394648889281

0.347854845137453857373063949222
0.652145154862546142626936050778
0.652145154862546142626936050778
0.347854845137453857373063949222

 5

-0.90617984593866399279762687829939
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0
0.53846931010568309103631442070021
0.90617984593866399279762687829939

0.23692688505618908751426404071992
0.47862867049936646804129151483564
0.56888888888888888888888888888889
0.47862867049936646804129151483564
0.23692688505618908751426404071992

 6

-0.93246951420315202781230155449399
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0.93246951420315202781230155449399

0.17132449237917034504029614217273
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0.46791393457269104738987034398955
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0.36076157304813860756983351383772
0.17132449237917034504029614217273

 7

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0
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0.27970539148927666790146777142378
0.12948496616886969327061143267908

 8

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0.79666647741362673959155393647583
0.96028985649753623168356086856947

0.10122853629037625915253135430996
0.22238103445337447054435599442624
0.3137066458778872873379622019866
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0.3137066458778872873379622019866
0.22238103445337447054435599442624
0.10122853629037625915253135430996

 9

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0.83603110732663579429942978806973
0.96816023950762608983557620290367

0.081274388361574411971892158110524
0.18064816069485740405847203124291
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0.31234707704000284006863040658444
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0.26061069640293546231874286941863
0.18064816069485740405847203124291
0.081274388361574411971892158110524

10

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0.67940956829902440623432736511487
0.86506336668898451073209668842349
0.97390652851717172007796401208445

0.066671344308688137593568809893332
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0.21908636251598204399553493422816
0.26926671930999635509122692156947
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0.26926671930999635509122692156947
0.21908636251598204399553493422816
0.1494513491505805931457763396577
0.066671344308688137593568809893332

11

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0
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0.88706259976809529907515776930393
0.97822865814605699280393800112286

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0.055668567116173666482753720442549

12

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0.98156063424671925069054909014928

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0.2031674267230659217490644558098
0.16007832854334622633465252954336
0.106939325995318430960254718194
0.047175336386511827194615961485017

13

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0
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0.64234933944034022064398460699552
0.80157809073330991279420648958286
0.91759839922297796520654783650072
0.98418305471858814947282944880711

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0.22628318026289723841209018603978
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0.13887351021978723846360177686887
0.092121499837728447914421775953797
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14

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0.12151857068790318468941480907248
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0.035119460331751863031832876138192

15

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0
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0.84820658341042721620064832077422
0.93727339240070590430775894771021
0.98799251802048542848956571858661

0.030753241996117268354628393577204
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0.13957067792615431444780479451103
0.16626920581699393355320086048121
0.18616100001556221102680056186642
0.19843148532711157645611832644384
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0.10715922046717193501186954668587
0.070366047488108124709267416450667
0.030753241996117268354628393577204

16

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0.45801677765722738634241944298358
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0.86563120238783174388046789771239
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0.98940093499164993259615417345033

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17

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0.9506755217687677612227169578958
0.99057547531441733567543401994067

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18

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19

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20

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21

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0.10176238974840550459642895216855
0.10285265289355884034128563670542
0.10285265289355884034128563670542
0.10176238974840550459642895216855
0.099593420586795267062780282103569
0.09636873717464425963946862635181
0.092122522237786128717632707087619
0.086899787201082979802387530715126
0.08075589522942021535469493846053
0.073755974737705206268243850022191
0.065974229882180495128128515115962
0.057493156217619066481721689402056
0.048402672830594052902938140422808
0.038799192569627049596801936446348
0.028784707883323369349719179611292
0.018466468311090959142302131912047
0.0079681924961666056154658834746736

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Holistic Numerical Methods Open Course Ware:

Numerical Methods for the STEM undergraduate at http://nm.MathForCollege.com;

Introduction to Matrix Algebra for the STEM undergraduate at http://ma.MathForCollege.com

the textbooks on

Numerical Methods with Applications

Introduction to Programming Concepts Using MATLAB

the Massive Open Online Course (MOOCs) available at

Numerical Methods 

Introduction to Matrix Algebra

An example of Gaussian quadrature rule by using two approaches

Here is an example of using Gaussian quadrature rule through two approaches:

EITHER

by applying it on the original integrand by updating the argument of the integrand

OR

by applying it to the equivalent integrand because of the need to change the limits of integration to: -1 to 1.

http://nm.MathForCollege.com/blog/3pointquadruleexample.pdf

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A video tutorial on Simpson’s 1/3 rule

Simpson’s 1/3 rule is a popular method of conducting numerical integration.  We have recorded a series of short videos on this topic and they are avilable as a playlist at http://www.youtube.com/user/numericalmethodsguy#g/c/2A25C3DC6D8E5616.

This post is brought to you by Holistic Numerical Methods: Numerical Methods for the STEM undergraduate at http://nm.mathforcollege.com, the textbook on Numerical Methods with Applications available from the lulu storefront, and the YouTube video lectures available at http://nm.mathforcollege.com/videos and http://www.youtube.com/numericalmethodsguy

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Experiment for spline interpolation and integration

Background:

The motivation behind the experiment is to understand spline interpolation and numerical integration by finding the volume of water that can be held by a champagne glass.

What does the student do in the lab:

The student chooses one of the odd-shaped champagne glasses (Figure 1). The student measures the outer radius of the champagne glass at different known locations along the height. The student measures the thickness of the glass, so that he/she will be able to find the inner radius of the champagne glass at the locations he/she measured the outer radius. The student pours water to the brim in the champagne glass and checks how much volume the champagne glass holds.

Champagne GlassExercises assigned to the students:
Use MATLAB to solve problems. Use comments, display commands and fprintf statements, sensible variable names and units to explain your work. Staple all the work in the following sequence. Use USCS system of units throughout.

  1. Attach the data sheet on which you collected the data in class.
  2. Find the spline interpolant that curve fits the radius vs height data.
  3. Show the individual points and the spline interpolant of radius vs height on a single plot.
  4. Find how much volume of water the champagne glass would hold.
  5. Compare the above result from problem#4 to the actual volume.
  6. In 100-200 words, type out your conclusions using a word processor. Any formulas should be shown using an equation editor. Any sketches need to be drawn using a drawing software such as Word Drawing. Any plots can be imported from MATLAB.

What materials do you need; where do I buy it; how much do the materials costs?

  1. Champagne Glasses: These glasses, called the Hurricane Plastic Glasses, are available at www.poolsidepineapple.com, part nos. HUR-105, HUR-106, YAR-114. We used glasses made of plastic to avoid breakage. http:/www.poolsidepineapple.com/cart_pages/shopping%20page%20tropical.htm. You can try other places to buy the champagne glasses. About $40 or so for about six pieces including S&H. Better yet, go to a cruise and get souvenir glasses. Whenever you do the experiment, you will remember the good times.
  2. Graduated Cylinder: The graduated cylinder is available at http://scientificsonline.com/, part number 3036286. The cost of the cylinder is $20+S&H.
  3. Vernier Caliper: The caliper is available at http://mcmaster.com part number 20265A49. The cost of the vernier caliper is $60+S&H.
  4. Scale: Need to buy a thin scale for this. Any art-supplies store for $2 or so.

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This post is brought to you by Holistic Numerical Methods: Numerical Methods for the STEM undergraduate at http://nm.mathforcollege.com

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