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integration 127✞/ / Trap . java trapezoid−rule integration of parabola☎public class Trap {public static final double A = 0. , B =3.; / / Constant endpointspublic static final int N = 100; // N points (not intervals )public static void main ( String [] args ) { / / Main does summationdouble sum , h , t , w;int i;h = (B − A) /(N − 1) ; // Initializationsum = 0 . ;for ( i=1 ; i
128 chapter 6evaluate the function at the two endpoints and in the middle (Table 6.1):∫ xi+hx i−h∫ xi+h ∫ xif(x) dx = f(x) dx + f(x) dxx i x i−h≃ h 3 f i−1 + 4h 3 f i + h 3 f i+1. (6.16)Simpson’s rule requires the elementary integration to be over pairs of intervals,which in turn requires that the total number of intervals be even or that the numberof points N be odd. In order to apply Simpson’s rule to the entire interval, we addup the contributions from each pair of subintervals, counting all but the first andlast endpoints twice:∫ baf(x)dx ≃ h 3 f 1 + 4h 3 f 2 + 2h 3 f 3 + 4h 3 f 4 + ···+ 4h 3 f N−1 + h 3 f N. (6.17)In terms of our standard integration rule (6.3), we have{ hw i =3 , 4h 3 , 2h 3 , 4h 3 , ..., 4h 3 , h }3(Simpson’s rule). (6.18)The sum of these weights provides a useful check on your integration:N∑w i =(N − 1)h. (6.19)i=1Remember, the number of points N must be odd for Simpson’s rule.6.2.3 Integration Error (Analytic Assessment)In general, you should choose an integration rule that gives an accurate answerusing the least number of integration points. We obtain a crude estimate of theapproximation or algorithmic error E and the relative error ɛ by expanding f(x) ina Taylor series around the midpoint of the integration interval. We then multiplythat error by the number of intervals N to estimate the error for the entire region[a, b]. For the trapezoid and Simpson rules this yields( ) [b − a]3E t = ON 2 f (2) ,( ) [b − a]5E s = ON 4 f (4) ,ɛ t,s = E t,sf . (6.20)−101COPYRIGHT 2008, PRINCET O N UNIVE R S I T Y P R E S SEVALUATION COPY ONLY. NOT FOR USE IN COURSES.ALLpup_06.04 — 2008/2/15 — Page 128
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−101COPYRIGHT 2008, PRINCET O N U
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Copyright © 2008 by Princeton Univ
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viiicontents2.3 Experimental Error
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xcontents6.3 Experimentation 1356.4
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xiicontents9.7.1 Friction: Model an
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xvicontents14.15.2Exercise 2: Cache
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xviiicontents18.4 Waves for Variabl
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xxcontentsC.3 DX Tools Summary 576C
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xxivprefacewhich includes understan
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2 chapter 1PhysicsCPFigure 1.1 A re
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4 chapter 1Scientific LibrariesPerf
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6 chapter 1C DWe have tried to make
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8 chapter 1possibly when installing
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10 chapter 11.4.1 Structured Progra
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12 chapter 17. Revise Area.java so
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14 chapter 1argv). Because main met
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16 chapter 1Package Class Tree Depr
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18 chapter 1Memory and storage size
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20 chapter 1TABLE 1.4The IEEE 754 S
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22 chapter 1gives 24-bit precision
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24 chapter 1TABLE 1.6Representation
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26 chapter 1The computer fetches th
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28 chapter 1Your problem is to use
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2Errors & Uncertainties in Computat
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32 chapter 2purposes, let us consid
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34 chapter 2can be avoided by combi
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36 chapter 2means that a program ru
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38 chapter 2To be more specific, le
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40 chapter 2Let us assume that an a
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42 chapter 2To see if these assumpt
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44 chapter 2computation. Accordingl
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46 chapter 3is regular practice to
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48 chapter 3to plot for x and y, we
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50 chapter 3Sample PtPlot Data file
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52 chapter 3TABLE 3.1Text Files Gra
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54 chapter 3Figure 3.4 Left: A plot
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56 chapter 3TABLE 3.2Grace Menu and
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58 chapter 33.4.1 Gnuplot Input Dat
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60 chapter 3gnuplot> set output "pl
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62 chapter 3gnuplot> set terminal e
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64 chapter 3By setting terminal to
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66 chapter 33.6 Texturing and 3-D I
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68 chapter 4R R 2RL L 2LC C 2CFigur
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70 chapter 44.3 Resistance Becomes
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72 chapter 4be either static or dyn
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74 chapter 4On line 8 we see a meth
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Page 183 and 184: 156 chapter 7Figure 7.3 Two example
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178 chapter 8Lagrange interpolation
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180 chapter 8Figure 8.3 Three fits
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182 chapter 8if you have the abilit
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184 chapter 840fitNumber20dataN(t)0
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186 chapter 8theoretical curve went
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188 chapter 8This is a measure of t
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190 chapter 88.7.4 Linear Quadratic
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192 chapter 8Your problem here is t
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9Differential Equation Applications
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196 chapter 9V(x)HarmonicAnharmonic
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198 chapter 9B(t) are solutions of
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200 chapter 9This expresses the acc
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202 chapter 9derivative:dy(t)dt≃
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204 chapter 9As an example of the u
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206 chapter 9C Dhigh-precision work
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208 chapter 9Amplitude Dependence,
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210 chapter 9Here N is the normal f
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212 chapter 99.9 Unit II. Binding A
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214 chapter 9the Schrödinger equat
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216 chapter 99.11.1 Numerov Algorit
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218 chapter 9}}xl = xl0+i∗h;ul[i]
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220 chapter 9public static double d
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222 chapter 99.13 Unit III. Scatter
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224 chapter 9The equations for the
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226 chapter 9Figure 9.9 The traject
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228 chapter 9fFigure 9.10 Left: The
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230 chapter 99.17.1.1 EXPLORATION:
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232 chapter 10y(t)10Y( )10-1t-10 20
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234 chapter 10As seen in the b n co
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236 chapter 1010.4 Fourier Transfor
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238 chapter 10Regardless of the act
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240 chapter 10for the exponential a
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242 chapter 10102 4 6-1Figure 10.2
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244 chapter 10coefficients a k . If
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246 chapter 104. As always, check t
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248 chapter 10A special case of the
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256 chapter 1010.8 Unit III. Fast F
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258 chapter 10Figure 10.9 The basic
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260 chapter 10TABLE 10.1Binary-Reve
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262 chapter 10✞/∗ FFT. java : F
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11Wavelet Analysis & Data Compressi
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266 chapter 11the second derivative
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268 chapter 1111.2.1 Wave Packet As
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270 chapter 11localized in time, su
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272 chapter 11201 Y100Signal0-10-10
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274 chapter 11you have been using f
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276 chapter 11}return Math . sin (8
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278 chapter 11FrequencyTimeFigure 1
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280 chapter 11N SamplesInputN/2(1)c
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282 chapter 11Figure 11.10 The filt
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wavelet analysis & data compression
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288 chapter 113. Reproduce the scal
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290 chapter 12discrete decay law,
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292 chapter 120.80.400 10 20Ax nn n
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296 chapter 121b=1.0b=4.0b=5.0X0-1-
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300 chapter 12✞/ / LyapLog . java
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302 chapter 1212.10 Unit II. Pendul
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304 chapter 12In Chapter 9, “Diff
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306 chapter 12Figure 12.6 The data
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308 chapter 12rotating solutionsθ2
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310 chapter 12θ(t)8040.θ(0)=0.314
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312 chapter 12Figure 12.11 Top row:
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314 chapter 12|θ(t)|200 1 2Figure
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Angular Velocity of Lower Pendulum3
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318 chapter 12some fraction of a ch
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320 chapter 1212.19 Lotka-Volterra
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322 chapter 12numbers that accounts
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324 chapter 12400200pPopulationpP00
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13Fractals & Statistical GrowthIt i
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328 chapter 1330010,000 points20010
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330 chapter 13Figure 13.2 Left: A f
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332 chapter 13copy of a frond, and
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334 chapter 13The results of this t
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336 chapter 13log N(r) = log C −
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338 chapter 13to determine the slop
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340 chapter 13✞☎import java . i
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342 chapter 13Figure 13.8 Number 8
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344 chapter 13Conway in the 1970s,
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346 chapter 13(x 0, y 1)(x 0, y 0)(
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348 chapter 13yxFigure 13.13 After
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350 chapter 13gradient // Vertical
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14High-Performance Computing Hardwa
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354 chapter 14A(1)A(2)A(3)CPUA(N)M(
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356 chapter 14ADBCFigure 14.4 Multi
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358 chapter 14as Fortran or C, tran
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360 chapter 14TABLE 14.2Computation
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362 chapter 14The processors in a p
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364 chapter 14Figure 14.7 Two views
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366 chapter 14Speedup8Amdahl's Lawp
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368 chapter 1414.11 Parallelization
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370 chapter 14The problem affects p
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372 chapter 14• A race condition
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374 chapter 14yet in order to obtai
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376 chapter 14slightly faster or sm
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378 chapter 14You see (Listing 14.1
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380 chapter 14/optimize:4/optimize:
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382 chapter 14• As indicated in
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384 chapter 1410,000Execution Time
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386 chapter 14as high-performance c
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388 chapter 14✞Dimension Vec( idi
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16Simulating Matter withMolecular D
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426 chapter 16+ +Figure 16.1 The mo
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428 chapter 16the molecules stay cl
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430 chapter 162 3 2 3 2 31 4 1 4 1
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432 chapter 16Velocity-Verlet Algor
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434 chapter 16Energy vs Timefor 36
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436 chapter 16Figure 16.7 A simulat
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thermodynamic simulations & feynman
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simulating matter with molecular dy
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17PDEs for Electrostatics & Heat Fl
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pdes for electrostatics & heat flow
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θpde waves: string, quantum packet
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pde waves: string, quantum packet,
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solitons & computational fluid dyna
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✞solitons & computational fluid d
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integral equations in quantum mecha
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✞integral equations in quantum me
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Appendix A: Glossaryabsolute value
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glossary 557control character — A
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glossary 559log in (on) — To sign
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glossary 561supercomputer — The c
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installing ptplot & java developer
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industrial-strength data visualizat
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Appendix D: An MPI TutorialIn this
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an mpi tutorial 595• Open a shell
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an mpi tutorial 597of all the compu
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an mpi tutorial 599TABLE D.1Some Co
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an mpi tutorial 601jobs to MPI. 2 A
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an mpi tutorial 603✞> cd $HOME Do
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an mpi tutorial 605✞/ / MPIhello
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an mpi tutorial 607Argument Namemsg
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an mpi tutorial 609D.3.4 Broadcast
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an mpi tutorial 611D.4 Parallel Tun
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an mpi tutorial 613✝}MPI_Recv ( &
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an mpi tutorial 615✞/ / Code list
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an mpi tutorial 617-1) does not sen
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an mpi tutorial 619D.6.1 Nonblockin
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an mpi tutorial 621D.9 List of MPI
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Appendix E: Calling LAPACK from CCa
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calling LAPACK from C 625E.2 Compil
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software on the CD 627JavaCodes Con
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software on the CD 629JavaCodes Con
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software on the CD 631Applets Direc
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software on the CD 633Fortran77code
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Appendix G: Compression via DWTwith
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compression via DWT with thresholdi
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compression via DWT with thresholdi
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BIBLIOGRAPHY[Abar 93] Abarbanel, H.
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ibliography 643[Erco] Ercolessi, F.
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ibliography 645[L&F 93] Landau, R.
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ibliography 647[P&W 91] Pinson, L.
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ibliography 649[VdeV 94] van de Vel
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IndexAbstract data structures,70Abs
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index 653drand, 114Driving force, 2
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index 655Libraries, see Subroutines
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index 657structured, 10for virtual
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