NUMERICAL METHODS AND STATISTICAL TECHNIQUES USING C

NUMERICAL METHODS AND STATISTICAL TECHNIQUES USING ‘C’ NUMERICAL METHODS AND STATISTICAL TECHNIQUES USING ‘C’ By MANISH GOYAL M.Sc. (Mathematics)...
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NUMERICAL METHODS AND

STATISTICAL TECHNIQUES USING ‘C’

NUMERICAL METHODS AND

STATISTICAL TECHNIQUES USING ‘C’

By MANISH GOYAL M.Sc. (Mathematics) Programme Coordinator Department of Applied Sciences & Humanities G.L.A. Institute of Technology and Management Mathura (U.P.)

LAXMI PUBLICATIONS (P) LTD BANGALORE

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CHENNAI

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COCHIN

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GUWAHATI

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HYDERABAD

JALANDHAR

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KOLKATA

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LUCKNOW

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MUMBAI

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RANCHI

NEW DELHI

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Contents Chapter

Pages

1.

1—24

INTRODUCTION 1.1. 1.2. 1.3. 1.4. 1.5. 1.6. 1.7. 1.8. 1.9. 1.10. 1.11. 1.12. 1.13. 1.14. 1.15. 1.16. 1.17. 1.18.

2.

Introduction to Computers ........................................................................................ 4 Definitions ................................................................................................................... 4 Introduction to ‘‘C’’ Language ................................................................................... 6 Advantages/Features of ‘C’ Language ...................................................................... 6 ‘C’ Character Set ......................................................................................................... 6 ‘C’ Constants ............................................................................................................... 7 ‘C’ Variables ................................................................................................................ 8 ‘C’ Key Words .............................................................................................................. 9 ‘C’ Instructions ............................................................................................................ 9 Hierarchy of Operations .......................................................................................... 10 Escape Sequences ..................................................................................................... 10 Basic Structure of ‘C’ Program ................................................................................ 10 Decision Making Instructions in ‘C’– ...................................................................... 12 Loop Control Structure ............................................................................................ 14 Arrays and String ..................................................................................................... 15 Pointers ..................................................................................................................... 16 Structure and Unions ............................................................................................... 16 Storage Classes in ‘C’ ............................................................................................... 17

ERROR ANALYSIS AND ESTIMATION 2.1. 2.2. 2.3. 2.4. 2.5. 2.6. 2.7. 2.8. 2.9. 2.10. 2.11. 2.12. 2.13.

25—60

Errors and their Analysis ........................................................................................ 27 Accuracy of Numbers ............................................................................................... 28 Errors ........................................................................................................................ 29 A General Error Formula ........................................................................................ 36 Errors in Numerical Computations ........................................................................ 36 Error in a Series Approximation ............................................................................. 45 Mathematical Preliminaries .................................................................................... 48 Floating Point Representation of Numbers ........................................................... 48 Arithmetic Operations with Normalized Floating Point Numbers ...................... 50 Numerical Instability ............................................................................................... 56 Ill-Posed Problem ..................................................................................................... 56 Machine Computation .............................................................................................. 57 Computer Software .................................................................................................. 57 ( vii )

( viii ) 3.

ALGEBRAIC AND TRANSCENDENTAL EQUATIONS 3.1. 3.2. 3.3. 3.4. 3.5. 3.6. 3.7. 3.8. 3.9. 3.10. 3.11. 3.12. 3.13. 3.14. 3.15. 3.16. 3.17. 3.18. 3.19. 3.20. 3.21. 3.22. 3.23. 3.24. 3.25. 3.26. 3.27. 3.28. 3.29. 3.30. 3.31. 3.32. 3.33. 3.34. 3.35. 3.36. 3.37. 3.38. 3.39. 3.40.

61—162

Bisection Method ...................................................................................................... 63 Algorithm .................................................................................................................. 64 Flow-chart ................................................................................................................. 65 Program Writing ....................................................................................................... 66 Order of Convergence of Iterative Methods ........................................................... 66 Order of Convergence of Bisection Method ............................................................ 66 Convergence of a Sequence ...................................................................................... 66 Prove that Bisection Method Always Converges ................................................... 67 Program to Implement Bisection Method .............................................................. 69 Iteration Method—(Successive Approximation Method) ...................................... 78 Sufficient Condition for Convergence of Iterations ............................................... 78 Theorem .................................................................................................................... 78 Convergence of Iteration Method ............................................................................ 78 Algorithm for Iteration Method .............................................................................. 79 Flow-chart for Iteration Method ............................................................................. 80 Computer Program ................................................................................................... 80 The Method of Iteration for System of Non-Linear Equations ............................. 90 Method of False Position or Regula-Falsi Method ................................................. 91 Algorithm .................................................................................................................. 92 Flow-chart ................................................................................................................. 93 Convergence of Regula-Falsi Method ................................................................... 105 Secant Method ........................................................................................................ 107 Convergence of Secant Method ............................................................................. 108 Polynomial Equations ............................................................................................ 110 Descartes’ Rule of Signs ......................................................................................... 110 Sturm Sequence ...................................................................................................... 110 Lin-Bairstow’s Method or Method for Complex Root .......................................... 112 Muller’s Method ...................................................................................................... 117 Algorithm of Muller’s Method ............................................................................... 117 Flow-chart for Muller’s Method............................................................................. 119 The Quotient-difference Method ........................................................................... 126 Horner’s Method ..................................................................................................... 129 Newton-Raphson Method ...................................................................................... 131 Convergence ............................................................................................................ 131 Order of Convergence............................................................................................. 132 Geometrical Interpretation ................................................................................... 133 Algorithm of Newton-Raphson Method ................................................................ 133 Flow-chart of Newton–Raphson Method .............................................................. 134 Newton’s Iterative Formulae for Finding Inverse, Square Root etc. ................. 135 Chebyshev Method ................................................................................................. 149

( ix ) 3.41. 3.42. 3.43. 3.44. 3.45. 3.46. 3.47. 3.48. 4.

Aitken’s ∆2 Method ................................................................................................. 150 Definitions ............................................................................................................... 151 Methods for Multiple Roots ................................................................................... 151 Nearly Equal Roots ................................................................................................ 154 Comparison of Newton’s Method with Regula-Falsi Method ............................. 156 Comparison of Iterative Methods .......................................................................... 156 Graeffe’s Root-Squaring Method ........................................................................... 157 Ramanujan’s Method ............................................................................................. 161

INTERPOLATION

163—308

4.1. 4.2. 4.3. 4.4. 4.5. 4.6. 4.7. 4.8. 4.9.

Introduction ............................................................................................................ 165 Assumptions for Interpolation .............................................................................. 165 Errors in Polynomial Interpolation....................................................................... 165 Finite Differences ................................................................................................... 167 Other Difference Operators ................................................................................... 169 Relation Between Operators ................................................................................. 169 Differences of a Polynomial ................................................................................... 170 Factorial Notation .................................................................................................. 183 To show that (i) ∆n [x]n = n ! (ii) ∆n+1[x]n = 0 ......................................................... 183

4.10.

To show that X(–n) =

4.11. 4.12. 4.13. 4.14. 4.15. 4.16. 4.17. 4.18. 4.19. 4.20. 4.21. 4.22. 4.23. 4.24. 4.25. 4.26. 4.27. 4.28. 4.29. 4.30.

Missing Term Technique ....................................................................................... 184 Method of Separation of Symbols .......................................................................... 189 Detection of Errors by use of Difference Tables .................................................. 189 Newton’s Formulae for Interpolation ................................................................... 195 Newton’s Gregory Forward Interpolation Formula ............................................ 195 Newton’s Gregory Backward Interpolation Formula .......................................... 210 Central Difference Interpolation Formulae ......................................................... 224 Gauss’ Forward Difference Formula ..................................................................... 224 Gauss’ Backward Difference Formula .................................................................. 234 Stirling’s Formula................................................................................................... 243 Bessel’s Interpolation Formula ............................................................................. 253 Laplace-Everett’s Formula .................................................................................... 264 Interpolation by Unevenly Spaced Points ............................................................ 273 Lagrange’s Interpolation Formula ........................................................................ 273 Error in Lagrange’s Interpolation Formula ......................................................... 285 Expression of Rational Function as a Sum of Partial Fractions ........................ 287 Inverse Interpolation ............................................................................................. 287 Divided Differences ................................................................................................ 288 Properties of Divided Differences ......................................................................... 288 Newton’s General Interpolation Formula ............................................................ 289

1 ( x + n )( n )

, the Interval of Differencing being Unity ......... 183

(x)

4.31. 4.32. 4.33. 4.34. 5.

NUMERICAL INTEGRATION AND DIFFERENTIATION 5.1. 5.2. 5.3. 5.4. 5.5. 5.6. 5.7. 5.8. 5.9. 5.10. 5.11. 5.12. 5.13. 5.14. 5.15. 5.16. 5.17. 5.18. 5.19. 5.20. 5.21. 5.22. 5.23. 5.24. 5.25. 5.26. 5.27. 5.28. 5.29.

6.

Or Newton’s Divided Difference Interpolation Formula ........................................... 289 Relation Between Divided Differences and Ordinary Differences ..................... 290 Merits and Demerits of Lagrange’s Formula ....................................................... 291 Hermite’s Interpolation Formula .......................................................................... 301 Interpolation by Iteration (Aitken’s Method) ....................................................... 308 309—382

Introduction ............................................................................................................ 311 Numerical Differentiation ..................................................................................... 311 Formulae for Derivatives ....................................................................................... 312 Maxima and Minima of a Tabulated Function .................................................... 317 Errors in Numerical Differentiation ..................................................................... 331 The Cubic Spline Method ....................................................................................... 332 Numerical Integration ........................................................................................... 334 Newton-Cote’s Quadrature Formula .................................................................... 334 Trapezoidal Rule (n = 1) ......................................................................................... 335 Simpson’s One-Third Rule (n = 2) ......................................................................... 335 Simpson’s Three-Eighth Rule (n = 3) .................................................................... 336 Boole’s Rule ............................................................................................................. 336 Weddle’s Rule (n = 6) .............................................................................................. 337 Errors in Quadrature Formulae ............................................................................ 338 Algorithm of Trapezoidal Rule .............................................................................. 340 Flow Chart for Trapezoidal Rule........................................................................... 341 Program to Implement Trapezoidal Method of Numerical Integrations ........... 341 Algorithm of Simpson’s 3/8th Rule ......................................................................... 343 Flow Chart of Simpson’s 3/8th Rule ....................................................................... 344 Program to Implement Simpson’s 3/8th Method of Numerical Integration ....... 344 Algorithm of Simpson’s 1/3rd Rule ......................................................................... 346 Flow Chart of Simpson’s 1/3rd Rule ....................................................................... 347 Program to Implement Simpson’s 1/3rd Method of Numerical Integration ....... 348 Euler-Maclaurin’s Formula ................................................................................... 366 Gaussian Quadrature Formula ............................................................................. 367 Numerical Evaluation of Singular Integrals ........................................................ 369 Romberg Integration .............................................................................................. 370 Gauss-Legendre Two and Three Point Formulae ................................................ 371 Numerical Double Integration .............................................................................. 380

SOLUTION OF A SYSTEM OF SIMULTANEOUS LINEAR ALGEBRAIC EQUATIONS 383—428 6.1. 6.2.

Introduction ............................................................................................................ 385 Direct Methods ....................................................................................................... 386

( xi ) 6.3. 6.4. 6.5. 6.6. 6.7. 6.8. 6.9. 6.10. 6.11. 6.12. 7.

MATRIX INVERSION 7.1. 7.2. 7.3. 7.4. 7.5. 7.6.

8.

445—484

Introduction ............................................................................................................ 447 Eigen Values and Eigen Vectors of a Matrix ....................................................... 447 Bounds on Eigen Values ........................................................................................ 451 Power Method ......................................................................................................... 452 Jacobi’s Rotation Method for Symmetric Matrices .............................................. 465 Eigen Values of Complex Matrices ....................................................................... 470 Eigen Values of a Symmetric Tridiagonal Matrix ............................................... 470 Given’s Rotation Method........................................................................................ 471 Householder’s Method ............................................................................................ 475 Matrix Decomposition ............................................................................................ 480 Singular Value Decomposition (SVD) ................................................................... 482 Generalized Inverse of a Singular Matrix ............................................................ 484

NUMERICAL SOLUTION OF ORDINARY DIFFERENTIAL EQUATIONS 9.1. 9.2. 9.3. 9.4. 9.5. 9.6. 9.7.

429—444

Gauss-Jordan Elimination Method ....................................................................... 431 Triangularization Method...................................................................................... 433 Choleski’s Method .................................................................................................. 435 Iterative Method ..................................................................................................... 437 Escalator Method .................................................................................................... 438 Inverse of a Complex Matrix ................................................................................. 441

THE EIGEN VALUE PROBLEM 8.1. 8.2. 8.3. 8.4. 8.5. 8.6. 8.7. 8.8. 8.9. 8.10. 8.11. 8.12.

9.

Gauss Elimination Method .................................................................................... 387 Ill-Conditioned System of Equations .................................................................... 401 Refinement of Solution .......................................................................................... 402 Gauss-Jordan Elimination Method ....................................................................... 403 Triangularization Method...................................................................................... 410 Cholesky Method .................................................................................................... 414 Iterative Methods ................................................................................................... 415 Jacobi’s Iterative Method or Method of Simultaneous Displacements .............. 416 Gauss-Seidel Iterative Method or Method of Successive Displacements .......... 418 Successive Over Relaxation (SOR) Method .......................................................... 426

485—560

Introduction ............................................................................................................ 487 Initial-Value and Boundary-Value Problems ....................................................... 487 Single-Step and Multi-Step Methods .................................................................... 487 Comparison of Single-Step and Multi-Step Methods .......................................... 488 Numerical Methods of Solution of O.D.E. ............................................................ 488 Picard’s Method of Successive Approximations ................................................... 488 Picard’s Method for Simultaneous First Order Differential Equations ............. 494

( xii ) 9.8. 9.9. 9.10. 9.11. 9.12. 9.13. 9.14. 9.15. 9.16. 9.17. 9.18. 9.19. 9.20. 9.21. 9.22. 9.23. 9.24. 9.25. 9.26. 9.27. 9.28. 9.29. 9.30. 9.31. 9.32. 9.33. 9.34. 9.35.

Picard’s Method for Second Order Differential Equations ................................. 497 Euler’s Method ........................................................................................................ 500 Algorithm of Euler’s Method ................................................................................. 501 Flow Chart of Euler’s Method ............................................................................... 502 Program of Euler’s Method .................................................................................... 502 Improved Euler’s Method ...................................................................................... 503 Algorithm of Improved Euler’s Method ................................................................ 504 Flow Chart of Improved Euler’s Method .............................................................. 505 Program of Improved Euler’s Method .................................................................. 506 Modified Euler’s Method ........................................................................................ 507 Taylor’s Series Method ........................................................................................... 512 Taylor’s Series Method for Simultaneous I Order Diff. Equations .................... 513 Taylor’s Series Method for Second Order Differential Equations ...................... 514 Runge-Kutta Methods ............................................................................................ 519 Fourth Order Runge-Kutta Method ...................................................................... 521 Runge-Kutta Method for Simultaneous First Order Equations ......................... 524 Runge-Kutta Method for Second Order Differential Equations ......................... 525 Predictor-Corrector Methods ................................................................................. 534 Milne’s Method ....................................................................................................... 535 Adams-Moulton (or Adams-Bashforth) Formula ................................................. 546 Stability ................................................................................................................... 549 Stability in the Solution of Ordinary Differential Equation ............................... 550 dy = Ay with Initial dx Condition y(x0) = y0 ................................................................................................. 550 Automatic Error Monitoring.................................................................................. 550 Boundary-Value Problems ..................................................................................... 554 Finite-Difference Method ....................................................................................... 554 Shooting Method (Initial Value Problem Method) .............................................. 557 Numerov’s Method ................................................................................................. 559

Stability of I Order Linear Diff. Equation of form

10. CURVE-FITTING, CUBIC SPLINE AND APPROXIMATION 10.1. 10.2. 10.3. 10.4. 10.5. 10.6. 10.7.

561—634

Curve Fitting .......................................................................................................... 563 Method of Least Squares ....................................................................................... 563 Fitting a Straight Line ........................................................................................... 564 Algorithm for Fitting a Straight Line of The Form y = a + bx for a Given Set of Data Points ............................................................................... 565 Flow Chart for Fitting a Straight Line y = a + bx for a given Set of Data Points ................................................................................................... 566 Program to Implement Curve Fitting to Fit a Straight Line ............................. 567 Fitting of An Exponential Curve y = aebx ............................................................. 573

( xiii ) 10.8. 10.9. 10.10. 10.11.

Fitting of the Curve y = axb ................................................................................... 574 Fitting of the Curve y = abx ................................................................................... 574 Fitting of the Curve pvr = k ................................................................................... 574 Fitting of the Curve of Type xy = b + ax ............................................................... 574

b ........................................................................... 575 x 10.13. Fitting of the Curve y = ax + bx2 ........................................................................... 575

10.12. Fitting of the Curve y = ax2 +

10.14. Fitting of the Curve y = ax + 10.15. Fitting of the Curve y = a +

c b + 2 ..................................................................... 576 x x

c0 + c1 x ........................................................................ 577 x Fitting of the Curve 2x = ax2 + bx + c .................................................................... 578 Fitting of the Curve y = ae–3x + be–2x ..................................................................... 578 Most Plausible Solution of a System of Linear Equations .................................. 588 Polynomial Fit : Non-Linear Regression .............................................................. 590 Algorithm of Second Degree Parabolic Curve Fitting ......................................... 596 Flow Chart of Second Degree Parabolic Curve Fitting ....................................... 598 Program in ‘C’ for Second Degree Parabolic Curve Fitting ................................ 599 Multiple Linear Regression ................................................................................... 601 Algorithm of Multiple Linear Regression ............................................................. 601 Curve-Fitting by Sum of Exponentials ................................................................. 604 Spline Interpolation ............................................................................................... 608 Spline Function ....................................................................................................... 609 Cubic Spline Interpolation ..................................................................................... 609 Steps to Obtain Cubic Spline for Given Data ....................................................... 611 Approximation of Functions .................................................................................. 616 Taylor Series Approximation ................................................................................. 617 Least Squares Approximation ............................................................................... 617 Orthogonal Polynomials ......................................................................................... 620 Gram-Schmidt Orthogonalization Process ........................................................... 621 Legendre and Chebyshev Polynomials ................................................................. 622 Legendre Polynomials ............................................................................................ 622 Chebyshev Polynomials ......................................................................................... 623 Special Values of Chebyshev Polynomials ........................................................... 623 Orthogonal Properties ............................................................................................ 624 Recurrence Relations ............................................................................................. 624 Aliter to Find Chebyshev Polynomials ................................................................. 624 Expression of Powers of x Interms of Chebyshev Polynomials .......................... 625

10.16. Fitting of the Curve y = 10.17. 10.18. 10.19. 10.20. 10.21. 10.22. 10.23. 10.24. 10.25. 10.26. 10.27. 10.28. 10.29. 10.30. 10.31. 10.32. 10.33. 10.34. 10.35. 10.36. 10.37. 10.38. 10.39. 10.40. 10.41. 10.42. 10.43.

b ............................................................................. 576 x

( xiv ) 10.44. 10.45. 10.46. 10.47.

Properties of Chebyshev Polynomials ................................................................... 625 Chebyshev Polynomial Approximation ................................................................. 625 Lanczos Economization of Power Series for a General Function ....................... 626 Rational Approximation ......................................................................................... 632

11. NUMERICAL SOLUTION OF PARTIAL DIFFERENTIAL EQUATIONS 11.1. 11.2. 11.3. 11.4. 11.5. 11.6. 11.7. 11.8. 11.9. 11.10. 11.11. 11.12. 11.13.

Introduction ............................................................................................................ 637 Classification of Linear Partial Differential Equation of II Order ..................... 637 Finite-Difference Approximations to Derivatives ................................................ 638 Elliptic Partial Differential Equations ................................................................. 641 Laplace’s Equation ................................................................................................. 641 Poisson’s Equation .................................................................................................. 653 Parabolic Partial Differential Equations .............................................................. 657 Explicit and Implicit Methods ............................................................................... 657 Bender–Schmidt Explicit Finite Difference Scheme ........................................... 658 Crank-Nicolson Implicit Finite Difference Scheme ............................................. 658 Iterative Methods for the Solution ........................................................................ 667 Hyperbolic Partial Differential Equations ........................................................... 668 Explicit Finite Difference Method ......................................................................... 668

12. STATISTICAL COMPUTATION 12.1. 12.2. 12.3. 12.4. 12.5. 12.6. 12.7. 12.8. 12.9. 12.10. 12.11. 12.12. 12.13. 12.14. 12.15. 12.16. 12.17. 12.18. 12.19. 12.20.

635—672

673—766

The Statistical Methods ......................................................................................... 675 Limitation of Statistical Methods ......................................................................... 675 Frequency Charts ................................................................................................... 675 Regression Analysis ............................................................................................... 679 Curve of Regression and Regression Equation .................................................... 679 Linear Regression ................................................................................................... 679 Lines of Regression................................................................................................. 679 Derivation of Lines of Regression ......................................................................... 680 Use of Regression Analysis .................................................................................... 682 Comparison of Correlation and Regression Analysis .......................................... 682 Properties of Regression Co-efficients .................................................................. 683 Angle Between two Lines of Regression ............................................................... 684 Algorithm for Linear Regression ........................................................................... 684 Program to Implement Least Square Fit of a Regression Line of y on x ........... 685 Program to Implement Least Square Fit of a Regression Line of x on y ........... 686 Statistical Quality Control ..................................................................................... 700 Advantages of Statistical Quality Control............................................................ 701 Reasons for Variations in the Quality of a Product ............................................. 702 Techniques of Statistical Quality Control ............................................................ 702 Control Chart .......................................................................................................... 702

( xv ) 12.21. 12.22. 12.23. 12.24. 12.25. 12.26. 12.27. 12.28. 12.29. 12.30. 12.31. 12.32. 12.33. 12.34. 12.35. 12.36. 12.37. 12.38. 12.39. 12.40. 12.41. 12.42. 12.43. 12.44. 12.45. 12.46. 12.47. 12.48. 12.49. 12.50. 12.51. 12.52. 12.53.

Objectives of Control Charts ................................................................................. 703 Construction of Control Charts for Variables ...................................................... 703 Control Charts for Attributes ................................................................................ 705 Population or Universe .......................................................................................... 713 Sampling ................................................................................................................. 714 Parameters of Statistics ......................................................................................... 714 Standard Error ....................................................................................................... 714 Test of Significance ................................................................................................ 715 Testing of Statistical Hypothesis .......................................................................... 715 Errors in Sampling ................................................................................................. 717 Test of Significance for Large Samples ................................................................ 717 Test of Significance of Small Samples .................................................................. 726 Student’s t-distribution .......................................................................................... 726 Critical Value of t ................................................................................................... 726 Test I : t-test of Significance of the Mean of a Random Sample ......................... 726 Test II : t-test for Difference of Means of Two Small Samples (From a Normal Population) ................................................................................. 729 Snedecor’s Variance Ratio Test or F-Test ............................................................ 732 Chi-square (χ2) Test ................................................................................................ 736 The χ2 Distribution ................................................................................................. 737 χ2 Test as a Test of Goodness of Fit ...................................................................... 738 χ2 Test as a Test of Independence ......................................................................... 742 Analysis of Variance (ANOVA) ............................................................................. 745 One Way Classification .......................................................................................... 746 Two Way Classification .......................................................................................... 748 Time-series and Forecasting.................................................................................. 754 Components of Time-series.................................................................................... 754 Analysis of Time-series .......................................................................................... 755 Importance of Time-series ..................................................................................... 756 Time-series Graph .................................................................................................. 756 Measurement of Trend ........................................................................................... 757 Limitations of the Method of Moving Averages ................................................... 759 Measurement of Seasonal Fluctuations ............................................................... 760 Measurement of Cyclical Variation ...................................................................... 765 Tables ................................................................................................................. 767 Model Test Papers ............................................................................................ 773 Index ................................................................................................................... 787