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Chapter 1 Homework All Calculations Were Performed Standard Double
Floating Point Arithmetic 1 1.4 Floating Point Arithmetic 1. Determine the value of each of the following expressions using 4-digit rounding and 4-digit chopping arithmetic. For each quantity, compute the absolute and the relative error. (a) π+e−cos 22◦ (b) e/7 […]
Chapter 1 Homework Convergence 12 Convergence Compute Each The Following
1.2 Convergence 1. Compute each of the following limits and determine the corresponding rate of convergence. (a) limn→∞ n−1 n3+2 (b) limn→∞ √n+ 1 −√n (c) limn→∞ sin n n (d) limn→∞ 3n2−1 7n2+n+2 (a) For n > 1, […]
Chapter 1 Homework Multiplication Performed The Output Step Because The
Floating Point Number Systems 1 1.3 Floating Point Number Systems 1. Provide the floating point equivalent for each of the following numbers from the floating point number system F(10,4,0,4). Consider both chopping and round- ing. Compute the absolute and relative […]
Chapter 1 Homework Use the statistics algorithm from the text to compute the
Algorithms 1 Solutions Chapter 1 Getting Started 1.1 Algorithms 1. Use the statistics algorithm from the text to compute the mean, ¯x, and the standard deviation, s, of the data set: −5,−3,2,−2,1. The inputs are 2. With n= 4, use […]
Chapter 2 Homework Bisection Method Solutions Root finding 21 Bisection
Bisection Method 1 Solutions Chapter 2 Rootfinding 2.1 Bisection Method 1. Verify that each of the following equations has a root on the interval (0,1). Next, perform the bisection method to determine p3, the third approximation to the location of […]
Chapter 2 Homework cancellation error will primarily influence the second term
Accelerating Convergence 1 2.6 Accelerating Convergence 1. Show that the equation for Aitken’s ∆2-method can be rewritten as ˆpn=pnpn−2−p2 n−1 pn−2pn−1+pn−2 . Explain why this formula is inferior to the one used in the text. Combining the terms on the […]
Chapter 2 Homework Each of the following equations has a root on the
2.4 Newton’s Method 1. Each of the following equations has a root on the interval (0,1). Perform New- ton’s method to determine p4, the fourth approximation to the location of the root. (a) ln(1 + x)−cos x= 0 (b) x5+ […]
Chapter 2 Homework Note That The Error The Fifth False
2.2 The Method of False Position 1. Each of the following equations has a root on the interval (0,1). Perform the method of false position to determine p3, the third approximation to the location of the root, and to determine […]
Chapter 2 Homework Secant Method 25 Secant Method Each The
2.5 Secant Method 1. Each of the following equations has a root on the interval (0,1). Perform the secant method to determine p4, the fourth approximation to the location of the root. (a) ln(1 + x)−cos x= 0 (b) x5+ […]
Chapter 2 Homework Section 23 The Order Convergence Least Finally
2.3 Fixed Point Iteration Schemes 1. Suppose the sequence {pn}is generated by the fixed point iteration scheme pn=g(pn−1). Further, suppose that the sequence converges linearly to the fixed point p. (a) Show that g′(p)≈pn−pn−1 pn−1−pn−2 . (b) Show that |en| […]
Chapter 2 Homework The Hermite Polynomials X Are Special Class
2.7 Roots of Polynomials 1. Use synthetic division to deflate the given polynomial by the indicated root. (a) p(x) = x4−2.25×3−25.75×2+ 28.5x+ 126, x∗= 3 (b) p(x) = x4+ 1.83×3−0.081×2+ 1.83x−1.081, x∗=−2.3 (c) p(x) = x4+ 20.5×3+ 129.5×2+ 230x−150, x∗= […]
Chapter 3 Homework Bac And Have Arrived Contradiction Thus
Linear Algebra Review 1 Solutions Chapter 3 Systems of Equations 3.0 Linear Algebra Review In Exercises 1 – 9, compute the indicated matrices given A=1−1 3 205, B = 2 1 0 −3−1 5 1 3 4 , […]
Chapter 3 Homework Decomposition 35 Decomposition A Show That The
3.5 LU Decomposition 1. (a) Show that the algorithm to obtain an LU decomposition based on Gaussian elimination requires 2 3n3−1 2n2−1 6narithmetic operations. (b) Show that the solve step – forward substitution followed by backward sub- stitution – requires […]
Chapter 3 Homework Decomposition For The Third Pass Note That
LU Decomposition 17 For the third pass, we note that The larger value corresponds to r3, so again there is no need to modify the contents of the row vector. After the last pass of Gaussian elimination, the coefficient matrix […]
Chapter 3 Homework Direct Factorization Now Back Substitution Applied The
3.6 Direct Factorization In Exercises 1 – 6, determine the Crout decomposition of the given matrix, and then solve the system Ax=bfor each of the given right-hand side vectors. 1. A= 2 7 5 6 20 10 4 3 […]
Chapter 3 Homework First Calculate 1 Error Estimates And
3.4 Error Estimates and Condition Number 1. Let Aand Bbe n×nmatrices, and let αbe a non-zero real number. (a) Show that κ(AB)≤κ(A)κ(B). (b) Show that κ(αA) = κ(A). (a) κ(AB) = kABk k(AB)−1k ≤ kAk kBk kB−1A−1k ≤kAk kA−1kkBk kB−1k=κ(A)κ(B). […]
Chapter 3 Homework Iterative Methods Basic Concepts 38 Iterative Methods
Iterative Methods, Basic Concepts 1 3.8 Iterative Methods, Basic Concepts In Exercises 1 – 4: (a) Compute Tjac and Tgs for the given matrix. (b) Determine the spectral radius of each iteration matrix from part (a). (c) Will the Jacobi […]
Chapter 3 Homework Newtons Method Produce The Solution
12 Section 3.10 With x(0) =1 0 Tand a convergence tolerance of 5×10−6, Newton’s method converges in four iterations: nx(n)T 10.550388 0.155039 20.502720 0.150613 30.502379 0.150579 40.502379 0.150579 With the same initial vector and convergence tolerance, […]
Chapter 3 Homework Solving Requires Divisions Thus The Entire Solve
Direct Factorization 21 For the second pass, we multiply the second row of Lwith the second and third columns of U. Equating each product with the corresponding element from Agen- erates the equations Substituting the values determined from the previous […]
Chapter 3 Homework Sor Method Good Better Than The Performance
Iterative Methods, Basic Concepts 13 kx(k) 00.000000 0.000000 0.000000 0.000000 0.000000 T 10.571429 −0.666667 1.000000 0.700000 0.333333 T 20.285714 −0.587302 1.455556 0.933333 0.800000 T 13 0.238154 −0.777855 1.715579 1.369809 1.246065 T 14 0.238062 −0.777902 1.715888 1.369984 1.246539 T 15 0.238042 […]
Chapter 3 Homework Take X0 And Use Convergence Tolerance 107
Conjugate Gradient Method 1 3.9 Conjugate Gradient Method In Exercises 1 – 4, solve the indicated linear system using the conjugate gradient method in exact arithmetic. Show that the exact solution is obtained in each case in three or fewer […]
Chapter 3 Homework The corresponding augmented matrix is
3.1 Gaussian Elimination In Exercises 1 – 5, write out the augmented matrix for the indicated linear system of equations and then obtain the solution using Gaussian elimination with back substitution. 1. 2×1−x2+x3=−1 4×1+ 2×2+x3= 4 6×1−4×2+ 2×3=−2 The corresponding […]
Chapter 3 Homework The First Pass Gaussian Elimination Transforms The
3.2 Pivoting Strategies 1. For each of the following augmented matrices, identify the entry which would serve as the first pivot element for (i) Gaussian elimination with no pivoting; (ii) Gaussian elimination with partial pivoting; and (iii) Gaussian elimination with […]
Chapter 3 Homework The Only Way This Polynomial Can Take
Special Matrices 1 3.7 Special Matrices 1. Classify each of the following matrices as strictly diagonally dominant, symmet- ric positive definite, both or neither. (a) 2−1 0 −142 0 2 6 (b) 120 4 6 […]
Chapter 3 Homework Use the initial vector indicated below.
Nonlinear Systems of Equations 1 3.10 Nonlinear Systems of Equations 1. For each of the following nonlinear systems, write out the vector-valued function Fassociated with the system and compute the Jacobian of F. (a) x1−x2−x3 1= 0 x1+x2−x3 2= 0 […]
Chapter 3 Homework Vector And Matrix Norms B Compute The
Vector and Matrix Norms 1 3.3 Vector and Matrix Norms 1. Verify that the l∞-norm, kxk∞= max 1≤i≤n|xi|, satisfies the properties of a vector norm. In what follows, let xand ybe arbitrary n-vectors, and let αbe an arbitrary real number. […]
Chapter 3 Homework while back substitution applied to the system LT x
Special Matrices 13 Thus, forward substitution applied to the system Lz=byields z1=b1 =15 while back substitution applied to the system LTx=zgives x4=z4 l44 =−19/4 2=−19 8; x3=z3−l43x4 l33 =13/2−0(−19/8) 1=13 2; x2=z2−l32x3−l42 x4 l22 =9/2−1(−19/8) −1(13/2) 4=3 32;and x1=z1−l21x2−l31 x3−l41x4 […]
Chapter 3 Homework We therefore interchange the first and third
18 Section 3.2 12. x1−2×2+x3−x4=−5 x1+ 5×2−7×3+ 2×4= 2 3×1+x2−5×3+ 3×4= 1 2×1+ 3×2−5×3= 17 The initial augmented matrix for the system is (a) Initialize the row vector to r=1234T.Among the values |ar1,1|= 1,|ar2,1|= 1,|ar3,1|= 3,|ar4,1|= 2, the largest corresponds […]
Chapter 5 Homework Data For Group Objects Made From The
Regression 1 5.8 Regression 1. One of the following data sets follows an exponential law and the other follows a power law. Which is which? x2.0 2.5 3.0 3.5 4.0 4.5 5.0 y114.79 27.75 47.09 74.07 109.99 156.10 213.69 x2.0 […]
Chapter 5 Homework Indicate how to construct each of the following interpolating polynomials.
Neville’s Algorithm 1 5.2 Neville’s Algorithm 1. Indicate how to construct each of the following interpolating polynomials. (a) P0,1,2,3(x) from P0,1,2(x) and P1,2,3(x) (b) P0,1,2,3(x) from P0,2,3(x) and P0,1,3(x) (c) P0,1,2,3(x) from P1,2,3(x) and P0,2,3(x) (d) P0,1,2,3(x) from P0,1,3(x) and […]
Chapter 5 Homework Optimal Points For Interpolation 54 Optimal Points
5.4 Optimal Points for Interpolation 1. Prove each of the following properties of the Chebyshev polynomials: (a) for each n,Tn(1) = 1. (b) for each n,Tn(−1) = (−1)n. (c) for all j > k ≥0, Tj(x)Tk(x) = 1 2[Tj+k(x) + […]
Chapter 5 Homework Section 57 Letting Follows From The
14 Section 5.7 9. Repeat Exercise 8 using the Hermite interpolating polynomial. (a) The coefficients of the Newton form of the Hermite interpolating polynomial are, in order from left to right and from top to bottom, 0.290864 −0.16405 −8.26 414.7−1.555688 […]
Chapter 5 Homework The Complete Set Nota knot Spline Coefficients
Cubic Spline Interpolation 1 5.6 Cubic Spline Interpolation For Exercises 1 through 3, use the values given below for the temperature, T, pressure, p, and density, ρ, of the standard atmosphere as a function of altitude. This data was drawn […]
Chapter 5 Homework The First And Second Divided Differences Were
5.3 Newton Form of the Interpolating Polynomial 1. Assess the accuracy of the values in the relative viscosity table developed earlier in this section by plotting the values from the table and the six given data values on the same […]
Chapter 5 Homework Using The Result From Part A 15
5.7 Hermite and Hermite Cubic Interpolation 1. Show that the polynomials Hiand ˆ Hidefined by Hi(x) = [1 −2L′ n,i(xi)(x−xi)]L2 n,i(x) ˆ Hi(x) = (x−xi)L2 n,i(x), where Ln,i is the Lagrange polynomial associated with the point x=xisatisfy the relations Hi(xj) […]
Chapter 5 Homework The Lagrange form of the polynomial passing through the points
Lagrange Form of the Interpolating Polynomial 1 Solutions Chapter 5 Interpolation (and Curve Fitting) 5.1 Lagrange Form of the Interpolating Polynomial 1. Let x0=−1, x1= 1 and x2= 2. (a) Determine formulas for the Lagrange polynomials L2,0(x), L2,1(x) and L2,2(x) […]
Chapter 6 Homework Applying The Weighted Mean value Theorem For Integrals
Newton-Cotes Quadrature 1 6.4 Newton-Cotes Quadrature 1. Approximate the value of each of the following integrals using the trapezoidal rule. Verify that the theoretical error bound holds in each case. (a) R2 1 1 xdx (b) R1 0e−xdx (c) R1 […]
Chapter 6 Homework Because Numerical Evidence Suggests That The Rate
Composite Newton-Cotes Quadrature 13 17. R2 1 sin x xdx Let f(x) = sin x x. Then max x∈[1,2] |f′′(x)| ≈ 0.24 and max x∈[1,2] |f(4)(x)| ≈ 0.14. The smallest number of subintervals needed to guarantee an absolute error not […]
Chapter 6 Homework Compute the value of the indicated definite integral
Improper Integrals and Other Discontinuities 1 6.9 Improper Integrals and Other Discontinuities In Exercises 1 – 3: (a) Compute the value of the indicated definite integral using the trapezoidal rule, Simpson’s rule, the Midpoint Rule and the two-point Gaussian quadrature […]
Chapter 6 Homework Consider The Definite Integral If
Composite Newton-Cotes Quadrature 1 6.5 Composite Newton-Cotes Quadrature 1. Provide the details of the transformation of the error term associated with the composite Simpson’s rule from h5 90 m X j=1 f(4)(ξj) to (b−a)h4 180 f(4)(ξ). Suppose fhas four continuous […]
Chapter 6 Homework Gaussian Quadrature 66 Gaussian Quadrature Approximate The
6.6 Gaussian Quadrature 1. Approximate the value of each of the following integrals using the two-point Gaussian quadrature rule (the basic formula, not the composite rule). Verify that the theoretical error bound holds in each case. (a) R1 −1e−xdx (b) […]
Chapter 6 Homework Insulation Thickness Mm Insulation Thickness Mm Now
Numerical Differentiation, Part I 1 Solutions Chapter 6 Differentiation and Integration 6.1 Numerical Differentiation, Part I 1. Rework the coefficient of friction problem from the data in Table 6-1 using a not-a-knot cubic spline interpolant rather than a 10-th degree […]
Chapter 6 Homework The Four Row Romberg Integration Table 07071
Romberg Integration 1 6.7 Romberg Integration 1. Romberg integration approximates the value of the integral Z1 0 1 1 + x2dx with an error of 1.2113 ×10−11 using only 33 function evaluations. How many function evaluations would be needed to […]
Chapter 6 Homework The Solution This Inequality 417454 Therefore
12 Section 6.8 using 81 function evaluations. To guarantee an absolute error of no greater than 11. R2 0e−xsin(x2cos e−x)dx Using the adaptive Simpson’s rule with ǫ= 5 ×10−7, we find Z2 0 e−xsin(x2cos e−x)dx ≈0.2813862 using 61 function evaluations. […]
Chapter 6 Homework Thus Provided Has Two Continuous Derivatives Near
Numerical Differentiation, Part II 1 6.2 Numerical Differentiation, Part II 1. Derive the second-order central difference approximation for the first derivative, including error term: f0(x0) = f(x0+h)−f(x0−h) 2h−h2 6f000 (ξ). Let x0−h,x0and x0+hbe the interpolating points. Using the Lagrange form […]
Chapter 6 Homework What step size would be needed in the first-order
6.3 Richardson Extrapolation 1. In the last example, extrapolation was used to obtain an approximation to the first derivative of f(x) = tan−1xat x0= 2 with an error of 2.78 ×10−5. The smallest step size used in the construction of […]
Chapter 6 Homework which compares favorably with the actual error
Adaptive Quadrature 1 6.8 Adaptive Quadrature 1. For each of the following integrals, compute S(a, b), S(a, c) and S(c, b), where c= (a+b)/2. Compute the estimate for the error in S(a, c) + S(c, b) and compare this to […]
Chapter 6 Homework To guarantee an absolute error of no greater than
16. R4 0x√x2+ 9dx Consider the definite integral I(f) = Z4 0 xpx2+ 9 dx. The table below lists composite two-point Gaussian quadrature rule approximations and composite three-point Gaussian quadrature rule approximations to I(f)for several values of h. Observe that […]