 Chapter 1.1: Review of Calculus
 Chapter 1.2: Roundoff Errors and Computer Arithmetic
 Chapter 1.3: Algorithms and Convergence
 Chapter 10.1: Fixed Points for Functions of Several Variables
 Chapter 10.2: Newton's Method
 Chapter 10.3: QuasiNewton Methods
 Chapter 10.4: Steepest Descent Techniques
 Chapter 10.5: Homotopy and Continuation Methods
 Chapter 11.1: The Linear Shooting Method
 Chapter 11.2: The Shooting Method for Nonlinear Problems
 Chapter 11.3: FiniteDifference Methods for Linear Problems
 Chapter 11.4: FiniteDifference Methods for Nonlinear Problems
 Chapter 11.5: The RayleighRitz Method
 Chapter 12.1: Elliptic Partial Differential Equations
 Chapter 12.2: Parabolic Partial Differential Equations
 Chapter 12.3: Hyperbolic Partial Differential Equations
 Chapter 12.4: An Introduction to the FiniteElement Method
 Chapter 2.1: The Bisection Method
 Chapter 2.2: FixedPoint Iteration
 Chapter 2.3: Newton's Method and Its Extensions
 Chapter 2.4: Error Analysis for Iterative Methods
 Chapter 2.5: Accelerating Convergence
 Chapter 2.6: Zeros of Polynomials and Muller's Method
 Chapter 3.1: Interpolation and the Lagrange Polynomial
 Chapter 3.2: Data Approximation and Neville's Method
 Chapter 3.3: Divided Differences
 Chapter 3.4: Hermite Interpolation
 Chapter 3.5: Cubic Spline Interpolation
 Chapter 3.6: Parametric Curves
 Chapter 4.1: Numerical Differentiation
 Chapter 4.2: Richardson's Extrapolation
 Chapter 4.3: Elements of Numerical Integration
 Chapter 4.4: Composite Numerical Integration
 Chapter 4.5: Romberg Integration
 Chapter 4.6: Adaptive Quadrature Methods
 Chapter 4.7: Gaussian Quadrature
 Chapter 4.8: Multiple Integrals
 Chapter 4.9: Improper Integrals
 Chapter 5.1: The Elementary Theory of InitialValue Problems
 Chapter 5.10: Stability
 Chapter 5.11: Stiff Differential Equations
 Chapter 5.2: Euler's Method
 Chapter 5.3: HigherOrder Taylor Methods
 Chapter 5.4: RungeKutta Methods
 Chapter 5.5: Error Control and the RungeKuttaFehlberg Method
 Chapter 5.6: Multistep Methods
 Chapter 5.7: Variable StepSize Multistep Methods
 Chapter 5.8: Extrapolation Methods
 Chapter 5.9: HigherOrder Equations and Systems of Differential Equations
 Chapter 6.1: Linear Systems of Equations
 Chapter 6.2: Pivoting Strategies
 Chapter 6.3: Linear Algebra and Matrix Inversion
 Chapter 6.4: The Determinant of a Matrix
 Chapter 6.5: Matrix Factorization
 Chapter 6.6: Special Types of Matrices
 Chapter 7.1: Norms of Vectors and Matrices
 Chapter 7.2: Eigenvalues and Eigenvectors
 Chapter 7.3: The Jacobi and GaussSiedel Iterative Techniques
 Chapter 7.4: Relaxation Techniques for Solving Linear Systems
 Chapter 7.5: Error Bounds and Iterative Refinement
 Chapter 7.6: The Conjugate Gradient Method
 Chapter 8.1: Discrete Least Squares Approximation
 Chapter 8.2: Orthogonal Polynomials and Least Squares Approximates
 Chapter 8.3: Chebyshev Polynomials and Economization of Power Series
 Chapter 8.4: Rational Function Approximation
 Chapter 8.5: Trigonometric Polynomial Approximation
 Chapter 8.6: Fast Fourier Transforms
 Chapter 9.1: Linear Algebra and Eigenvalues
 Chapter 9.2: Orthogonal Matrices and Similarity Transformations
 Chapter 9.3: The Power Method
 Chapter 9.4: Householder's Method
 Chapter 9.5: The QR Algorithm
 Chapter 9.6: Singular Value Decomposition
Numerical Analysis 9th Edition  Solutions by Chapter
Full solutions for Numerical Analysis  9th Edition
ISBN: 9780538733519
Numerical Analysis  9th Edition  Solutions by Chapter
Get Full SolutionsSince problems from 73 chapters in Numerical Analysis have been answered, more than 2950 students have viewed full stepbystep answer. Numerical Analysis was written by Patricia and is associated to the ISBN: 9780538733519. The full stepbystep solution to problem in Numerical Analysis were answered by Patricia, our top Math solution expert on 03/16/18, 03:30PM. This textbook survival guide was created for the textbook: Numerical Analysis, edition: 9. This expansive textbook survival guide covers the following chapters: 73.

Affine transformation
Tv = Av + Vo = linear transformation plus shift.

Big formula for n by n determinants.
Det(A) is a sum of n! terms. For each term: Multiply one entry from each row and column of A: rows in order 1, ... , nand column order given by a permutation P. Each of the n! P 's has a + or  sign.

Column space C (A) =
space of all combinations of the columns of A.

Conjugate Gradient Method.
A sequence of steps (end of Chapter 9) to solve positive definite Ax = b by minimizing !x T Ax  x Tb over growing Krylov subspaces.

Diagonalizable matrix A.
Must have n independent eigenvectors (in the columns of S; automatic with n different eigenvalues). Then SI AS = A = eigenvalue matrix.

Diagonalization
A = S1 AS. A = eigenvalue matrix and S = eigenvector matrix of A. A must have n independent eigenvectors to make S invertible. All Ak = SA k SI.

Elimination.
A sequence of row operations that reduces A to an upper triangular U or to the reduced form R = rref(A). Then A = LU with multipliers eO in L, or P A = L U with row exchanges in P, or E A = R with an invertible E.

Factorization
A = L U. If elimination takes A to U without row exchanges, then the lower triangular L with multipliers eij (and eii = 1) brings U back to A.

Full row rank r = m.
Independent rows, at least one solution to Ax = b, column space is all of Rm. Full rank means full column rank or full row rank.

Krylov subspace Kj(A, b).
The subspace spanned by b, Ab, ... , AjIb. Numerical methods approximate A I b by x j with residual b  Ax j in this subspace. A good basis for K j requires only multiplication by A at each step.

Nullspace matrix N.
The columns of N are the n  r special solutions to As = O.

Orthogonal matrix Q.
Square matrix with orthonormal columns, so QT = Ql. Preserves length and angles, IIQxll = IIxll and (QX)T(Qy) = xTy. AlllAI = 1, with orthogonal eigenvectors. Examples: Rotation, reflection, permutation.

Rotation matrix
R = [~ CS ] rotates the plane by () and R 1 = RT rotates back by (). Eigenvalues are eiO and eiO , eigenvectors are (1, ±i). c, s = cos (), sin ().

Semidefinite matrix A.
(Positive) semidefinite: all x T Ax > 0, all A > 0; A = any RT R.

Spanning set.
Combinations of VI, ... ,Vm fill the space. The columns of A span C (A)!

Spectral Theorem A = QAQT.
Real symmetric A has real A'S and orthonormal q's.

Spectrum of A = the set of eigenvalues {A I, ... , An}.
Spectral radius = max of IAi I.

Sum V + W of subs paces.
Space of all (v in V) + (w in W). Direct sum: V n W = to}.

Trace of A
= sum of diagonal entries = sum of eigenvalues of A. Tr AB = Tr BA.

Vector v in Rn.
Sequence of n real numbers v = (VI, ... , Vn) = point in Rn.
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