List of Figures

Fig. 0.1 NLALIB hierarchy. xxv

Fig. 1.1 Matrix multiplication. 3

Fig. 1.2 Rotating the xy-plane. 7

Fig. 1.3 Rotated line 8

Fig. 1.4 Rotate three-dimensional coordinate system. 9

Fig. 1.5 Translate a point in two dimensions. 9

Fig. 1.6 Rotate a line about a point. 10

Fig. 1.7 Rotation about an arbitrary point. 11

Fig. 1.8 Undirected graph. 12

Fig. 2.1 Polynomial passing through four points. 26

Fig. 2.2 Inconsistent equations. 31

Fig. 2.3 Truss. 38

Fig. 2.4 Electrical circuit. 39

Fig. 2.5 Truss problem. 45

Fig. 2.6 Circuit problem. 45

Fig. 3.1 Subspace spanned by two vectors. 48

Fig. 4.1 Geometrical interpretation of the determinant. 75

Fig. 5.1 Direction of eigenvectors. 80

Fig. 5.2 Circuit with an inductor. 89

Fig. 5.3 Currents in the RL circuit. 92

Fig. 5.4 Digraph of an irreducible matrix. 93

Fig. 5.5 Hanowa matrix. 101

Fig. 6.1 Distance between points. 104

Fig. 6.2 Equality, addition, and subtraction of vectors. 104

Fig. 6.3 Scalar multiplication of vectors. 104

Fig. 6.4 Vector length. 106

Fig. 6.5 Geometric interpretation of the inner product. 106

Fig. 6.6 Law of cosines. 106

Fig. 6.7 Triangle inequality. 112

Fig. 6.8 Signal comparison. 112

Fig. 6.9 Projection of one vector onto another. 115

Fig. 7.1 Effect of an orthogonal transformation on a vector. 122

Fig. 7.2 Spherical coordinates. 123

Fig. 7.3 Orthonormal basis for spherical coordinates. 124

Fig. 7.4 Point in spherical coordinate basis and Cartesian coordinates. 125

Fig. 7.5 Function specified in spherical coordinates. 126

Fig. 7.6 Effect of a matrix on vectors. 128

Fig. 7.7 Unit spheres in three norms. 129

Fig. 7.8 Image of the unit circle. 133

Fig. 8.1 Floating-point number system. 149

Fig. 8.2 Map of IEEE double-precision floating-point. 150

Fig. 9.1 Matrix multiplication. 165

Fig. 10.1 Forward and backward errors. 184

Fig. 10.2 The Wilkinson polynomial. 187

Fig. 10.3 Ill-conditioned Cauchy problem. 188

Fig. 10.4 Conditioning of a problem. 189

Fig. 11.1 LU decomposition of a matrix. 206

Fig. 11.2 k × k submatrix. 215

Fig. 11.3 Gaussian elimination flop count. 217

Fig. 12.1 Square wave with period 2π. 244

Fig. 12.2 Fourier series converging to a square wave. 244

Fig. 12.3 The heat equation: a thin rod insulated on its sides. 245

Fig. 12.4 Numerical solution of the heat equation: subdivisions of the x and t axes. 245

Fig. 12.5 Numerical solution of the heat equation:locally related points in the grid. 246

Fig. 12.6 Grid for the numerical solution of the heat equation. 246

Fig. 12.7 Graph of the solution for the heat equation problem. 247

Fig. 12.8 Linear least-squares approximation. 250

Fig. 12.9 Quadratic least-squares approximation. 251

Fig. 12.10 Estimating absolute zero. 252

Fig. 12.11 Linear interpolation. 253

Fig. 12.12 Cubic splines. 253

Fig. 12.13 Cubic spline approximation. 256

Fig. 12.14 Sawtooth wave with period 2π. 258

Fig. 13.1 Conductance matrix. 270

Fig. 14.1 Vector orthogonal projection. 282

Fig. 14.2 Removing the orthogonal projection. 282

Fig. 14.3 Result of the first three steps of Gram-Schmidt. 283

Fig. 15.1 The four fundamental subspaces of a matrix. 305

Fig. 15.2 SVD rotation and distortion. 308

Fig. 15.3 (a) Lena (512 × 512) and (b) lena using 35 modes. 312

Fig. 15.4 Lena using 125 modes. 312

Fig. 15.5 Singular value graph of lena. 313

Fig. 15.6 SVD image capture. 314

Fig. 16.1 Geometric interpretation of the least-squares solution. 322

Fig. 16.2 An overdetermined system. 322

Fig. 16.3 Least-squares estimate for the power function. 329

Fig. 16.4 The reduced SVD for a full rank matrix. 330

Fig. 16.5 Velocity of an enzymatic reaction. 332

Fig. 16.6 Underdetermined system. 339

Fig. 17.1 Givens matrix. 353

Fig. 17.2 Givens rotation. 354

Fig. 17.3 Householder reflection. 363

Fig. 17.4 Linear combination associated with Householder reflection. 363

Fig. 17.5 Householder reflection to a multiple of e1. 366

Fig. 17.6 Transforming an m × n matrix to upper triangular form using householder reflections. 369

Fig. 17.7 Householder reflections and submatrices. 369

Fig. 17.8 Householder reflection for a submatrix. 369

Fig. 18.1 Tacoma Narrows Bridge collapse. 380

Fig. 18.2 Mass-spring system. 380

Fig. 18.3 Solution to a system of ordinary differential equations. 382

Fig. 18.4 Populations using the Leslie matrix. 387

Fig. 18.5 Column buckling. 387

Fig. 18.6 Deflection curves for critical loads P1, P2, and P3. 389

Fig. 18.7 Reduced Hessenberg matrix. 401

Fig. 18.8 Inductive step in Schur’s triangularization. 407

Fig. 18.9 Schur’s triangularization. 407

Fig. 18.10 Eigenvalues of a 2 × 2 matrix as shifts. 413

Fig. 18.11 Springs problem. 430

Fig. 19.1 Bisection. 454

Fig. 19.2 Interlacing. 454

Fig. 19.3 Bisection method: λk located to the left. 456

Fig. 19.4 Bisection method: λk located to the right. 457

Fig. 19.5 Bisection and multiple eigenvalues. 458

Fig. 19.6 Secular equation. 460

Fig. 20.1 SOR spectral radius. 479

Fig. 20.2 Region in the plane. 479

Fig. 20.3 Five-point stencil. 480

Fig. 20.4 Poisson’s equation. (a) Approximate solution and (b) analytical solution. 481

Fig. 20.5 One-dimensional Poisson equation grid. 484

Fig. 20.6 One-dimensional red-black GS. 486

Fig. 21.1 Examples of...