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182 <strong>GNU</strong> <strong>Octave</strong>[housv, beta, zer] = housh (x, j, z)Function FileComputes householder reflection vector housv to reflect x to be jth column of identity,i.e., (I - beta*housv*housv’)x =e(j) inputs x: vector j: index into vector z: thresholdfor zero (usually should be the number 0) outputs: (see Golub and Van Loan) beta: Ifbeta = 0, then no reflection need be applied (zer set to 0) housv: householder vector[u, h, nu] = krylov (a, v, k, eps1, pflg);Function Fileconstruct orthogonal basis U of block Krylov subspace; [v a*v a^2*v ... a^(k+1)*v];method used: householder reflections to guard against loss of orthogonality eps1:threshhold for 0 (default: 1e-12) pflg: flag to use row pivoting (improves numericalbehavior) 0 [default]: no pivoting; prints a warning message if trivial null space iscorrupted 1 : pivoting performedoutputs: u: orthogonal basis of block krylov subspace h: Hessenberg matrix; if vis a vector then a u = u h otherwise h is meaningless nu: dimension of span ofkrylov subspace (based on eps1) if b is a vector and k > m-1, krylov returns h = theHessenberg decompostion of a.Reference: Hodel and Misra, "Partial Pivoting in the Computation of Krylov Subspaces",to be submitted to Linear Algebra and its Applications20.3 Functions of a Matrixexpm (a)Loadable FunctionReturn the exponential of a matrix, defined as the infinite Taylor seriesexp(A) = I + A + A22! + A33! + · · ·The Taylor series is not the way to compute the matrix exponential; see Moler andVan Loan, Nineteen Dubious Ways to Compute the Exponential of a Matrix, SIAMReview, 1978. This routine uses Ward’s diagonal Padé approximation method withthree step preconditioning (SIAM Journal on Numerical Analysis, 1977). DiagonalPadé approximations are rational polynomials of matrices D q (a) −1 N q (a) whose Taylorseries matches the first 2q + 1 terms of the Taylor series above; direct evaluation ofthe Taylor series (with the same preconditioning steps) may be desirable in lieu ofthe Padé approximation when D q (a) is ill-conditioned.logm (a)Function FileCompute the matrix logarithm of the square matrix a. Note that this is currentlyimplemented in terms of an eigenvalue expansion and needs to be improved to bemore robust.[result, error estimate] = sqrtm (a)Loadable FunctionCompute the matrix square root of the square matrix a.Ref: Nicholas J. Higham. A new sqrtm for MATLAB. Numerical Analysis ReportNo. 336, Manchester Centre for Computational Mathematics, Manchester, England,January 1999.
Chapter 20: Linear Algebra 183kron (a, b)Form the kronecker product of two matrices, defined block by block asx = [a(i, j) b]For example,kron (1:4, ones (3, 1))⇒ 1 2 3 41 2 3 41 2 3 4Function Filex = syl (a, b, c)Solve the Sylvester equationLoadable FunctionAX + XB + C = 0using standard Lapack subroutines. For example,syl ([1, 2; 3, 4], [5, 6; 7, 8], [9, 10; 11, 12])⇒ [ -0.50000, -0.66667; -0.66667, -0.50000 ]
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GNU OctaveA high-level interactive
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iTable of ContentsPreface . . . . .
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10 Expressions . . . . . . . . . .
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19 Arithmetic . . . . . . . . . . .
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Appendix A Tips and Standards . . .
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Preface 1PrefaceOctave was original
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Preface 3• Tony Richardson richar
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Chapter 1: A Brief Introduction to
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Chapter 2: Getting Started 132 Gett
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Chapter 2: Getting Started 15argvBu
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Chapter 2: Getting Started 17helpCo
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Chapter 2: Getting Started 19clc ()
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Chapter 2: Getting Started 21C-rC-s
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Chapter 2: Getting Started 23‘\h
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Chapter 2: Getting Started 25Anothe
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Chapter 2: Getting Started 27usage:
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Chapter 3: Data Types 293 Data Type
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Chapter 3: Data Types 31size (a, n)
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Chapter 4: Numeric Data Types 334 N
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Chapter 4: Numeric Data Types 35war
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Chapter 4: Numeric Data Types 37Emp
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Chapter 5: Strings 395 StringsA str
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Chapter 5: Strings 41string fill ch
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Chapter 5: Strings 435.3 String Con
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Chapter 5: Strings 45toupper (s)Bui
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Chapter 6: Data Structures 476 Data
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Chapter 6: Data Structures 49octave
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Chapter 7: Containers 517 Container
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Chapter 8: I/O Streams 538 I/O Stre
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Chapter 9: Variables 559 VariablesV
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Chapter 9: Variables 57persistent p
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Chapter 9: Variables 59exist (name,
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Chapter 9: Variables 61default_save
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Chapter 9: Variables 63warn_divide_
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Chapter 10: Expressions 6510 Expres
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Chapter 10: Expressions 67sqrt (arg
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Chapter 10: Expressions 69x - ySubt
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Chapter 10: Expressions 7110.5.1 El
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Chapter 10: Expressions 73z = 1Afte
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Chapter 10: Expressions 75++x--xThi
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Chapter 11: Evaluation 7711 Evaluat
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Chapter 12: Statements 7912 Stateme
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Chapter 12: Statements 81warn assig
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Chapter 12: Statements 83while (con
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Chapter 12: Statements 85for [ val,
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Chapter 12: Statements 87# print el
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Chapter 12: Statements 89For contin
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Chapter 13: Functions and Script Fi
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Chapter 14: Error Handling 10514 Er
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Chapter 15: Debugging 10715 Debuggi
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Chapter 16: Input and Output 10916
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Chapter 16: Input and Output 111for
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Chapter 16: Input and Output 113bit
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Chapter 16: Input and Output 115-ma
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Chapter 16: Input and Output 117The
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Chapter 16: Input and Output 119‘
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Chapter 16: Input and Output 121Thi
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Chapter 16: Input and Output 123‘
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Chapter 16: Input and Output 12516.
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Chapter 16: Input and Output 127max
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Chapter 16: Input and Output 129[nr
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Page 173 and 174: Chapter 19: Arithmetic 16319 Arithm
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Chapter 29: Control Theory 23329 Co
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Chapter 29: Control Theory 23529.1.
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Chapter 29: Control Theory 237outli
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Chapter 29: Control Theory 239nnzst
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Chapter 29: Control Theory 241[zer,
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Chapter 29: Control Theory 243Examp
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Chapter 29: Control Theory 245A mat
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Chapter 29: Control Theory 247clstu
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Chapter 29: Control Theory 249_____
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Chapter 29: Control Theory 251dsysa
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Chapter 29: Control Theory 253retsy
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Chapter 29: Control Theory 25529.5
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Chapter 29: Control Theory 257dlyap
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Chapter 29: Control Theory 25929.6
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Chapter 29: Control Theory 261retva
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Chapter 29: Control Theory 263sysef
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Chapter 29: Control Theory 265d2c (
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Chapter 29: Control Theory 267[y, t
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Chapter 29: Control Theory 269absys
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Chapter 29: Control Theory 273[k, p
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Chapter 29: Control Theory 275gmaxg
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Chapter 29: Control Theory 277syssi
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Chapter 29: Control Theory 279[rlda
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Chapter 29: Control Theory 281[a, b
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Chapter 30: Signal Processing 28330
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Chapter 30: Signal Processing 285wh
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Chapter 30: Signal Processing 287[1
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Chapter 30: Signal Processing 289hu
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Chapter 31: Image Processing 29131
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Chapter 31: Image Processing 293hsv
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Chapter 32: Audio Processing 29532
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Chapter 33: Quaternions 29733 Quate
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Chapter 34: System Utilities 29934
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Chapter 34: System Utilities 301asc
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Chapter 34: System Utilities 303eti
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Chapter 34: System Utilities 305[er
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Chapter 34: System Utilities 307fnm
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Chapter 34: System Utilities 309in
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Chapter 34: System Utilities 311[er
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Chapter 34: System Utilities 31334.
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Chapter 34: System Utilities 315isi
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Appendix A: Tips and Standards 317A
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Appendix A: Tips and Standards 319
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Appendix A: Tips and Standards 321i
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Appendix B: Known Causes of Trouble
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Appendix C: Installing Octave 329Ap
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Appendix C: Installing Octave 331On
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Appendix C: Installing Octave 333gc
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Appendix D: Emacs Octave Support 33
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Appendix E: Grammar 343Appendix E G
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Appendix F: GNU GENERAL PUBLIC LICE
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Concept Index 351Concept Index#‘#
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Concept Index 353Iif statement . .
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Variable Index 355Variable IndexAan
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Function Index 357Function Index(([
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Function Index 359errorbar . . . .
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Function Index 361isnumeric . . . .
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Function Index 363qzval . . . . . .
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Function Index 365vertcat. . . . .
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Operator Index 367Operator Index!!.
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