Numerical Methods Library for OCTAVE

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3.2.1 Normal equation[X, RES] = NORMALEQ(A,B) computes the solution of linear least squares problemmin(norm(A*X-B,2)) solving associated normal equation A’*A*X = A’*B.ParametersA a matrix.B right hand side vector.ReturnsX computed solution.RES value of norm(A*X-B) with X solution computed.3.2.2 Householder[X, RES] = LLS_HQR(A,B) computes the solution of linear least squares problemmin(norm(A*X-B,2)) using Householder QR.ParametersA a matrix.B right hand side vector.ReturnsX computed solution.RES value of norm(A*X-B) with X solution computed.3.2.3 SVD[X, RES] = SVD_LEAST_SQUARES(A,B) computes the solution of linear least squaresproblem min(norm(A*X-B,2)) using the singular value decomposition of A.ParametersA a matrix.B right hand side vector.ReturnsX computed solution.RES value of norm(A*X-B) with X solution computed.3.3 Nonlinear equationsWe will now consider methods for solving nonlinear equations. Given a nonlinear function f ,we seek a value x for whichf (x) = 0. (2)Such a solution value for x is called a root of the equation, and a zero of the function f .Though technically they have distinct meanings, these two terms are informally used more or9
less interchangeably, with the obvious meaning. Thus, this problem is often referred to as rootfinding or zero finding. In discussing numerical methods for solving nonlinear equations, wewill distinguish two cases:andf : R → R (scalar), (3)f : R n → R n (vector). (4)The latter is referred to as a system of nonlinear equations, in which we seek a vector xsuch that all the component functions of f (x) are zero simultaneously.3.3.1 BisectionX = BISECTION(FUN,A,B,ITMAX,TOL) tries to find a zero X of the continuousfunction FUN in the interval [A, B] using the bisection method. FUN acceptsreal scalar input x and returns a real scalar value. If the search failsan error message is displayed. FUN can also be an inline object.[X, RES, NBIT] = BISECTION(FUN,A,B,ITMAX,TOL) returns the value of the residualin X solution and the iteration number at which the solution was computed.ParametersFUN evaluated function.A,B [A,B] interval where the solution is computed, A < B and sign(FUN(A))= - sign(FUN(B)).ITMAX maximal number of iterations.TOL tolerance on the stopping criterion.ReturnsX computed solution.NBIT number of iterations to find the solution.RES value of the residual in X solution.3.3.2 Fixed-pointX = FIXED_POINT(FUN,X0,ITMAX,TOL) solves the scalar nonlinear equation suchthat ’FUN(X) == X’ with FUN continuous function. FUN accepts real scalarinput X and returns a real scalar value. If the search fails an error messageis displayed. FUN can also be inline objects.[X, RES, NBIT] = FIXED_POINT(FUN,X0,ITMAX,TOL) returns the norm of the residualin X solution and the iteration number at which the solution was computed.ParametersFUN evaluated function.DFUN f’s derivate.10
Page 3 and 4: 3.8.2 Implicit Euler . . . . . . .
Page 6 and 7: • Euler• Implicit Euler• Modi
Page 8 and 9: X0 initial guess.ReturnsX computed
Page 12 and 13: X0 initial point.ITMAX maximal numb
Page 14 and 15: 3.4.1 Monomial basisP = ITPOL_MONOM
Page 16 and 17: N number of subdivisions.ReturnsRES
Page 18 and 19: 3.6.4 Orthogonal iteration[LAMBDA,
Page 20 and 21: H f (xk)s k = f (x k ) (12)for s k
Page 22 and 23: 3.8.1 Euler[TT,Y] = ODE_EULER(ODEFU
Page 24 and 25: condition Y0 using the fourth-order
Page 26 and 27: Figure 2: Shooting method for a two
Page 28 and 29: X equispaced grid of NX intervals.U
Page 30 and 31: ReturnsT grid of NT intervals.X gri
Page 32: A, BC, D rectangle (A,B)x(C,D) wher

Numerical Methods Library for OCTAVE

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