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Various methods derived from single equation solutions can be used to solve simultaneous nonlinear equations. This section extends the Newton-Raphson method to handle systems of n equations in n variables. The procedure involves concepts like matrices, vectors, norms, and solutions of linear systems. Starting with an initial estimate vector, the Newton-Raphson method seeks an improved estimate based on truncated Taylor series. The multivariable Newton-Raphson method follows steps outlined in Procedure 3.5. Numerical integration techniques like the Trapezoidal rule and Simpson’s rule are introduced for evaluating integrals of polynomials.
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3.2 SYSTEMS FOR NONLINEAR EQUATIONS • Several methods for solving simultaneous nonlinear equations may be derived from methods that are used for a single equation. The Newton-Raphson method of §3.1.5 is extended in this section to solve systems of nonlinear equations. These systems consist of n equations in n variables (x1, x2, …, xn) and have the general form
• Our development of the procedure for solving systems of nonlinear equations requires the concepts of matrices, vectors, norms, and solutions of linear systems that are discussed in Chapter 2. We begin by defining a column vector x in transpose form as • • And rewrite the system of equations more compactly as
• As in the Newton-Raphson method, we start the solution process with an initial estimate x0of the solution vector. We write the initial vector as • And seek an improved estimate The procedure for determining h is based on the truncated Taylor series for a function of several variables. If the truncation takes place immediately following the first derivative terms of series, we have
( ) fk x h We then obtain the vector h for which is zero from the linear 0 system of equations given by The matrix form of Eq. [3.10a] is In which f is the column vector given by
• And J is the partial derivative matrix known as the Jacobian and is given by
The vector h in Eq. [3.10b] may be easily found by any of several methods when n is small. • The procedure for the multivariable Newton-Raphson method is now given. It follows the same sequence of steps given in procedure 3.5 of §3.1.5 • • Procedure 3.7 The Newton-Raphson method for systems of equations. 1. Choose a starting vector x0. 2. Compute fk(x0) for k =1, 2, …, m 3. If ║f ║≤ ε, x0is the estimated solution, otherwise, go on to Step4. 4. Obtain h from Eq.[3.10b] and compute ximpfrom Eq.[3.9] 5. Set x0equal to ximpand return to Step 2. • • • • •
7.1 Integration of functions Integrals of elementary functions could not, in general, be • computed analytically. Numerical integration (Quadrature) is widely used in such situations. The evaluation of the integral I is equivalent to solving for the differential equation Q with the boundary condition B. where,
7.1.3 Trapezoidal rule This rule evaluates the integrals of polynomials up to and • including degree 1. This rule is a two-point formula. • The error term O(amount ) implies that the true answer differs from the estimated one by the value amount .
7.1.4 Simpson’s rule This rule evaluates the integrals of polynomials up to and • including degree 2. This rule is a three-point formula.
7.1.6 Trapezoidal rule implementation The following subroutine implements the trapezoidal rule to evaluate the integral • external func read*,a,b,n call trapzd(func,a,b,s) print*,s End SUBROUTINE trapzd(func,a,b,s,n ( INTEGER n REAL a,b,s,func EXTERNAL func INTEGER it,j REAL del,sum,tnm,x If (n.eq.1) then s=0.5*(b-a)*(func(a)+func(b)) else it=2**(n-2) tnm=it del=(b-a)/tnm x=a+0.5*del sum=0.0 • • • • • • • • • • • • • • • • • • • •
do j=1,it sum=sum+func(x( x=x+del Enddo s=0.5*(s+(b-a)*sum/tnm) endif return END function func(x) func=5 return end • • • • • • • • • • • •
