By John H. Mathews, Russell W. Howell
This article is meant for undergraduate, scholars in arithmetic, physics, and engineering. it's meant to stability among the natural and utilized elements of complicated research and to provide thoughts in a transparent writing sort that's comprehensible to scholars on the junior or senior undergraduate point. A wealth of workouts that change in either trouble and substance supplies the textual content flexibility. adequate functions are incorporated to demonstrate how complicated research is utilized in technological know-how and engineering.
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Extra info for Complex Analysis for Mathematics and Engineering
4022 G3 = FN F(X(I»: REM G3 RECALCULATED AT EACH NODE 4025 A(I)=G2 4030 B(I)=-G3 4035 C(I)=G1 4040 D(I)=-G4 4045 NEXT I 4050 B(2)=B(2)-A(2)*Y(1) 4055 B(N-1)=B(N-1)-C(N-1)*Y(N) 4060 FOR 1=3 TO N-1 4065 M1=A(I)/D(I-1) 4070 D(I)=D(I)-M1*C(I-1) 4075 B(I)=B(I)-M1*B(I-1) 4080 NEXT I 4082 Y(N-1)=B(N-1)/D(N-1) 4084 FOR I=N-2 TO 2 STEP -1 4086 O=Y(I) 4088 Y(I)=(B(I)-C(I)*Y(I+1»/D(I) 4090 D3=D3+ABS(O-Y(I» 4095 NEXT I 4100 C3 = C3 + 1: REM INCREASE COUNTER 4110 REM PRINT OUT TYPICAL VALUE AND THE "RESIDUAL" 4120 PRINT "RESIDUAL IS "1031" AFTER "lC3;" ITERATIONS" 4130 PRINT "Y("l INT (N / 2)1") = "lY( INT (N / 2» 4200 REM IF NOT CONVERGED GO BACK TO 4015 4210 IF 03 > IE - 6 THEN GOTO 4015 4300 PRINT "HIT RETURN TO CONTINUE" 4310 INPUT QS: REM APPLE IS BETTER WITH "GET QS" 4999 RETURN Better Solution Methods 39 The remaining subroutines (5000-5999 and 9000-9999) are exactly as in Program 4.
A2nX n This scheme is convergent if the sum of the absolute values of the coefficients on the right-hand side of the equations is less than 1, except for perhaps one equation, in which the coefficients may add up to 1. This is known as the 'diagonal dominance criterion'. The proof that this condition is sufficient to ensure convergence is beyond the scope of this book. 8 Use Program 6 to prove that the diagonal dominance criterion is not necessary for convergence. 2 The idea behind adding 1E-6 to the denominator of the fraction in 4035 is to ensure that if all the values of Y happen to be zero (as they are at the outset in BASIC) 4035 will not give a 'DNISION BY ZERO' error.
In fact it matters a great deal. We can write a very simple program to compute the solution of two simultaneous equations Al*X + Bl*Y = Cl A2*X + B2*Y = C2 by the two possible iterative methods. 2). 99609375 1 3 7 15 31 63 127 255 511 1023 1 3 7 15 31 63 127 255 511 1023 Scheme 2 1 2 3 4 5 6 7 8 9 10 Clearly the first scheme converges and the second diverges. , see if you can work out the rule for determining which scheme will converge and which will diverge. ] It is possible to show that if we arrange the equations in the form x = Ml*Y + Kl Y = M2*X + K2 32 Microcomputer Modelling by Finite Differences where both M1 and M2 are between -1 and +1, then the scheme we get will truly converge.
Complex Analysis for Mathematics and Engineering by John H. Mathews, Russell W. Howell