355w2003_lab06

355w2003_lab06 - MLC Lab Visit Lab 06 Maple Mth 355(a.k.a...

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MLC Lab Visit - Lab 06 - Maple Mth 355 (a.k.a. Mth 399) Feb 12, 2003 Maple 7 Bent E. Petersen [email protected] There are 3 problems below. Problem solutions are due Feb 19, 2003. Email your solutions to me as Maple worksheet attachments. Your worksheet must execute correctly for full credit. In this week’s lab we investigate a few randomly chosen features of Maple. Partial Fractions > restart; Partial fraction decomposition in Maple is carried out with the convert() command. > f:=(3*x^2-2*x+7)/( (x^2+1)*(x-3)^3*(x^2+x-1) ); := f + 3 x 2 2 x 7 () + x 2 1( ) x 3 3 + x 2 x 1 > convert(f,parfrac,x); + + 14 55 1 x 3 3 512 3025 1 x 3 2 30197 332750 x 3 1 250 + 91 3 x + x 2 1 5 1331 + 75 38 x + x 2 x 1 Note since all the coefficients here are rational, Maple attempts the expansion over the rational numbers. This is the reason the last term is not what you were led to expect in calculus since x^2+x-1 does factor over the rationals. We can force a factorization by telling Maple to use the extension field of the rationals obtained by appending the square root of 5. > convert(f,parfrac,x,sqrt(5)); 2 1331 56 59 5 + + 2 x 1 5 2 1331 + 56 5 + 2 x 1 5 1 250 + 3 x + x 2 1 30197 332750 x 3 512 3025 1 x 3 2 + + 14 55 x 3 3 +

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This expression should agree with what you learned in calculus, but it is inconvenient, and not really general, to have to specifiy the radicals to append. A more general and more convenient procedure is to use RootOf:: > h:=convert(f,parfrac,x,RootOf(x^2+x-1)); h 1 1331 + 151 112 ( ) RootOf + _Z 2 _Z 1 x () RootOf + _Z 2 _Z 1 1 1331 + 39 112 ( ) RootOf + _Z 2 _Z 1 + + x 1( ) RootOf + _Z 2 _Z 1 + := 1 250 + 91 3 x + x 2 1 30197 332750 x 3 512 3025 1 x 3 2 14 55 x 3 3 + + + Cool, but not very familiar. We need to evaluate the RootOf() expressions. This may be done with the allvalues() command, but it will evaluate the full expression for each root (and so give us 2 copies): > allvalues(h); 1 1331 + 56 59 5 + x 1 2 1 2 5 1 1331 56 5 + + x 1 2 1 2 5 1 250 + 3 x + x 2 1 30197 332750 x 3 512 3025 1 x 3 2 + + + 14 55 x 3 3 + 1 1331 95 56 5 + + x 1 2 1 2 5 1 1331 95 56 5 + x 1 2 1 2 5 1 250 + 3 x + x 2 1 30197 332750 x 3 + + + , 512 3025 1 x 3 2 14 55 x 3 3 + As expected we got 2 copies of the answer. Just pick one: > allvalues(h)[1]; 1 1331 + 56 5 + x 1 2 1 2 5 1 1331 56 5 + + x 1 2 1 2 5 1 250 + 3 x + x 2 1 30197 332750 x 3 512 3025 1 x 3 2 + + +
14 55 () x 3 3 + We can also request that Maple perform the factorization over the real numbers. In this case Maple will use floating point arithmetic. > convert(f,parfrac,x,real); .02270458805 1 + x 1.618033989 .1654544002 x .6180339887 .2545454546 x 3. 3 .1692561984 x 3. 2 + .09074981217 x 3. + .03599999996 .05200000000 x + x 2 1. + + You can make a rational approximation in the obvious way: > convert(convert(f,parfrac,x,real),rational); 6119 269505 1 + x 28657 17711 2070 12511 1 x 17711 28657 14 55 x 3 3 512 3025 1 x 3 2 27659 304783 x 3 + + + 3599996 99999889 13 250 x + x 2 1 + Let’s look at complex roots: > g:=( (2*x^3-x-1)/((x^2+2*x+5)^2*(x^2-1)) ); := g 2 x 3 x 1 + + x 2 2 x 5 2 x 2 1 > convert(g,parfrac,x); + 1 16 1 + x 1 1 16 + x 1 + + x 2 2 x 5 1 4 − + 17 x + + x 2 2 x 5 2 Exactly what we expected! Of course we may prefer to expand over the complex numbers: > convert(g,parfrac,x,complex);

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+ .1250000000 .2187500000 I () + x + 1.000000000 2.000000000 I 2 + .03125000000 .06250000000 I + x + 1. 2. I .06250000000 + x 1.
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355w2003_lab06 - MLC Lab Visit Lab 06 Maple Mth 355(a.k.a...

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