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{SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 30 "Overview of Numerical Ana
lysis" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 23 "Floating points numbers
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{MPLTEXT 1 0 31 "plot(exp(x),x=0..10); # ..100 ?" }}}{EXCHG {PARA 0 ">
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{MPLTEXT 1 0 29 "plot(10^39*sin(x),x=0..2*Pi);" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 9 "underflow" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "plot(
10^(-10)*x,x=1..2); # how low can we go ?" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 50 "plot(10^(-42)*x,x=0..10,view=[0..10,0..10^(-38)]);
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"" {MPLTEXT 1 0 37 "plot(1+10^(-7)*x,x=0..10); #4,5,6,..." }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT 
-1 32 "Failure of usual algebraic rules" }}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 28 "This is always true, right ?" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 14 "x + y - x = y;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
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(x)(y);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 84 "In other words, floati
ng point addition is not associative (but it is commutative): " }
{XPPEDIT 18 0 "(x+y)+z <> x+(y+z)" "6#0,(%\"xG\"\"\"%\"yGF&%\"zGF&,&F%
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0 "> " 0 "" {MPLTEXT 1 0 28 "h := exp(x+1/exp(x))-exp(x);" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 39 "Related problem \"massive cancellation\":
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{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "evalb(0. = -0.);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "evalb(Float(infinity) = Float(-infi
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),x=0,order=1..10,view=[-3..2,-2..2]);" }}}{EXCHG {PARA 0 "> " 0 "" 
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{TEXT -1 15 "Differentiation" }}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 28 "U
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0 35 "p := piecewise(x=0,0,x^2*sin(1/x));" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 36 "NewtonQuotient( p, x = 0, h = 'h' );" }}}{EXCHG 
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