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SandBoxDoublePowerSeries

Edit detail for SandBoxDoublePowerSeries revision 15 of 19

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Editor: Bill Page Time: 2014/08/29 23:10:06 GMT+0
Note: GSERIES

changed:
-aa:DMP([k,l], FRAC INT) := 1+reduce(+,concat [[a(i,j)*k^i*l^j for i in 0..2] for j in 1..2])
aa:DMP([k,l], FRAC INT) := 1+reduce(+,concat [[a(i,j)*k^i*l^j for i in 0..3] for j in 1..3])

changed:
-Because sqrt(x) does not have a Taylor series. Even though cosh(sqrt(x))
Because 'sqrt(x)' does not have a Taylor series. Even though 'cosh(sqrt(x))'

changed:
-poly2(cosh(sqrt x)*sinh(y),5,5)::DMP(['x,'y],FRAC INT)
poly2(cosh(sqrt x)*sinh(sqrt y)/sqrt(y),3,3)::DMP(['x,'y],FRAC INT)
aa

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a!

$\label{eq1}a !$

(1)

Type: Variable(a!)

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!:=operator '!

$\label{eq2}!$

(2)

Type: BasicOperator?

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a(i,j)==(1+(j-1)*i)/!(2*i+4*j)

Type: Void

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a0:=matrix [[a(i,j)*k^i*l^j for i in 0..5] for j in 1..4]

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Compiling function a with type (NonNegativeInteger,PositiveInteger)
       -> Expression(Integer)

$\label{eq3}\left[ \begin{array}{cccccc} {l \over{! \left({4}\right)}}&{{k \ l}\over{! \left({6}\right)}}&{{{{k}^{2}}\ l}\over{! \left({8}\right)}}&{{{{k}^{3}}\ l}\over{! \left({1 0}\right)}}&{{{{k}^{4}}\ l}\over{! \left({12}\right)}}&{{{{k}^{5}}\ l}\over{! \left({14}\right)}} \ {{{l}^{2}}\over{! \left({8}\right)}}&{{2 \ k \ {{l}^{2}}}\over{! \left({10}\right)}}&{{3 \ {{k}^{2}}\ {{l}^{2}}}\over{! \left({1 2}\right)}}&{{4 \ {{k}^{3}}\ {{l}^{2}}}\over{! \left({14}\right)}}&{{5 \ {{k}^{4}}\ {{l}^{2}}}\over{! \left({16}\right)}}&{{6 \ {{k}^{5}}\ {{l}^{2}}}\over{! \left({18}\right)}} \ {{{l}^{3}}\over{! \left({12}\right)}}&{{3 \ k \ {{l}^{3}}}\over{! \left({14}\right)}}&{{5 \ {{k}^{2}}\ {{l}^{3}}}\over{! \left({1 6}\right)}}&{{7 \ {{k}^{3}}\ {{l}^{3}}}\over{! \left({18}\right)}}&{{9 \ {{k}^{4}}\ {{l}^{3}}}\over{! \left({20}\right)}}&{{{11}\ {{k}^{5}}\ {{l}^{3}}}\over{! \left({22}\right)}} \ {{{l}^{4}}\over{! \left({16}\right)}}&{{4 \ k \ {{l}^{4}}}\over{! \left({18}\right)}}&{{7 \ {{k}^{2}}\ {{l}^{4}}}\over{! \left({2 0}\right)}}&{{{10}\ {{k}^{3}}\ {{l}^{4}}}\over{! \left({22}\right)}}&{{{1 3}\ {{k}^{4}}\ {{l}^{4}}}\over{! \left({24}\right)}}&{{{16}\ {{k}^{5}}\ {{l}^{4}}}\over{! \left({26}\right)}}$

(3)

Type: Matrix(Expression(Integer))

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a(i:INT,j:INT):FRAC INT == (1+(j-1)*i)/Gamma(1+2*i+4*j)

   Function declaration a : (Integer,Integer) -> Fraction(Integer) has 
      been added to workspace.
   Compiled code for a has been cleared.
   1 old definition(s) deleted for function or rule a

Type: Void

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aa:DMP([k,l], FRAC INT) := 1+reduce(+,concat [[a(i,j)*k^i*l^j for i in 0..3] for j in 1..3])

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Compiling function a with type (Integer,Integer) -> Fraction(Integer
      )

$\label{eq4}\begin{array}{@{}l} \displaystyle {{1 \over{914624815104000}}\ {{k}^{3}}\ {{l}^{3}}}+{{1 \over{2 1794572800}}\ {{k}^{3}}\ {{l}^{2}}}+ \ \ \displaystyle {{1 \over{3628800}}\ {{k}^{3}}\ l}+{{1 \over{4184557977600}}\ {{k}^{2}}\ {{l}^{3}}}+{{1 \over{159667200}}\ {{k}^{2}}\ {{l}^{2}}}+ \ \ \displaystyle {{1 \over{40320}}\ {{k}^{2}}\ l}+{{1 \over{29059430400}}\ k \ {{l}^{3}}}+{{1 \over{1814400}}\ k \ {{l}^{2}}}+{{1 \over{7 20}}\ k \ l}+ \ \ \displaystyle {{1 \over{479001600}}\ {{l}^{3}}}+{{1 \over{40320}}\ {{l}^{2}}}+{{1 \over{24}}\ l}+ 1$

(4)

Type: DistributedMultivariatePolynomial?([k,l],Fraction(Integer))

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eval(aa,[k=0,l=l])

$\label{eq5}{{1 \over{479001600}}\ {{l}^{3}}}+{{1 \over{40320}}\ {{l}^{2}}}+{{1 \over{24}}\ l}+ 1$

(5)

Type: DistributedMultivariatePolynomial?([k,l],Polynomial(Fraction(Integer)))

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eval(aa,[k=k,l=0])

$\label{eq6}1$

(6)

Type: DistributedMultivariatePolynomial?([k,l],Polynomial(Fraction(Integer)))

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eval(aa,[k=1.0,l=1.0])

$\label{eq7}1.0431059938_809735313$

(7)

Type: Polynomial(Float)

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eval(aa,[k=0,l=0])

$\label{eq8}1$

(8)

Type: DistributedMultivariatePolynomial?([k,l],Polynomial(Fraction(Integer)))

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eval(aa,[k=-1,l=-1])::Float

$\label{eq9}0.9597219508_1267275382$

(9)

Type: Float

Q: Why doesn't this work?

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x:TaylorSeries FRAC INT

Type: Void

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y:TaylorSeries FRAC INT

Type: Void

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cosh(sqrt x)

   >> Error detected within library code:
   ^: rational power does not exist

Because sqrt(x) does not have a Taylor series. Even though cosh(sqrt(x)) does have a Taylor series, it can not be constructed this way.

Ans: Use GSERIES

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x:GeneralUnivariatePowerSeries(FRAC INT,'x,0)

Type: Void

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cosh(sqrt x)

$\label{eq10}1 +{{1 \over 2}\ x}+{{1 \over{24}}\ {{x}^{2}}}+{{1 \over{720}}\ {{x}^{3}}}+{{1 \over{40320}}\ {{x}^{4}}}+{{1 \over{3628800}}\ {{x}^{5}}}+{O \left({{x}^{{11}\over 2}}\right)}$

(10)

Type: GeneralUnivariatePowerSeries?(Fraction(Integer),x,0)

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poly(s,n)==reduce(+,[ [(i.c)*variable(s)^(i.k) for i in terms s].j for j in 1..n])

Type: Void

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poly(sinh(sqrt x),5)

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Compiling function poly with type (GeneralUnivariatePowerSeries(
      Fraction(Integer),x,0),PositiveInteger) -> Expression(Integer)

$\label{eq11}{{\left({{x}^{4}}+{{72}\ {{x}^{3}}}+{{3024}\ {{x}^{2}}}+{{604 80}\ x}+{362880}\right)}\ {\sqrt{x}}}\over{362880}$

(11)

Type: Expression(Integer)

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y:GeneralUnivariatePowerSeries(GeneralUnivariatePowerSeries(FRAC INT,'x,0),'y,0)

Type: Void

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poly2(ss,n,m)==reduce(+,[ [poly(i.c,n)*variable(ss)^(i.k) for i in terms ss].j for j in 1..m])

Type: Void

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poly2(cosh(sqrt x)*sinh(sqrt y)/sqrt(y),3,3)::DMP(['x,'y],FRAC INT)

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Compiling function poly2 with type (GeneralUnivariatePowerSeries(
      GeneralUnivariatePowerSeries(Fraction(Integer),x,0),y,0),
      PositiveInteger,PositiveInteger) -> Expression(Integer)

$\label{eq12}\begin{array}{@{}l} \displaystyle {{1 \over{2880}}\ {{x}^{2}}\ {{y}^{2}}}+{{1 \over{144}}\ {{x}^{2}}\ y}+{{1 \over{24}}\ {{x}^{2}}}+{{1 \over{240}}\ x \ {{y}^{2}}}+{{1 \over{12}}\ x \ y}+{{1 \over 2}\ x}+ \ \ \displaystyle {{1 \over{120}}\ {{y}^{2}}}+{{1 \over 6}\ y}+ 1$

(12)

Type: DistributedMultivariatePolynomial?([x,y],Fraction(Integer))

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aa

$\label{eq13}\begin{array}{@{}l} \displaystyle {{1 \over{914624815104000}}\ {{k}^{3}}\ {{l}^{3}}}+{{1 \over{2 1794572800}}\ {{k}^{3}}\ {{l}^{2}}}+ \ \ \displaystyle {{1 \over{3628800}}\ {{k}^{3}}\ l}+{{1 \over{4184557977600}}\ {{k}^{2}}\ {{l}^{3}}}+{{1 \over{159667200}}\ {{k}^{2}}\ {{l}^{2}}}+ \ \ \displaystyle {{1 \over{40320}}\ {{k}^{2}}\ l}+{{1 \over{29059430400}}\ k \ {{l}^{3}}}+{{1 \over{1814400}}\ k \ {{l}^{2}}}+{{1 \over{7 20}}\ k \ l}+ \ \ \displaystyle {{1 \over{479001600}}\ {{l}^{3}}}+{{1 \over{40320}}\ {{l}^{2}}}+{{1 \over{24}}\ l}+ 1$

(13)

Type: DistributedMultivariatePolynomial?([k,l],Fraction(Integer))

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)lib GDRAW

   GnuDraw is now explicitly exposed in frame initial 
   GnuDraw will be automatically loaded when needed from 
      /var/aw/var/LatexWiki/GDRAW.NRLIB/GDRAW
gnuDraw(aa,k=-30..60,l=-30..30,"SandBoxDoublePowerSeries1.dat",title=="Generating Function")

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Compiling function %B with type (DoubleFloat,DoubleFloat) -> 
      DoubleFloat 
   Transmitting data...

Type: Void

[terminal=pslatex,terminaloptions=color,scale=1.3]
set view 60, 30, 0.85, 1.1
set samples 20, 20
set isosamples 21, 21
set contour base
set cntrparam levels auto 20
set xlabel "k axis"
set ylabel "l axis"
load "SandBoxDoublePowerSeries1.dat"