Polynomial inversion

This is a DACE example showing polynomial inversion, demonstrating:

How to load DACE.jl
How to initialise the DACE library
How to create a DA object
How to create an AlgebraicVector
How to invert a Taylor polynomial

Install dependencies

Make sure the required packages are installed

using Pkg
Pkg.add("DACE")

Using DACE

Write

using DACE

to load DACE functions and objects into our script.

Initialise DACE for 10th-order computations in 1 variable

DACE.init(10, 1)

Initialise x as a DA object

x = DACE.DA(1, 1)

     I  COEFFICIENT              ORDER EXPONENTS
     1    1.0000000000000000e+00   1   1
------------------------------------------------

Create y as AlgebraicVector of type DA and size 1

y = AlgebraicVector{DA}(1)

1-element DACE.AlgebraicVectorAllocated{DA}:
         ALL COEFFICIENTS ZERO
------------------------------------------------

Store the Taylor expansion of sin(x) in the first element of y

y[1] = sin(x)

     I  COEFFICIENT              ORDER EXPONENTS
     1    1.0000000000000000e+00   1   1
     2   -1.6666666666666666e-01   3   3
     3    8.3333333333333332e-03   5   5
     4   -1.9841269841269841e-04   7   7
     5    2.7557319223985893e-06   9   9
------------------------------------------------

Invert the Taylor polynomial

inv_y = DACE.invert(y)

1-element DACE.AlgebraicVectorAllocated{DA}:
      I  COEFFICIENT              ORDER EXPONENTS
     1    1.0000000000000000e+00   1   1
     2    1.6666666666666666e-01   3   3
     3    7.4999999999999997e-02   5   5
     4    4.4642857142857137e-02   7   7
     5    3.0381944444444437e-02   9   9
------------------------------------------------

Finally compare the polynomial inversion of sin(x)

println("polynomial inversion of sin(x)")
println(inv_y)

polynomial inversion of sin(x)
[[[ 1 vector
     I  COEFFICIENT              ORDER EXPONENTS
     1    1.0000000000000000e+00   1   1
     2    1.6666666666666666e-01   3   3
     3    7.4999999999999997e-02   5   5
     4    4.4642857142857137e-02   7   7
     5    3.0381944444444437e-02   9   9
------------------------------------------------

]]]

with asin(x)

println("asin(x)")
println(asin(x))

asin(x)
     I  COEFFICIENT              ORDER EXPONENTS
     1    1.0000000000000000e+00   1   1
     2    1.6666666666666669e-01   3   3
     3    7.5000000000000067e-02   5   5
     4    4.4642857142857206e-02   7   7
     5    3.0381944444444586e-02   9   9
------------------------------------------------

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