PLib

Used as argument for a non rational functions.

Used as argument for a non rational functions.

Returns the Binomial Cnp. N should be <= BSplCLib::MaxDegree().

Computes the derivatives of a ratio at order <N> in dimension <Dimension>.
<Ders> is an array containing the values of the input derivatives from 0 to std::min(<N>,<Degree>). For orders higher than <Degree> the inputcd /s2d1/BMDL/ derivatives are assumed to be 0.
Content of <Ders>:
x(1),x(2),...,x(Dimension),w x'(1),x'(2),...,x'(Dimension),w' x''(1),x''(2),...,x''(Dimension),w''
If <All> is false, only the derivative at order <N> is computed. <RDers> is an array of length Dimension which will contain the result:
x(1)/w , x(2)/w , ... derivated <N> times
If <All> is true all the derivatives up to order <N> are computed. <RDers> is an array of length Dimension * (N+1) which will contains:
x(1)/w , x(2)/w , ... x(1)/w , x(2)/w , ... derivated <1> times x(1)/w , x(2)/w , ... derivated <2> times ... x(1)/w , x(2)/w , ... derivated <N> times
Warning: <RDers> must be dimensioned properly.

Computes DerivativesRequest derivatives of a ratio at of a BSpline function of degree <Degree> dimension <Dimension>.
<PolesDerivatives> is an array containing the values of the input derivatives from 0 to <DerivativeRequest> For orders higher than <Degree> the input derivatives are assumed to be 0.
Content of <PoleasDerivatives> :
x(1),x(2),...,x(Dimension) x'(1),x'(2),...,x'(Dimension) x''(1),x''(2),...,x''(Dimension)
WeightsDerivatives is an array that contains derivatives from 0 to <DerivativeRequest> After returning from the routine the array RationalDerivatives contains the following x(1)/w , x(2)/w , ... x(1)/w , x(2)/w , ... derivated once x(1)/w , x(2)/w , ... twice x(1)/w , x(2)/w , ... derivated <DerivativeRequest> times
The array RationalDerivatives and PolesDerivatives can be same since the overwrite is non destructive within the algorithm
Warning: <RationalDerivates> must be dimensioned properly.

Performs Horner method with synthetic division for derivatives parameter , with <Degree> and <Dimension>. PolynomialCoeff are stored in the following fashion.

c0(1)c0(2)....c0(Dimension) c1(1)c1(2)....c1(Dimension) cDegree(1)cDegree(2)....cDegree(Dimension)

where the polynomial is defined as :

2Degree c0+c1X+c2X+....cDegreeX

Results stores the result in the following format

f(1)f(2)....f(Dimension) (1)(1)(1) f(1)f(2)....f(Dimension) (DerivativeRequest)(DerivativeRequest) f(1)f(Dimension)

this just evaluates the point at parameter U
Warning: <Results> and <PolynomialCoeff> must be dimensioned properly

Same as above with DerivativeOrder = 0;.

Applies EvalPolynomial twice to evaluate the derivative of orders UDerivativeOrder in U, VDerivativeOrder in V at parameters U,V.
PolynomialCoeff are stored in the following fashion

c00(1)....c00(Dimension) c10(1)....c10(Dimension) .... cm0(1)....cm0(Dimension) .... c01(1)....c01(Dimension) c11(1)....c11(Dimension) .... cm1(1)....cm1(Dimension) .... c0n(1)....c0n(Dimension) c1n(1)....c1n(Dimension) .... cmn(1)....cmn(Dimension)

where the polynomial is defined as :

2m c00+c10U+c20U+....+cm0U 2m +c01V+c11UV+c21UV+....+cm1UV nmn +....+c0nV+....+cmnUV

with m = UDegree and n = VDegree
Results stores the result in the following format

f(1)f(2)....f(Dimension)

Warning: <Results> and <PolynomialCoeff> must be dimensioned properly

Performs the Lagrange Interpolation of given series of points with given parameters with the requested derivative order Results will store things in the following format with d = DerivativeOrder.

[0],[Dimension-1]:value [Dimension],[Dimension+Dimension-1]:firstderivative [d*Dimension],[d*Dimension+Dimension-1]:dthderivative

Performs the Cubic Hermite Interpolation of given series of points with given parameters with the requested derivative order. ValueArray stores the value at the first and last parameter. It has the following format :

[0],[Dimension-1]:valueatfirstparam [Dimension],[Dimension+Dimension-1]:valueatlastparam

Derivative array stores the value of the derivatives at the first parameter and at the last parameter in the following format

[0],[Dimension-1]:derivativeat

first param

[Dimension],[Dimension+Dimension-1]:derivativeat

last param
ParameterArray stores the first and last parameter in the following format :

[0]:firstparameter [1]:lastparameter

Results will store things in the following format with d = DerivativeOrder

[0],[Dimension-1]:value [Dimension],[Dimension+Dimension-1]:firstderivative [d*Dimension],[d*Dimension+Dimension-1]:dthderivative

This build the coefficient of Hermite's polynomes on [FirstParameter, LastParameter].
if j <= FirstOrder+1 then
MatrixCoefs[i, j] = ith coefficient of the polynome H0,j-1
else
MatrixCoefs[i, j] = ith coefficient of the polynome H1,k with k = j - FirstOrder - 2
return false if

• |FirstParameter| > 100
• |LastParameter| > 100
• |FirstParameter| +|LastParameter| < 1/100
• |LastParameter - FirstParameter| / (|FirstParameter| +|LastParameter|) < 1/100

Compute the coefficients in the canonical base of the polynomial satisfying the given constraints at the given parameters The array FirstContr(i,j) i=1,Dimension j=0,FirstOrder contains the values of the constraint at parameter FirstParameter idem for LastConstr.

Compute the number of points used for integral computations (NbGaussPoints) and the degree of Jacobi Polynomial (WorkDegree). ConstraintOrder has to be GeomAbs_C0, GeomAbs_C1 or GeomAbs_C2 Code: Code d' init. des parametres de discretisation. = -5 = -4 = -3 = -2 = -1 = 1 calcul rapide avec precision moyenne. = 2 calcul rapide avec meilleure precision. = 3 calcul un peu plus lent avec bonne precision. = 4 calcul lent avec la meilleure precision possible.

translates from GeomAbs_Shape to Integer

translates from Integer to GeomAbs_Shape

Initialize the polynomial class Degree has to be <= 30 ConstraintOrder has to be GeomAbs_C0 GeomAbs_C1 GeomAbs_C2.

This method computes the maximum error on the polynomial W(t) Q(t) obtained by missing the coefficients of JacCoeff from NewDegree +1 to Degree.

Compute NewDegree <= MaxDegree so that MaxError is lower than Tol. MaxError can be greater than Tol if it is not possible to find a NewDegree <= MaxDegree. In this case NewDegree = MaxDegree.

Convert the polynomial P(t) = H(t) + W(t) Q(t) in the canonical base.

Compute the values of the basis functions in u.

Compute the values and the derivatives values of the basis functions in u.

Compute the values and the derivatives values of the basis functions in u.

Compute the values and the derivatives values of the basis functions in u.

returns WorkDegree

returns NivConstr

Initialize the polynomial class Degree has to be <= 30 ConstraintOrder has to be GeomAbs_C0 GeomAbs_C1 GeomAbs_C2.

returns the Jacobi Points for Gauss integration ie the positive values of the Legendre roots by increasing values NbGaussPoints is the number of points chosen for the integral computation. TabPoints (0,NbGaussPoints/2) TabPoints (0) is loaded only for the odd values of NbGaussPoints The possible values for NbGaussPoints are : 8, 10, 15, 20, 25, 30, 35, 40, 50, 61 NbGaussPoints must be greater than Degree

returns the Jacobi weights for Gauss integration only for the positive values of the Legendre roots in the order they are given by the method Points NbGaussPoints is the number of points chosen for the integral computation. TabWeights (0,NbGaussPoints/2,0,Degree) TabWeights (0,.) are only loaded for the odd values of NbGaussPoints The possible values for NbGaussPoints are: 8, 10, 15, 20, 25, 30, 35, 40, 50, 61 NbGaussPoints must be greater than Degree

this method loads for k=0,q the maximum value of abs ( W(t)Jk(t) ) for t bellonging to [-1,1] This values are loaded is the array TabMax(0,myWorkDegree-2(myNivConst+1)) MaxValue ( me ; TabMaxPointer : in out Real );

This method computes the maximum error on the polynomial W(t) Q(t) obtained by missing the coefficients of JacCoeff from NewDegree +1 to Degree.

Compute NewDegree <= MaxDegree so that MaxError is lower than Tol. MaxError can be greater than Tol if it is not possible to find a NewDegree <= MaxDegree. In this case NewDegree = MaxDegree.

Convert the polynomial P(t) = R(t) + W(t) Q(t) in the canonical base.

Compute the values of the basis functions in u.

Compute the values and the derivatives values of the basis functions in u.

Compute the values and the derivatives values of the basis functions in u.

Compute the values and the derivatives values of the basis functions in u.

returns WorkDegree

returns NivConstr