Arbitrary precision evaluation, over the complex plane, of a wider range of special functions, including Elliptic and Inverse Elliptic, family of Bessel functions.

Bessel family: AiryAi, AiryBi, BesselI, BesselJ, BesselK, BesselY, HankelH1, HankelH2, KelvinBer, KelvinBei, KelvinKer, KelvinKei, KelvinHer, KelvinHei, StruveH, StruveL, AngerJ, WeberE:

z:=evalf(1+2/3*I):

AiryAi(z), AiryBi(z);

nu:=evalf(2/3+4/5*I):

BesselI(nu,z), BesselJ(nu,z), BesselK(nu,z), BesselY(nu,z);

HankelH1(nu,z), HankelH2(nu,z);

KelvinBer(nu,z), KelvinBei(nu,z), KelvinKer(nu,z), KelvinKei(nu,z),
KelvinHer(nu,z), KelvinHei(nu,z);

StruveH(nu,z), StruveL(nu,z);

AngerJ(nu,z), WeberE(nu,z);

FresnelC, FresnelS:

FresnelC(z),FresnelS(z);

dawson:

dawson(z);

Elliptic and Inverse Elliptic: EllipticK, EllipticE, EllipticPi, EllipticF, EllipticE, EllipticPi, EllipticCK, EllipticCE, EllipticCPi, EllipticNome

k:=-0.1+2.3*I:

Complete elliptic integrals of the first, second and third kind:

EllipticK(k),EllipticE(k),EllipticPi(nu,k);

Incomplete elliptic integrals of the first, second and third kind:

EllipticF(z,k),EllipticE(z,k),EllipticPi(z,nu,k);

Complementary complete elliptic integrals of the first, second and third kind:

EllipticCK(k),EllipticCE(k),EllipticCPi(nu,k);

Elliptic Nome:

q:=EllipticNome(k);

Jacobi Amplitude and  Elliptic functions: JacobiAM, JacobiSN, JacobiCN, JacobiDN, JacobiNS, JacobiNC, JacobiND, JacobiSC, JacobiCS, JacobiSD, JacobiDS, JacobiCD, JacobiDC

JacobiAM(z,k);

JacobiSN(z,k), JacobiCN(z,k), JacobiDN(z,k);

JacobiNS(z,k), JacobiNC(z,k), JacobiND(z,k);

JacobiSC(z,k), JacobiCS(z,k);

JacobiSD(z,k), JacobiDS(z,k);

JacobiCD(z,k), JacobiDC(z,k);

Jacobi Theta Functions: JacobiTheta1, JacobiTheta2, JacobiTheta3, JacobiTheta4

JacobiTheta1(z,q), JacobiTheta2(z,q),
JacobiTheta3(z,q), JacobiTheta4(z,q);

Weierstrass P, P', zeta and sigma functions: WeierstrassP, WeierstrassPPrime, WeierstrassZeta, WeierstrassSigma

g2:=-0.11-0.7*I: g3:=0.22-0.4*I:

WeierstrassP(z,g2,g3), WeierstrassPPrime(z,g2,g3),
WeierstrassZeta(z,g2,g3), WeierstrassSigma(z,g2,g3);

General Pochhammer function over the complex plane: pochhammer

pochhammer(1.0+0.2*I,-0.3+0.7*I);

Many extensions to dsolve/numeric extending Maple's power in developing numerical solutions to differential equations.

Classical: dsolve[classical]

restart;

deq1 := diff(y(t),t$3)=y(t)+diff(x(t),t),diff(x(t),t$2)=
x(t)*y(t)-diff(y(t), t$2):

init1 := y(0)=1,D(y)(0)=2,(D@@2)(y)(0)=-1,x(0)=4,D(x)(0)=4.3:

ans1 := dsolve({deq1,init1},{x(t),y(t)},numeric,method=classical[abmoulton],
corrections=2):

ans1(1.0);

Gear: dsolve[gear] - a C.W.Gear single-step extrapolation method.

Digits := 10:

deq1 := {diff(y(x), x$3) = y(x)*diff(y(x), x)-x}:

init1 := {(D@@2)(y)(1) = 4, D(y)(1) = 3, y(1) = 2.4 }:

ans1 := dsolve(deq1 union init1, y(x), type=numeric,
method=gear[polyextr], stepsize=0.015, minstep=Float(1,-11),
errorper=Float(1,-5)):

ans1(1.01);

Mgear: dsolve[mgear] -  a C.W.Gear multiple-step extrapolation method.

Digits := 12:

deq3 := diff(y(t), t$2) = 100*(exp(-10*t)+exp(10*t)):

ans3 := dsolve({deq3}, y(t), numeric, method=mgear[msteppart],
initial=array([2,0]), start=0):

Error, (in dsolve/numeric) the 'mgear' method has been removed, please use 'rosenbrock' or 'lsode' instead

ans3(0.7653);

Lsode: dsolve[lsode] - the "Livermore Stiff ODE Solver"

Digits := 10:

deq1 := {diff(y(x), x$3) = y(x)*diff(y(x), x) - x }:

init1 := { (D@@2)(y)(1) = 4, D(y)(1) = 3, y(1) = 2.4 }:

ans1 := dsolve(deq1 union init1, y(x), type=numeric,
method=lsode[adamsfull]):

ans1(1.5);

Numerical computations for matrix-decomposition routines: linalg[cholesky], linalg[QRdecomp], linalg[LUdecomp]

with(linalg):

A := matrix([[1.,2.,3.],[4.,5.,6.]]):

R := QRdecomp(A,Q='q',rank='r');

eval(q);

r;

linalg[eigenvals], linalg[inverse]: use hardware floating-point when possible.

numapprox[chebmult]: Tools to manipulate Chebyshev series:

with(numapprox):  Digits:=3:

a:=chebyshev(sin(x),x):

b:=chebyshev(exp(x),x):

chebmult(a,b);

ANOVA (oneway): stats[anova]

with(stats[anova],oneway):

example with variance ratio of 0.1 with 1 and 4 degrees of freedom. Level of significance is 0.23 < 0.95 (non-signif)

oneway( [ [1.0,2.0,3.0],[1.0,2.0,4.0] ] );

least median of squares regression: fit[leastmediansquare]

with(stats[fit]):

leastmediansquare[[x,y]]( [[1,2,3,4],[2,3,5,5]] );

The package finance provides functions for calculations related to finance. Some of the functions are: annuity, cashflows, perpetuity.

Amortization table of a loan of 100 at 10% interest, with payments of 50 per period

A:=finance[ amortization ] ( 100, 50, .10 ):

amortization_table[N, Payment, Interest, Principal, Balance]=matrix(A[1]);

cost_of_loan:=A[2];

The routines factor, convert/parfrac, resultant and content have been extended to handle floating points (for content, it just works for simple cases).

f := (2.3*x)/(5.4*x^3-2.3*x+1):

convert(f,parfrac,x);

factor(x^3+5.0);

resultant( 1.1*x+1.2, 1.3*x+1.4, x);

content( 5.4*x*y, x);

The routine Maple_floats computes the values of various parameters and constants associated with the arbitrary precision floating point computation environment.

readlib(Maple_floats):

Maple_floats(MAX_FLOAT);

Maple_floats(sqrt(MAX_FLOAT)/LN_MAX_FLOAT);