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);

$0.1037571496-0.1044950325\mathrm{I},0.9630190393+0.5220371079\mathrm{I}$

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

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

$0.7244935146-0.06234310868\mathrm{I},0.5109964553-0.1847149940\mathrm{I},0.3598077053-0.2438873156\mathrm{I},\mathrm{-0.5709515960}-0.0002188584214\mathrm{I}$

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

$0.5112153138-0.7556665900\mathrm{I},0.5107775969+0.3862366020\mathrm{I}$

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

$\mathrm{-0.8089166307}-2.183698556\mathrm{I},2.267853794-0.8137995614\mathrm{I},\mathrm{-1.776971141}-0.9961751070\mathrm{I},\mathrm{-1.078568331}+1.766448520\mathrm{I},\mathrm{-0.6866379254}+1.124556055\mathrm{I},1.131254963+0.6341847698\mathrm{I}$

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

$0.3106257308+0.01505583261\mathrm{I},0.3511670099+0.07163120875\mathrm{I}$

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

$0.8531469480-0.9735866595\mathrm{I},0.9933652429+0.3464383081\mathrm{I}$

FresnelC, FresnelS:

FresnelC(z),FresnelS(z);

$1.682212498+0.4016272750\mathrm{I},0.07110045108+1.175585852\mathrm{I}$

dawson:

dawson(z);

$0.7494749046-0.1808917014\mathrm{I}$

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);

$0.9413942917-0.02100700736\mathrm{I},2.883535667+0.08445357415\mathrm{I},0.9032179896+0.4174342314\mathrm{I}$

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

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

$0.7242068410+0.3592870858\mathrm{I},0.4926925228+1.780640518\mathrm{I},0.5430830974+0.4039826151\mathrm{I}$

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

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

$0.6737225716+0.9153170422\mathrm{I},0.4248129606-1.954337561\mathrm{I},0.1660202658+0.9486420664\mathrm{I}$

Elliptic Nome:

q:=EllipticNome(k);

$q≔\mathrm{-0.1130622230}-0.004343557919\mathrm{I}$

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

JacobiAM(z,k);

$4.005054544+2.900860234\mathrm{I}$

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

$\mathrm{-6.933891559}-5.892090470\mathrm{I},\mathrm{-5.927814364}+6.892104554\mathrm{I},\mathrm{-15.37625018}-14.22897650\mathrm{I}$

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

$\mathrm{-0.08374711996}+0.07116430985\mathrm{I},\mathrm{-0.07173049406}-0.08339904633\mathrm{I},\mathrm{-0.03503417853}+0.03242016078\mathrm{I}$

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

$0.005976741231+1.000922504\mathrm{I},0.005965516634-0.9990427250\mathrm{I}$

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

$0.4339457152-0.01837332974\mathrm{I},2.300311961+0.09739556973\mathrm{I}$

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

$\mathrm{-0.01576703103}-0.4336399162\mathrm{I},\mathrm{-0.08373704817}+2.303016116\mathrm{I}$

Jacobi Theta Functions: JacobiTheta1, JacobiTheta2, JacobiTheta3, JacobiTheta4

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

$1.200435806-0.4743634443\mathrm{I},0.01315436257-1.009987991\mathrm{I},1.175151117+0.3717688091\mathrm{I},0.8212434355-0.3687317775\mathrm{I}$

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);

$0.3204287896-0.6325222283\mathrm{I},0.3493485146+1.127304155\mathrm{I},0.6741545530-0.4702886987\mathrm{I},0.9993746770+0.6595322141\mathrm{I}$

General Pochhammer function over the complex plane: pochhammer

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

$0.5216047723-0.3157393579\mathrm{I}$

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);

$\left[t=1.0\,x\left(t\right)=13.1246093792716\,\frac{\ⅆ}{\ⅆt}x\left(t\right)=18.7876915350014\,y\left(t\right)=3.76283413230601\,\frac{\ⅆ}{\ⅆt}y\left(t\right)=5.31760397608216\,\frac{{\ⅆ}^2}{\ⅆt^2}y\left(t\right)=10.2504683623596\right]$

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);

$\left[x=1.01\,y\left(x\right)=2.43020104070930\,\frac{\ⅆ}{\ⅆx}y\left(x\right)=3.04031295467142\,\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)=4.06288854913227\right]$

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);

$\mathrm{ans3}\left(0.7653\right)$

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);

$\left[x=1.5\,y\left(x\right)=4.59316878380163\,\frac{\ⅆ}{\ⅆx}y\left(x\right)=6.33685490613476\,\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)=11.0436021501495\right]$

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');

$R≔\left[?\right]$

eval(q);

$\left[?\right]$

r;

$2$

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);

$0.496T\left(0\,x\right)+1.22T\left(1\,x\right)+0.494T\left(2\,x\right)+0.0718T\left(3\,x\right)-0.00212T\left(4\,x\right)-0.00227T\left(5\,x\right)-0.000344T\left(6\,x\right)-0.0000390T\left(7\,x\right)+5.\times {10}^{\mathrm{-7}}T\left(8\,x\right)+1.37\times {10}^{\mathrm{-6}}T\left(9\,x\right)+1.36\times {10}^{\mathrm{-7}}T\left(10\,x\right)$

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] ] );

$\left[\left[1\&comma;0.16\&comma;0.16\right]\&comma;\left[4\&comma;6.67\&comma;1.668\right]\&comma;\left[5\&comma;6.83\right]\right],\left[1\&comma;4\&comma;0.09592\&comma;0.2278\right]$

least median of squares regression: fit[leastmediansquare]

with(stats[fit]):

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

$y=x+1$

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]);

Error, (in linalg:-matrix) invalid argument Vector[row](5, [n,Payment,Interest,Principal,Balance])

cost_of_loan:=A[2];

$\mathrm{cost\_of\_loan}≔\left[\right]$

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);

$-\frac{0.224}{x+0.809}+\frac{0.224x+0.0636}{x^2-0.810x+0.229}$

factor(x^3+5.0);

$\left(x+1.71\right)\left(x^2-1.71x+2.92\right)$

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

$\mathrm{-0.02}$

content( 5.4*x*y, x);

$5.4y$

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);

$1.\times {10}^{9223372036854775806}$

Maple_floats(sqrt(MAX_FLOAT)/LN_MAX_FLOAT);

$4.708630210\times {10}^{4611686018427387883}$