Scalar functions

Real and complex sine

ctxlib.sin(x)

where ctx is ctx_pm (see Python contexts for details), ctx53, ctxcpp, ctxflint (see .NET contexts for details). The corresponding ctx python lists are ctxlistreal and ctxlistcplx.

Returns the sine of \(x\), \(\sin(x)\). See also Wikipedia [1350], MathWorld [947], NIST [517], Ehrhardt [309] (4.2.55), Flint [812], Flint [802], Mpmath [584].

Quadratic equation

ctx.eval_quadratic(x, A, B, C)

Returns the value of a quadratic polynomial, \(A x^2 + B x + C\).

ctx.quadratic_equation(A, B, C)

Returns the roots \(x_1, x_2\) of the quadratic equation \(A x^2 + B x + C = 0\). See also Wikipedia [1189], Press et al. [499].

See also: https://dlmf.nist.gov/1.11#iii

\[x_1 = \frac{Q}{A}, \quad x_2 = \frac{C}{Q}, \quad \text{where } Q = -\frac{1}{2} \left(B + \sqrt{B^2 - 4AC}) \right)\]

The sign of the square root is chosen so as to make \(\displaystyle \Re(B^* \sqrt{B^2 - 4AC}) \ge 0\), where the asterisk denotes complex conjugation.

An example with real input:

>>> from xlcalcnet import dec, mpm, ipm
>>> mpm.dps = 40; x = '125'; n = '3'
>>> \mathrm{d}x = dec.nthroot(x, n); mx = mpm.nthroot(x, n); ix = ipm.nthroot(x, n)
>>> mpm.show([\mathrm{d}x, mx, ix])
dec:  5.000000000000000000000000000000000000000E+0
mpm:  5.000000000000000000000000000000000000000e+0
ipm:  5.000000000000000000000000000000000000000e+0 (2.755e-39%)

>>> from xlcalcnet import mpm, fpm, gmp, apm
>>> mpm.dps = 40; x = '125'; n = '3'
>>> fx = fpm.nthroot(x, n); gx = gmp.nthroot(x, n); ax = apm.nthroot(x, n)
>>> mpm.show([fx, gx, ax])
fpm:  5.00000000000000E+00
gmp:  5.000000000000000000000000000000000000000E+00
apm:  5.000000000000000000000000000000000000002e+0 (3.673e-39%)

Monic cubic equation

ctx.eval_monic_cubic(x, a, b, c)

Returns the value of a monic cubic polynomial, \(x^3 + a x^2 + b x + c\).

ctx.cubic_equation_monic(a, b, c)

Returns the roots \(x_1, x_2, x_3\) of the monic cubic equation \(x^3 + a x^2 + b x + c = 0\). See also Wikipedia [1180], Press et al. [499].

See also: https://dlmf.nist.gov/1.11#iii

\[Q = \frac{a^2 - 3b}{9}, \quad R = \frac{2a^3 - 9ab + 27c}{54}.\]

If \(Q\) and \(R\) are real and \(R^2 < Q^3\), then the cubic equation has three real roots, with \(\displaystyle \theta = \arccos \left( R Q^{-3/2} \right)\):

\[x_1 = -2 \sqrt{Q} \cos \left( \frac{\theta}{3} \right) - \frac{a}{3}, \quad x_2 = -2 \sqrt{Q} \cos \left( \frac{\theta + 2\pi}{3} \right) - \frac{a}{3}, \quad x_3 = -2 \sqrt{Q} \cos \left( \frac{\theta - 2\pi}{3} \right) - \frac{a}{3}.\]

Otherwise, \(\displaystyle A = - \left(R + \sqrt{R^2 - Q^3}) \right)^{1/3}\), where the sign of the square root is chosen so as to make \(\displaystyle \Re(R^* \sqrt{R^2 - Q^3}) \ge 0\), and the asterisk denotes complex conjugation. Define \(B = 0\) if \(A = 0\) and \(B = Q / A\) if \(A \ne 0\). Then the three roots are given by

\[x_1 = (A + B) - \frac{a}{3}, \quad x_2 = -\frac{1}{2} (A + B) - \frac{a}{3} + i \frac{\sqrt{3}}{2} (A - B), \quad x_3 = -\frac{1}{2} (A + B) - \frac{a}{3} - i \frac{\sqrt{3}}{2} (A - B).\]

Note that \(x_1\) is real if \(a, b, c\) are real.

An example with real input:

>>> from xlcalcnet import dec, mpm, ipm
>>> mpm.dps = 40; x = '125'; n = '3'
>>> \mathrm{d}x = dec.nthroot(x, n); mx = mpm.nthroot(x, n); ix = ipm.nthroot(x, n)
>>> mpm.show([\mathrm{d}x, mx, ix])
dec:  5.000000000000000000000000000000000000000E+0
mpm:  5.000000000000000000000000000000000000000e+0
ipm:  5.000000000000000000000000000000000000000e+0 (2.755e-39%)

>>> from xlcalcnet import mpm, fpm, gmp, apm
>>> mpm.dps = 40; x = '125'; n = '3'
>>> fx = fpm.nthroot(x, n); gx = gmp.nthroot(x, n); ax = apm.nthroot(x, n)
>>> mpm.show([fx, gx, ax])
fpm:  5.00000000000000E+00
gmp:  5.000000000000000000000000000000000000000E+00
apm:  5.000000000000000000000000000000000000002e+0 (3.673e-39%)

Cubic equation

ctx.eval_cubic(x, A, B, C, D)

Returns the value of a cubic polynomial, \(A x^3 + B x^2 + C x + D\).

ctx.cubic_equation(A, B, C, D)

Returns the roots \(x_1, x_2, x_3\) of the cubic equation \(A x^3 + B x^2 + C x + D = 0\). See also Wikipedia [1180], Press et al. [499].

See also: https://dlmf.nist.gov/1.11#iii

This just calls CubicEquationMonicRoots(z, a, b, c) with \(a = B / A, b = C / A, c = D / A\).

An example with real input:

>>> from xlcalcnet import dec, mpm, ipm
>>> mpm.dps = 40; x = '125'; n = '3'
>>> \mathrm{d}x = dec.nthroot(x, n); mx = mpm.nthroot(x, n); ix = ipm.nthroot(x, n)
>>> mpm.show([\mathrm{d}x, mx, ix])
dec:  5.000000000000000000000000000000000000000E+0
mpm:  5.000000000000000000000000000000000000000e+0
ipm:  5.000000000000000000000000000000000000000e+0 (2.755e-39%)

>>> from xlcalcnet import mpm, fpm, gmp, apm
>>> mpm.dps = 40; x = '125'; n = '3'
>>> fx = fpm.nthroot(x, n); gx = gmp.nthroot(x, n); ax = apm.nthroot(x, n)
>>> mpm.show([fx, gx, ax])
fpm:  5.00000000000000E+00
gmp:  5.000000000000000000000000000000000000000E+00
apm:  5.000000000000000000000000000000000000002e+0 (3.673e-39%)

Quartic equation

ctx.eval_quartic(x, A, B, C, D, E)

Returns the value of a quartic polynomial, \(A x^4 + B x^3 + C x^2 + D x + E\).

ctx.quartic_equation(A, B, C, D, E)

Returns the roots \(x_1, x_2, x_3, x_4\) of the quartic equation \(A x^4 + B x^3 + C x^2 + D x + E = 0\). See also Wikipedia [1190].

See also: https://dlmf.nist.gov/1.11#iii

Define \(\displaystyle a = \frac{-3 B^2}{8 A^2} + \frac{C}{A}, \quad b = \frac{ B^3}{8 A^3} - \frac{BC}{2 A^2} + \frac{D}{A}, \quad c = \frac{-3 B^4}{256 A^4} + \frac{CB^2}{16 A^3} - \frac{BD}{4 A^2} + \frac{E}{A}, \quad V = \frac{B}{4 A}\).

If \(b = 0\) then

\(\displaystyle x_1 = V + Z_1, \quad x_2 = V - Z_1, \quad x_3 = V + Z_2, \quad x_4 = V - Z_2\),

where \(\displaystyle W = \sqrt{a^2 - 4c}, \quad Z_1 = \sqrt{\tfrac{1}{2}(-a + W)}, \quad Z_2 = \sqrt{\tfrac{1}{2}(-a - W)}\).

If \(b \ne 0\) then

\(\displaystyle x_1 = V + \tfrac{1}{2}(W + Z_1), \quad x_2 = V + \tfrac{1}{2}(W - Z_1), \quad x_3 = V - \tfrac{1}{2}(W + Z_2), \quad x_4 = V -\tfrac{1}{2}(W - Z_2)\),

where \(\displaystyle W = \sqrt{a + 2y}, \quad Z_1 = \sqrt{-3a - 2y - \frac{2b}{W}}, \quad Z_2 = \sqrt{-3a - 2y + \frac{2b}{W}}\),

and \(y\) is any root of the monic cubic equation \(\displaystyle y^3 + ey^2+ fy + g =0\), with \(\displaystyle e = \frac{5a}{2}, \quad f = 2 a^2 -c, \quad g = \frac{a^3}{2} - \frac{a c}{2} - \frac{b^2}{8}\);

\(y\) is calculated as the first root returned by CubicEquationMonicRoots(y, e, f, g).

An example with real input:

>>> from xlcalcnet import dec, mpm, ipm
>>> mpm.dps = 40; x = '125'; n = '3'
>>> \mathrm{d}x = dec.nthroot(x, n); mx = mpm.nthroot(x, n); ix = ipm.nthroot(x, n)
>>> mpm.show([\mathrm{d}x, mx, ix])
dec:  5.000000000000000000000000000000000000000E+0
mpm:  5.000000000000000000000000000000000000000e+0
ipm:  5.000000000000000000000000000000000000000e+0 (2.755e-39%)

>>> from xlcalcnet import mpm, fpm, gmp, apm
>>> mpm.dps = 40; x = '125'; n = '3'
>>> fx = fpm.nthroot(x, n); gx = gmp.nthroot(x, n); ax = apm.nthroot(x, n)
>>> mpm.show([fx, gx, ax])
fpm:  5.00000000000000E+00
gmp:  5.000000000000000000000000000000000000000E+00
apm:  5.000000000000000000000000000000000000002e+0 (3.673e-39%)

Transscribed from Julia: complex elliptic functions in double precision

Text describing functions

Transscribed from Julia: complex hurwitz function, and related, in double precision

Text describing functions

Speeedups for iterative algorithms which require an initial guess

Text describing functions

Inverses of cdfs: inverses of noncentral functions

inverses livk gamma_inva

Examples for matrix functions

Text describing functions

Descriptive statistics via numpy in multiple precision

Multiple linear regression via numpy in multiple precision

Canonical correlation via numpy in multiple precision

Descriptive statistics via Eigen in multiple precision

Multiple linear regression via Eigen in multiple precision

Canonical correlation via Eigen in multiple precision

Levenberg-Marquardt algorithm via Eigen in multiple precision

L-BFGS algorithm via Eigen in multiple precision