.. |newpage| raw:: latex \newpage .. |cr| raw:: latex \hspace{0.0mm} |newpage| Scalar functions =============================================================================== Real and complex sine ------------------------------------------------------------------------------- .. method:: ctxlib.sin(x) where ``ctx`` is ``ctx_pm`` (see :ref:`Python contexts ` for details), ``ctx53``, ``ctxcpp``, ``ctxflint`` (see :ref:`.NET contexts ` for details). The corresponding ``ctx`` python lists are ``ctxlistreal`` and ``ctxlistcplx``. Returns the sine of `x`, `\sin(x)`. See also Wikipedia :cite:p:`WikipediaFun31`, MathWorld :cite:p:`WolframFun31`, NIST :cite:p:`DLMFun30`, :cite:t:`Ehrhardt2018` (4.2.55), Flint :cite:p:`FlintFun30`, Flint :cite:p:`FlintFun31`, Mpmath :cite:p:`MpmathFun31`. |newpage| .. _rst_mpm_quadratic_equation_roots: Quadratic equation ------------------------------------------------------------------------------- .. method:: ctx.eval_quadratic(x, A, B, C) Returns the value of a quadratic polynomial, `A x^2 + B x + C`. .. method:: 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 :cite:p:`WikipediaAlg02`, :cite:t:`Press2007`. See also: https://dlmf.nist.gov/1.11#iii .. math :: 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: .. code-block:: pycon >>> 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%) |newpage| .. _rst_mpm_cubic_equation_monic_roots: Monic cubic equation ------------------------------------------------------------------------------- .. method:: ctx.eval_monic_cubic(x, a, b, c) Returns the value of a monic cubic polynomial, `x^3 + a x^2 + b x + c`. .. method:: 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 :cite:p:`WikipediaAlg03`, :cite:t:`Press2007`. See also: https://dlmf.nist.gov/1.11#iii .. math :: 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)`: .. math :: 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 .. math :: 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: .. code-block:: pycon >>> 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%) |newpage| .. _rst_mpm_cubic_equation_roots: Cubic equation ------------------------------------------------------------------------------- .. method:: ctx.eval_cubic(x, A, B, C, D) Returns the value of a cubic polynomial, `A x^3 + B x^2 + C x + D`. .. method:: 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 :cite:p:`WikipediaAlg03`, :cite:t:`Press2007`. See also: https://dlmf.nist.gov/1.11#iii This just calls :ref:`CubicEquationMonicRoots(z, a, b, c) ` with `a = B / A, b = C / A, c = D / A`. An example with real input: .. code-block:: pycon >>> 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%) |newpage| .. _rst_mpm_quartic_equation_roots: Quartic equation ------------------------------------------------------------------------------- .. method:: 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`. .. method:: 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 :cite:p:`WikipediaAlg04`. 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 :ref:`CubicEquationMonicRoots(y, e, f, g) `. An example with real input: .. code-block:: pycon >>> 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%) |newpage| Transscribed from Julia: complex elliptic functions in double precision ------------------------------------------------------------------------------- Text describing functions |newpage| Transscribed from Julia: complex hurwitz function, and related, in double precision -------------------------------------------------------------------------------------------- Text describing functions |newpage| Speeedups for iterative algorithms which require an initial guess ------------------------------------------------------------------------------- Text describing functions Inverses of cdfs: inverses of noncentral functions inverses livk gamma_inva |newpage| 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