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CALCULATOR MODULE : Solve Simultaneous Equation ±
Calculate the solution vector (A) for the linear simultaneous equations, from the linear equation matrix (M), and the result vector (C). The simultaneous equations are solved using the Crout or Chelensky factorial `L · U = M `
`M · A = C `
`L · D = C `
`U · A = D ` where : L is a lower diagonal matrix
U is an upper diagonal matrix.
D is an intermediate result vector Enter each matrix row as a comma separated list, with a new line for each row. Matrices must have an equal number of elements in each row. The number of rows in the matrix must be equal to the number of columns. The matrix equations must be independent (ie well conditioned - determinant not equal to zero). If the matrix determinant equals zero, the equations are indeterminant. Enter vectors as a comma separated list. The number of elements in the vector must equal the number of matrix columns. Change Module : |
CALCULATOR MODULE : Maths Special Function ±
Calculate special function values. Elliptic integrals are calculated using Carlsons forms. Jacobi elliptic functions are calculated using Landens transformation. The Gamma function is calculated using the Lanczos approximation. Reference : Numerical Recipes, The Art Of Scientific Computing, Press, Teukolsky, Vetterling, Flannery, Cambridge University Press Change Module : |
CALCULATOR MODULE : Maths Elliptic Integral ±
Calculate the complete and incomplete elliptic integrals of the first, second and third kind from the elliptic k modulus. Elliptic integrals are calculated for an ellipse of the form `x^2 + (y / b)^2 = 1 `
`k = √(1 - 1 / b^2) ` where : k = the elliptic k modulus For a circle k = 0. k tends to 1 as b tends to infinity. Use the Result Plot option to plot the integrals versus the k modulus. Reference : Numerical Recipes, The Art Of Scientific Computing, Press, Teukolsky, Vetterling, Flannery, Cambridge University Press Change Module : |
CALCULATOR MODULE : Maths Elliptic Function ±
Calculate the elliptic functions and elliptic amplitude (sn, cn, dn and am). Elliptic functions are calculated for an ellipse of the form `x^2 + (y / b)^2 = 1 `
`k = √(1 - 1 / b^2) ` where : k = the elliptic k modulus For a circle k = 0. k tends to 1 as b tends to infinity. Use the Result Plot option to plot the elliptic functions versus the k modulus. Reference : Numerical Recipes, The Art Of Scientific Computing, Press, Teukolsky, Vetterling, Flannery, Cambridge University Press Change Module : |
CALCULATOR MODULE : Maths Gamma Function ±
Calculate the Gamma function Γ(z), the log Gamma function ln(Γ(z)), the incomplete lower Gamma function γ(z,x), the incomplete upper Gamma function Γ(z,x), the incomplete lower Gamma unit function PL(z,x), the incomplete upper Gamma unit function PU(z,x). The Gamma function is a continuous form of the integer factorial: `n! = Γ(n + 1) ` The Gamma function is recursive for values greater than 1 or less than 0 (z > 1 or z < 0). `Γ(z + 1) = z Γ(z) ` The Gamma function is invalid if z equals zero, or if z is a negative integer. The lower incomplete Gamma unit functions are defined as `PL(z,x) = γ(z,x) / Γ(z) `
`PU(z,x) = Γ(z,x) / Γ(z) ` Use the Result Plot option to plot the Gamma functions versus z. Reference : Numerical Recipes, The Art Of Scientific Computing, Press, Teukolsky, Vetterling, Flannery, Cambridge University Press Change Module : |
CALCULATOR MODULE : Maths Error Function ±
Calculate the error function (erf) and complementary error function (erfc). The error function asymptotes to 1 as x tends to infinity. The complementary error function asymptotes to 0 as x tends to infinity. Reference : Numerical Recipes, The Art Of Scientific Computing, Press, Teukolsky, Vetterling, Flannery, Cambridge University Press Change Module : |
CALCULATOR MODULE : Maths Probability Function ±
Calculate normal and log normal probability density, cumulative distribution function and complementary distribution function from the mean, standard deviation and percentile. The cumulative probability can be calculated from the percentile, or the percentile can be calculated from the cumulative probability. The cumulative distribution (cdf) function asymptotes to 1 as the percentile tends to infinity. The complementary distribution function asymptotes to 0 as the percentile x tends to infinity. Use the Result Plot option to plot the probability density, cumulative distribution, and complementary distribution (or tail) versus the percentile. Reference : Numerical Recipes, The Art Of Scientific Computing, Press, Teukolsky, Vetterling, Flannery, Cambridge University Press Change Module : |
CALCULATOR MODULE : Maths Beta Function ±
Calculate the Beta function B(z, w) versus z and w. The Beta function is calculated from the Gamma function `B(z, w) = (Γ(z) x Γ(w)) / Γ(z + w) ` The Beta function tends to infinity for z equals zero, or if z is a negative integer. Reference : Numerical Recipes, The Art Of Scientific Computing, Press, Teukolsky, Vetterling, Flannery, Cambridge University Press Change Module : |
CALCULATOR MODULE : Maths Trigonometry ±
Calculate the maths trigonometric functions (sin, cos, tan, cosec, sec and cot), and the inverse trigonometric functions (asin, acos, atan, acosec, asec, and acot). The trigonometric fuctions are circular functions calculated for a right angle triangle with a hypotenuse equals 1 and angle θ. The base length of the triangle = cos(θ). The height of the triangle equals sin(θ). The slope of the hypotenuse equals tan(θ), equals sin(θ) / cos(θ). From Pythagorus' formula cos(θ)^2 + sin(θ)^2 = 1. The functions cosec (1/sin), sec (1/cos) and cot (1/tan) were mainly used for navigation. Change Module : |
CALCULATOR MODULE : Maths Right Angle Triangle (Pythagorus) ±
Calculate maths triangle sides, heights and angles for a right angled triangle. The area is calculated from the base and height. Angles, sides and heights can be calculated using cos, sin, tan, and Pythagorus theorem. Change Module : |
CALCULATOR MODULE : Maths Scalene Triangle (Three Unequal Sides) ±
Calculate maths triangle sides, heights and angles for a scalene triangle. Scalene triangles have three unequal sides, and three unequal angles. The area is calculated from the base and height. Angles, sides and heights can be calculated using the sin rule and the cosine rule. Change Module : |
CALCULATOR MODULE : Maths Equilateral Triangle (Three Equal Sides) ±
Calculate maths triangle sides, heights and angles for an equilateral triangle. Equilateral triangles have three equal sides, and three equal angles (60 degrees). The area is calculated from the base and height. Change Module : |
CALCULATOR MODULE : Maths Isoceles Triangle (Two Equal Sides) ±
Calculate maths triangle sides, heights and angles for an isoceles triangle. Isoceles triangles have two equal sides, and two ewqqual angles. The area is calculated from the base and height. The sides and angles can be calculated using the sin rule and the cosine rule. Change Module : |
CALCULATOR MODULE : Maths Hyperbolic Functions ±
Calculate maths hyperbolic functions (sinh, cosh, tanh, csch (1/sinh), sech (1/cosh), and coth (1/tanh)), and hyperbolic inverse functions (asinh, acosh, atanh, asch, asech, acoth). The hyperbolic functions are defined as `cosh(x) = (e^x + e^-x) / 2 `
`sinh(x) = (e^x - e^-x) / 2 `
`tanh(x) = sinh(x) / cosh(x) `
`cosh(x)^2 - sinh(x)^2 = 1` Change Module : |
CALCULATOR MODULE : Maths Elliptic Functions ±
Calculate maths elliptic functions (sn, cn, and dn). The ellipse is of the form. `x^2 + (y/b)^2 = 1 `
`m = √(1 - 1/b^2) `
`x = (cn(u, m)) / (dn(u, m)) `
`y = (sn(u, m)) / (dn(u, m)) ` where : b = the semi major axis (y axis)
u = the elliptic arc length
m = the elliptic parameter The semi minor axis (x axis) a = 1, and the semi major axis (y axis) b ≥ 1. The elliptic parameter 0 ≤ m ≤ 1. There is an alternative form for m which is used in some methods. `m' = 1 - 1/b^2 `
`m' = m^2 ` Refer to the Keisan special function calculators for check values (https://keisan.casio.com/ : Home => Professional => Special Function => Elliptic Function). The Keisan functions use u and k instead of u and m. Reference : Numerical Calculation of Elliptic Integrals and Elliptic Functions, R. Bulirsch, 1965 Change Module : |