## RapidCalculator

Without our technology, most mobile devices would not have the computational capabilities (processor power) to carry out such intense calculations. We also support large scale computations.
The syntax for RapidCalculator, while not exactly one to one, is most parts similar or the same as the syntax for MATLAB, Mathematica or MAPLE.

Note: These are just examples, and are by no means exhaustive. RapidCalculator is a powerful tool, feel free to explore the computational possibilities!

### Sections

Mathematical formRapidCalculator syntax
RapidCalculator follows the order of operations:
((5X7)+6-2)÷8
((5*7)+6-2)/8
Modulo:
7 mod 5
7 % 5
Imaginary Numbers:
1. i
2. 5+i
3. 6-3i
1. 1i or 1j
2. 5+1i or 5+1j
3. 6-3i or 6-3j
Note: j is used in Electrical Engineering to mean $\sqrt{-1}$
Conjugate of complex numbers:
(8-i)*
(8-1j).conjugate()
Complex number operations:
1. $\sqrt{(8-i)(7-i)}$
2. $Re[\sqrt{(8-i)(7-i)}]$
3. $Im[\sqrt{(8-i)(7-i)}]$
1. sqrt((8-1j)*(7-8j))
2. sqrt((8-1j)*(7-8j)).real
3. sqrt((8-1j)*(7-8j)).imag
Power:
$6^{0.3}$
6^0.3
Square root:
$\sqrt{5}$
sqrt(5)
Exponentiation:
$e^{0.6}$
exp(0.6)
Natural Logarithm:
ln(0.7)
log(0.7)
$\pi$ pi
Trigonometric functions and their inverses:
1. sin(0.5)
2. cos(0.5)
3. tan(0.5)
4. arcsin(0.5)
5. arccos(0.5)
6. arctan(0.5)
1. sin(0.5) or _sin(0.5)
2. cos(0.5) or _cos(0.5)
3. tan(0.5) or _tan(0.5)
4. arcsin(0.5)
5. arccos(0.5)
6. arctan(0.5)
Hyperbolic functions and their inverses:
1. sinh(0.5)
2. cosh(0.5)
3. tanh(0.5)
4. arcsinh(0.5)
5. arccosh(0.5)
6. arctanh(0.5)
1. sinh(0.5) or _sinh(0.5)
2. cosh(0.5) or _cosh(0.5)
3. tanh(0.5) or _tanh(0.5)
4. arcsinh(0.5)
5. arccosh(0.5)
6. arctanh(0.5)
Defining a single variable function:
1. $x^2+2x+1$
2. $\cos(x)+\arctan(x)+x^2$
1. poly_x[x^2+2*x+1]
2. poly_x[cos(x)+arctan(x)+x^2]
Defining a multi-variable function (we support variables from a-z single letters only. Hence the maximum you can define is a 26-variable function):
$x^2+2xy+\cos(ab)+1$
poly_x[x^2+2*x*y+cos(a*b)+1]
Discrete summation of functions:
$\sum_{x=1}^{14} \left(x^2y+z\right)$
Note: The lower and upper bounds of the summation should be integers
poly_x[evalsum{x^2*y+z,x,1,14}]
Evaluation of single variable functions at fixed values:
1. $\lim_{n\to\infty}\left(1+\frac{1}{n}\right)^n$
2. $f(5)$ where $f(x)=e^x$
Note: This function can also be used to take limits!
1. poly_x[evalx{(1+1/n)^n,n,infty}]
2. poly_x[evalx{exp(x),x,5}]
Taylor series expansion:
1. Series expansion of $f(x)=e^x$ about point $x=0$ (Maclaurin series) up to the term $x^5$
2. Series expansion of $f(x,y)=y\sinh(x)$ about point $x=1$ up to the term $x^7$

1. poly_x[taylors{exp(x),x,0,5}]
2. poly_x[taylors{sinh(x)*y,x,1,7}]
Other function operations (including calculus):
1. $(x+1)^3$
2. $(2x^2+2x+1)(x-1)$
3. $\frac{d}{dx}(4x^5-3x+1)$
4. $\int 4x^2-3x+1 dx$
5. $\frac{d}{dx}(\cos x+\arctan x+x^2)$
6. $\frac{d}{dx}\left(x^2+3\right)+\frac{d}{dy}\left(y^2+3y\right)$
7. Partial derivatives: $\frac{\partial}{\partial x}\left(x^2+3xy\right)$
8. Multivariable integration: $\int x^2yz dx$
9. Higher order derivatives: $\frac{\partial^4}{\partial x^4}\left(x^{10}+3x^{12}y\right)$
10. Multiple integrals: $\int \int \left(x^{10}+3x^{12}y\right) dx dy$
1. poly_x[(x+1)^3]
2. poly_x[(2*x^2+2*x+1)*(x-1)]
3. poly_x[4*x^5-3*x+1].ddiff{x}
4. poly_x[4*x^2-3*x+1].cint
or
poly_x[(4*x^2-3*x+1).cint]
or
poly_x[(4*x^2-3*x+1).dint{x}]
or
poly_x[(4*x^2-3*x+1)].dint{x}
5. poly_x[cos(x)+arctan(x)+x^2].ddiff{x}
or
poly_x[(cos(x)+arctan(x)+x^2).ddiff{d}]
6. poly_x[(x^2+3).ddiff{x}+(y^2+3*y).ddiff{y}]
7. poly_x[(x^2+3*x*y).ddiff{x}]
8. poly_x[(x^2*y*z).dint{x}]
9. poly_x[(x^10+3*x^12*y).ddiff{x,4}]
10. poly_x[(x^10+3*x^12*y).dint{x}.dint{y}]
Factorization of polynomials (both monomials and multinomials):
1. $factorize(x^2+2x+1)=(x+1)^2$
2. $factorize(x^2+2xy+y^2)=(x+y)^2$
1. poly_x[factorize(x^2+2*x+1)]
2. poly_x[factorize(x^2+2*x*y+y^2)]
Solving equations (this is an exact solver):
1. $x^2+ax+(b-3c)=0$ where $x$ is the variable to solve for
2. $y^3+ax+(b-3c-\frac{\sinh(d)}{e})=0$ where $y$ is the variable to solve for
1. poly_x[eqsolve(x^2+a*x+(b-3*c),x)]
2. poly_x[eqsolve(y^3+a*x+(b-3*c-sinh(d)/e),y)]
Defining a row vector:
$\begin{pmatrix}5&4&3\end{pmatrix}$
matrix(5,4,3)
Defining a column vector:
$\begin{pmatrix}5\\4\\3\end{pmatrix}$
matrix(5;4;3)
Defining a matrix:
$\begin{pmatrix}1&2\\3&4\end{pmatrix}$
matrix(1,2;3,4)
Defining an identity matrix:
E.g. A 3X3 identity matrix
$\begin{pmatrix}1&0&0\\0&1&0\\0&0&1\end{pmatrix}$
identity(3) or eye(3)
Row vector with elements of equal intervals:
$\begin{pmatrix}0&0.5&1&1.5&2&2.5\end{pmatrix}$
Here, the row vector starts at 0, goes up to (but not including) 3, with interval 0.5
range(0,3,0.5)
Matrix transpose:
$\begin{pmatrix}1&2\\3&4\\5&6\end{pmatrix}^T$
matrix(1,2;3,4;5,6).T
Matrix inverse:
$\begin{pmatrix}1&2\\3&4\end{pmatrix}^{-1}$
inv(matrix(1,2;3,4))
Matrix multiplication:
$\begin{pmatrix}1&2\\3&4\end{pmatrix}\begin{pmatrix}2\\3\end{pmatrix}$
matmult(matrix(1,2;3,4),matrix(2;3))
Cholesky Factorization of positive semidefinite matrix:
$\begin{pmatrix}11&2&3\\4&55&6\\7&8&9\end{pmatrix}$
chol(matrix(11,2,3;4,55,6;7,8,99))
Moore-Penrose Pseudo-Inverse of a matrix:
$\begin{pmatrix}11&2&3\\4&55&6\end{pmatrix}^{\dagger}$
mpseudo(matrix(11,2,3;4,55,6))
Fast Fourier Transform of a vector:
By default, the n-point fft will be taken, where n is the number of columns in the row vector.
fft(1,2,4,7)
fft(matrix(1,2,4,7))
n-point Fast Fourier Transform of a vector:
fft(1,2,4,7) where n=3
fft(matrix(1,2,4,7),3)
Inverse Fast Fourier Transform of a vector:
ifft(1,2,4,7)
ifft(matrix(1,2,4,7))
n-point Inverse Fast Fourier Transform of a vector:
ifft(1,2,4,7) where n=2
ifft(matrix(1,2,4,7),2)
Plotting graphs of functions:
Note: When you plot a graph, RapidCalculator will go into plotting mode and you will not be able to both plot and evaluate a computation simultaneously. And note that you can only use the variable x as your unknown variable to plot
Example:
$y=\sin(x)+x^3-\pi$
plot[sin(x)+x^3-pi]