## Algebra

Solve algebraic equations and manipulate expressions efficiently.

Algebra functions are fundamental for solving equations and manipulating algebraic expressions. Calcul.io includes functions for solving linear and quadratic equations, polynomial operations, and symbolic computation. These functions are crucial for fields like mathematics, engineering, and economics.

Our algebra functions provide powerful tools for analyzing and simplifying complex expressions. Whether you are solving systems of equations, performing algebraic manipulations, or exploring mathematical theory, Calcul.io offers the algebraic tools necessary to achieve precise and efficient solutions.

derivative | Takes the derivative of an expression expressed in parser Nodes. The derivative will be taken over the supplied variable in the second parameter. If there are multiple variables in the expression, it will return a partial derivative. |

leafCount | Computes the number of leaves in the parse tree of the given expression |

lsolve | Finds one solution of the linear system L * x = b where L is an [n x n] lower triangular matrix and b is a [n] column vector. |

lsolveAll | Finds all solutions of the linear system L * x = b where L is an [n x n] lower triangular matrix and b is a [n] column vector. |

lup | Calculate the Matrix LU decomposition with partial pivoting. Matrix A is decomposed in three matrices (L, U, P) where P * A = L * U |

lusolve | Solves the linear system A * x = b where A is an [n x n] matrix and b is a [n] column vector. |

lyap | Solves the Continuous-time Lyapunov equation AP+PA'+Q=0 for P |

polynomialRoot | Finds the roots of a univariate polynomial given by its coefficients starting from constant, linear, and so on, increasing in degree. |

qr | Calculates the Matrix QR decomposition. Matrix `A` is decomposed in two matrices (`Q`, `R`) where `Q` is an orthogonal matrix and `R` is an upper triangular matrix. |

rationalize | Transform a rationalizable expression in a rational fraction. If rational fraction is one variable polynomial then converts the numerator and denominator in canonical form, with decreasing exponents, returning the coefficients of numerator. |

resolve | Recursively substitute variables in an expression tree. |

schur | Performs a real Schur decomposition of the real matrix A = UTU' |

simplify | Simplify an expression tree. |

simplifyConstant | Replace constant subexpressions of node with their values. |

simplifyCore | Perform simple one-pass simplifications on an expression tree. |

slu | Calculate the Matrix LU decomposition with full pivoting. Matrix A is decomposed in two matrices (L, U) and two permutation vectors (pinv, q) where P * A * Q = L * U |

sylvester | Solves the real-valued Sylvester equation AX+XB=C for X |

symbolicEqual | Returns true if the difference of the expressions simplifies to 0 |

usolve | Finds one solution of the linear system U * x = b where U is an [n x n] upper triangular matrix and b is a [n] column vector. |

usolveAll | Finds all solutions of the linear system U * x = b where U is an [n x n] upper triangular matrix and b is a [n] column vector. |