Symbol Reference API¶

Reference APIs for Scalar, Vector and Matrix as well as the diff facilities.

`Scalar`¶

Arithmetic Operations¶

Method	Description
`a + b`	Addition of two scalars
`a - b`	Subtraction of two scalars
`a * b`	Multiplication of two scalars
`a / b`	Division of two scalars
`a + val`	Add numeric constant
`a - val`	Subtract numeric constant
`a * val`	Multiply by numeric constant
`a / val`	Divide by numeric constant
`val + a`	Constant plus scalar
`val - a`	Constant minus scalar
`val * a`	Constant times scalar
`val / a`	Constant divided by scalar
`-a`	Unary negation

Assignment Operations¶

Method	Description
`a += b`	Add scalar and assign
`a -= b`	Subtract scalar and assign
`a *= b`	Multiply by scalar and assign
`a /= b`	Divide by scalar and assign
`a += val`	Add constant and assign
`a -= val`	Subtract constant and assign
`a *= val`	Multiply by constant and assign
`a /= val`	Divide by constant and assign

Mathematical Functions¶

Method	Description
`a.inv()`	Reciprocal (1/a)
`a.powN(n)`	Integer power
`a.powF(f)`	Floating point power
`a.sqrt()`	Square root
`a.exp()`	Exponential (e^x)
`a.ln()`	Natural logarithm
`a.log()`	Natural logarithm (alias for ln)
`a.log10()`	Base-10 logarithm
`a.sin()`	Sine
`a.cos()`	Cosine
`a.tan()`	Tangent
`a.asin()`	Arc sine
`a.acos()`	Arc cosine
`a.atan()`	Arc tangent

Free Function Versions¶

Method	Description
`powN(a, n)`	Integer power function
`powF(a, f)`	Floating point power function
`sqrt(a)`	Square root function
`exp(a)`	Exponential function
`ln(a)`	Natural logarithm function
`log(a)`	Natural logarithm function
`log10(a)`	Base-10 logarithm function
`sin(a)`	Sine function
`cos(a)`	Cosine function
`tan(a)`	Tangent function
`asin(a)`	Arc sine function
`acos(a)`	Arc cosine function
`atan(a)`	Arc tangent function

Conditional Operations¶

Method	Description
`branch(cond, pos, neg)`	Conditional expression with scalar branches
`branch(cond, pos, val)`	Conditional with scalar positive, constant negative
`branch(cond, val, neg)`	Conditional with constant positive, scalar negative
`branch(cond, val1, val2)`	Conditional with constant branches
`a > b`	Greater than comparison
`a < b`	Less than comparison
`a > val`	Greater than constant
`a < val`	Less than constant
`val > a`	Constant greater than scalar
`val < a`	Constant less than scalar
`abs(a)`	Absolute value
`sign(a)`	Sign function (-1, 0, or 1)
`min(a, b)`	Minimum of two scalars
`max(a, b)`	Maximum of two scalars

Value Operations¶

Method	Description
`set_value(val)`	Set the numeric value of a symbol
`get_value()`	Get the numeric value of a symbol
`eval()`	Evaluate the expression to a numeric value
`is_zero()`	Check if expression is exactly zero
`is_one()`	Check if expression is exactly one

Expression Structure¶

Method	Description
`left()`	Get left operand of binary operation
`right()`	Get right operand of binary operation
`has_right()`	Check if expression has a right operand
`has_not_right()`	Check if expression is unary
`has_branch()`	Check if expression is a conditional
`get_condition()`	Get condition of a branch expression

Symbol Information¶

Method	Description
`is_symbol()`	Check if expression is a basic symbol
`get_symbol_idx()`	Get the index of a symbol
`get_expression_graph()`	Get pointer to underlying expression graph
`get_checksum()`	Get hash checksum of the expression

Factory Methods¶

Method	Description
`get_zero()`	Create a zero constant
`get_one()`	Create a one constant
`make_constant(val)`	Create a constant with given value
`make_branch(c, p, n)`	Create a conditional expression

Utility¶

Method	Description
`print()`	Print expression for debugging (chainable)

`Vector`¶

Construction¶

Method	Description
`Vector(values)`	Construct from vector of scalars
`Vector::zero(size, seed)`	Create zero vector of given size

Properties¶

Method	Description
`size()`	Get the size of the vector
`values()`	Get reference to underlying scalar values

Element Access¶

Method	Description
`operator[](i)`	Access element by index
`operator()(i)`	Access element by index (alternative syntax)
`segment(begin, n)`	Extract a segment of the vector
`set_segment(begin, n, other)`	Set a segment from another vector

Vector Operations¶

Method	Description
`dot(other)`	Dot product with another vector
`cross2(other)`	2D cross product (returns scalar)
`cross3(other)`	3D cross product (returns vector)
`squared_norm()`	Squared Euclidean norm
`norm()`	Euclidean norm
`normalized()`	Return normalized vector
`normalize()`	Normalize this vector in-place

Arithmetic Operations¶

Method	Description
`v + other`	Vector addition
`v - other`	Vector subtraction
`v * val`	Scalar multiplication
`v * scalar`	Multiplication by symbolic scalar
`v / val`	Scalar division
`v / scalar`	Division by symbolic scalar
`v += other`	Vector addition assignment
`v -= other`	Vector subtraction assignment
`v *= val`	Scalar multiplication assignment
`v *= scalar`	Symbolic scalar multiplication assignment
`v /= val`	Scalar division assignment
`v /= scalar`	Symbolic scalar division assignment

Free Functions¶

Method	Description
`val * v`	Scalar times vector
`scalar * v`	Symbolic scalar times vector
`-v`	Vector negation
`dot(a, b)`	Dot product of two vectors

Value Operations¶

Method	Description
`set_values(v)`	Set numeric values from array-like object

`Matrix`¶

Construction¶

Method	Description
`Matrix(values, shape)`	Construct from values (row-major) and shape
`Matrix::zero(shape, seed)`	Create zero matrix of given shape
`Matrix::identity(size, seed)`	Create identity matrix

Properties¶

Method	Description
`rows()`	Get number of rows
`cols()`	Get number of columns
`shape()`	Get shape as array [rows, cols]
`values()`	Get reference to underlying scalar values

Element Access¶

Method	Description
`operator()(i, j)`	Access element at row i, column j
`block(r, c, nr, nc)`	Extract a block sub-matrix
`row(idx)`	Extract a row as a vector
`col(idx)`	Extract a column as a vector
`set_block(r, c, nr, nc, m)`	Set a block from another matrix
`set_row(idx, v)`	Set a row from a vector
`set_col(idx, v)`	Set a column from a vector

Matrix Operations¶

Method	Description
`transpose()`	Matrix transpose
`det()`	Determinant (for square matrices)
`inv()`	Matrix inverse
`inv_sym()`	Symmetric matrix inverse (optimized)
`trace()`	Matrix trace (sum of diagonal elements)
`frobenius_norm_sq()`	Squared Frobenius norm
`singular_values_2x2()`	Singular values for 2x2 matrix
`dot(other)`	Matrix multiplication with another matrix
`dot(vec)`	Matrix-vector multiplication

Arithmetic Operations¶

Method	Description
`m + other`	Matrix addition
`m - other`	Matrix subtraction
`m * val`	Scalar multiplication
`m * scalar`	Multiplication by symbolic scalar
`m * vec`	Matrix-vector multiplication
`m * other`	Matrix multiplication
`m / val`	Scalar division
`m / scalar`	Division by symbolic scalar
`m += other`	Matrix addition assignment
`m -= other`	Matrix subtraction assignment
`m *= val`	Scalar multiplication assignment
`m *= scalar`	Symbolic scalar multiplication assignment
`m /= val`	Scalar division assignment
`m /= scalar`	Symbolic scalar division assignment

Free Functions¶

Method	Description
`val * m`	Scalar times matrix
`scalar * m`	Symbolic scalar times matrix
`-m`	Matrix negation
`vec * m`	Vector-matrix multiplication (row vector)
`outer(a, b)`	Outer product of two vectors

Value Operations¶

Method	Description
`set_values_row_major(v)`	Set numeric values from row-major array

Differentiation (`diff`)¶

Basic Differentiation¶

Method	Description
`diff(expr, wrt)`	Compute derivative of scalar expression
`diff(expr, wrt, cache)`	Compute derivative with caching

Gradient Computation¶

Method	Description
`gradient(expr, wrt)`	Gradient of scalar w.r.t. vector of variables
`gradient(expr, wrt, cache)`	Gradient computation with caching
`gradient(vec, wrt, symmetric)`	Jacobian of vector w.r.t. variables
`gradient(vec, wrt, symmetric, cache)`	Jacobian computation with caching

Hessian Computation¶

Method	Description
`hessian(expr, wrt)`	Hessian matrix of scalar expression
`hessian(expr, wrt, cache)`	Hessian computation with caching

Combined Operations¶

Method	Description
`value_gradient(expr, wrt)`	Compute both value and gradient
`value_gradient(expr, wrt, cache)`	Value and gradient with caching
`value_gradient_hessian(expr, wrt)`	Compute value, gradient, and Hessian
`value_gradient_hessian(expr, wrt, cache)`	All three with caching