### AbstractPolynomialSolver Setup Method

Source: https://www.hipparchus.org/hipparchus-core/jacoco/org.hipparchus.analysis.solvers/AbstractPolynomialSolver.java.html

Details the `setup` method used to initialize the solver with evaluation parameters and the polynomial function.

```APIDOC
## void setup(int maxEval, PolynomialFunction f, double min, double max, double startValue)

### Description
Sets up the solver with the maximum number of evaluations, the polynomial function, and the search interval.

### Method
`protected void setup`

### Parameters
#### Path Parameters
*   None

#### Query Parameters
*   None

#### Request Body
*   `maxEval` (int) - Maximum number of evaluations allowed.
*   `f` (PolynomialFunction) - The polynomial function to solve.
*   `min` (double) - The lower bound of the search interval.
*   `max` (double) - The upper bound of the search interval.
*   `startValue` (double) - The initial guess for the root.
```

--------------------------------

### Get Start Point

Source: https://www.hipparchus.org/apidocs/org/hipparchus/geometry/euclidean/threed/Segment.html

Retrieves the start point of the segment.

```java
public Vector3D getStart()
```

--------------------------------

### Get Start Vertex of Edge

Source: https://www.hipparchus.org/hipparchus-geometry/jacoco/org.hipparchus.geometry.spherical.twod/Edge.java.html

Retrieves the starting vertex of the edge.

```java
public Vertex getStart() {
        return start;
    }
```

--------------------------------

### Setup Method

Source: https://www.hipparchus.org/apidocs/org/hipparchus/analysis/solvers/AbstractPolynomialSolver.html

Prepares the solver for computation. Subclasses must call this method if they override any of the solve methods.

```java
protected void setup​(int maxEval,
                     PolynomialFunction f,
                     double min,
                     double max,
                     double startValue)
```

--------------------------------

### Get Start Value

Source: https://www.hipparchus.org/hipparchus-optim/jacoco/org.hipparchus.optim.univariate/SearchInterval.java.html

Retrieves the start value for the search within the interval.

```java
public double getStartValue() {
    return start;
}
```

--------------------------------

### Method: setup

Source: https://www.hipparchus.org/hipparchus-core/apidocs/org/hipparchus/analysis/solvers/AbstractPolynomialSolver.html

Prepares the solver for computation. Subclasses must call this method if they override any of the solve methods.

```APIDOC
## setup

### Description
Prepare for computation. Subclasses must call this method if they override any of the solve methods.

### Parameters
#### Request Body
- **maxEval** (int) - Required - Maximum number of evaluations.
- **f** (PolynomialFunction) - Required - Function to solve.
- **min** (double) - Required - Lower bound for the interval.
- **max** (double) - Required - Upper bound for the interval.
- **startValue** (double) - Required - Start value to use.
```

--------------------------------

### Method: setup

Source: https://www.hipparchus.org/hipparchus-core/jacoco/org.hipparchus.analysis.integration/BaseAbstractUnivariateIntegrator.java.html

Prepares the integrator for computation by setting bounds, the function, and evaluation limits.

```APIDOC
## setup

### Description
Prepares the integrator for computation. Subclasses must call this method if they override any of the solve methods.

### Parameters
- **maxEval** (int) - Required - Maximum number of evaluations.
- **f** (UnivariateFunction) - Required - The integrand function.
- **lower** (double) - Required - The min bound for the interval.
- **upper** (double) - Required - The upper bound for the interval.

### Errors
- **NullArgumentException** - If f is null.
- **MathIllegalArgumentException** - If min >= max.
```

--------------------------------

### Get Start Index

Source: https://www.hipparchus.org/hipparchus-core/jacoco/org.hipparchus.util/ResizableDoubleArray.java.html

Retrieves the starting index of the first addressable element in the internal array.

```APIDOC
## GET /api/start-index

### Description
Retrieves the starting index of the internal array. This index is the position of the first addressable element in the internal storage array. Addressable elements are in the interval [startIndex, startIndex + numElements - 1].

### Method
GET

### Endpoint
/api/start-index

### Parameters
None

### Request Body
None

### Response
#### Success Response (200)
- **startIndex** (integer) - The start index of the first addressable element.

#### Response Example
```json
{
  "startIndex": 2
}
```
```

--------------------------------

### Set and Get Step Start

Source: https://www.hipparchus.org/hipparchus-ode/jacoco/org.hipparchus.ode/AbstractIntegrator.java.html

Manages the starting state of the current integration step. Includes methods to set and retrieve the step start information.

```Java
protected void setStepStart(final ODEStateAndDerivative stepStart) {
        this.stepStart = stepStart;
    }

```

```Java
public ODEStateAndDerivative getStepStart() {
        return stepStart;
    }

```

--------------------------------

### POST /setup

Source: https://www.hipparchus.org/apidocs/org/hipparchus/exception/class-use/MathIllegalArgumentException.html

Prepares an integrator for computation by setting the function and interval bounds.

```APIDOC
## POST /setup

### Description
Prepares the integrator instance for a specific computation task.

### Method
POST

### Endpoint
/setup

### Request Body
- **maxEval** (int) - Required - Maximum number of evaluations.
- **f** (UnivariateFunction) - Required - The function to be integrated.
- **lower** (double) - Required - The lower bound.
- **upper** (double) - Required - The upper bound.
```

--------------------------------

### Get Start Index

Source: https://www.hipparchus.org/apidocs/org/hipparchus/util/ResizableDoubleArray.html

Returns the starting index of the first addressable element within the internal storage array.

```java
protected int getStartIndex()
```

--------------------------------

### Solver Setup and Solving

Source: https://www.hipparchus.org/hipparchus-core/jacoco/org.hipparchus.analysis.solvers/BaseAbstractUnivariateSolver.java.html

Details the methods for setting up the solver with initial parameters and performing the actual solving process.

```APIDOC
## Solver Setup and Solving

### Description
Details the methods for setting up the solver with initial parameters and performing the actual solving process.

### Methods

- **setup(int maxEval, F f, double min, double max, double startValue)**
  - Description: Prepares the solver for computation by setting the function, interval, start value, and maximum evaluations. Subclasses must call this method if they override any `solve` methods.
  - Parameters:
    - **maxEval** (int) - Maximum number of evaluations allowed.
    - **f** (F) - The function to solve.
    - **min** (double) - The lower bound for the search interval.
    - **max** (double) - The upper bound for the search interval.
    - **startValue** (double) - The initial guess for the solution.
  - Throws:
    - `NullArgumentException` - If `f` is null.

- **solve(int maxEval, F f, double min, double max, double startValue)**
  - Description: Solves the equation using the provided parameters.
  - Parameters:
    - **maxEval** (int) - Maximum number of evaluations allowed.
    - **f** (F) - The function to solve.
    - **min** (double) - The lower bound for the search interval.
    - **max** (double) - The upper bound for the search interval.
    - **startValue** (double) - The initial guess for the solution.
  - Return Type: `double`
  - Throws:
    - `MathIllegalArgumentException` - If the initial search interval does not bracket a root and the solver requires it.
    - `MathIllegalStateException` - If the maximal number of evaluations is exceeded.

- **solve(int maxEval, F f, double min, double max)**
  - Description: Solves the equation with a default start value (midpoint of the interval).
  - Parameters:
    - **maxEval** (int) - Maximum number of evaluations allowed.
    - **f** (F) - The function to solve.
    - **min** (double) - The lower bound for the search interval.
    - **max** (double) - The upper bound for the search interval.
  - Return Type: `double`
  - Throws:
    - `MathIllegalArgumentException` - If the initial search interval does not bracket a root and the solver requires it.
    - `MathIllegalStateException` - If the maximal number of evaluations is exceeded.

- **solve(int maxEval, F f, double startValue)**
  - Description: Solves the equation with a default interval (NaN values indicate no specific interval).
  - Parameters:
    - **maxEval** (int) - Maximum number of evaluations allowed.
    - **f** (F) - The function to solve.
    - **startValue** (double) - The initial guess for the solution.
  - Return Type: `double`
  - Throws:
    - `MathIllegalArgumentException` - If the initial search interval does not bracket a root and the solver requires it.
    - `MathIllegalStateException` - If the maximal number of evaluations is exceeded.

- **doSolve()**
  - Description: Abstract method for implementing the actual optimization algorithms in derived classes.
  - Return Type: `double`
  - Throws:
    - `MathIllegalStateException` - If the maximal number of evaluations is exceeded.
    - `MathIllegalArgumentException` - If the initial search interval does not bracket a root and the solver requires it.
```

--------------------------------

### Get Step Start State

Source: https://www.hipparchus.org/apidocs/org/hipparchus/ode/ODEIntegrator.html

Retrieves the state at the start of the current step. Results are undefined if called outside of the integrate method.

```java
ODEStateAndDerivative getStepStart()
```

--------------------------------

### static void showExampleFrame(ExampleUtils.ExampleFrame frame)

Source: https://www.hipparchus.org/apidocs/org/hipparchus/samples/ExampleUtils.html

Displays a graphical frame for an example application.

```APIDOC
## showExampleFrame

### Description
Displays an example frame.

### Parameters
- **frame** (ExampleUtils.ExampleFrame) - Required - The frame to display.
```

--------------------------------

### Get State at Step Start Time

Source: https://www.hipparchus.org/hipparchus-ode/apidocs/org/hipparchus/ode/AbstractIntegrator.html

Retrieve the state of the system at the start time of the current step (`ti`). This method can be called during integration to access the current step's starting conditions.

```java
public ODEStateAndDerivative getStepStart()
```

--------------------------------

### setup

Source: https://www.hipparchus.org/apidocs/org/hipparchus/analysis/integration/BaseAbstractFieldUnivariateIntegrator.html

Prepares the integrator for computation. Subclasses must call this method if they override any of the solve methods.

```APIDOC
## setup

### Description
Prepare for computation. Subclasses must call this method if they override any of the solve methods.

### Method
protected void setup(int maxEval, CalculusFieldUnivariateFunction<T> f, T lower, T upper)

### Parameters
- **maxEval** (int) - Required - Maximum number of evaluations.
- **f** (CalculusFieldUnivariateFunction<T>) - Required - The integrand function.
- **lower** (T) - Required - The min bound for the interval.
- **upper** (T) - Required - The upper bound for the interval.

### Throws
- NullArgumentException - if f is null.
- MathIllegalArgumentException - if min >= max.
```

--------------------------------

### Get ODE Integration Step Start

Source: https://www.hipparchus.org/hipparchus-ode/xref/org/hipparchus/ode/AbstractIntegrator.html

Returns the starting state of the current integration step, which includes the time and derivatives at the beginning of the step.

```Java
@Override
public ODEStateAndDerivative getStepStart() {
    return stepStart;
}
```

--------------------------------

### Setup Solver Method

Source: https://www.hipparchus.org/apidocs/org/hipparchus/ode/events/AbstractFieldODEDetector.html

Sets up the root-finding algorithm to use for detecting state events. Returns a new instance with the updated solver.

```java
T withSolver (BracketedRealFieldUnivariateSolver<E> newSolver)
```

--------------------------------

### Setting up and Integrating Variational Equations

Source: https://www.hipparchus.org/hipparchus-ode/index.html

Demonstrates the workflow for setting up variational equations, including initializing the ODE, configuring initial states with derivatives, performing integration, and extracting Jacobian matrices. Ensure `setUpInitialState` is called before integration.

```java
// set up equations
ODEJacobiansProvider jode       = new MyODE(...);
ExpandableODE        expandable = new Expandable(jode);
VariationalEquation  ve         = new VariationalEquation(expandable, jode);

// set up initial state
ODEState initWithoutDerivatives = new ODEState(t0, y0);
ve.setInitialMainStateJacobian(dYdY0); // only needed if the default identity matrix is not suitable
ve.setInitialParameterJacobian(name, dYdP); // only needed if the default zero matrix is not suitable
ODEState initWithDerivatives = ve.setUpInitialState(initWithoutDerivatives);

// perform integration on the expanded equations with the expanded initial state
ODEStateAndDerivative finalState = integrator.integrate(expandable, initWithDerivatives, finalT);

// extract Jacobian matrices
dYdY0 = ve.extractMainSetJacobian(finalState);
dYdP  = ve.extractParameterJacobian(finalState, name);

```

--------------------------------

### Get First Arc Start

Source: https://www.hipparchus.org/hipparchus-geometry/jacoco/org.hipparchus.geometry.spherical.oned/ArcsSet.java.html

Identifies the node corresponding to the first arc start in the tree, returning null if no internal nodes exist.

```java
private BSPTree<Sphere1D, S1Point, LimitAngle, SubLimitAngle> getFirstArcStart() {

		// start search at the tree root
		BSPTree<Sphere1D, S1Point, LimitAngle, SubLimitAngle> node = getTree(false);
		if (node.getCut() == null) {
			return null;
		}

		// walk tree until we find the smallest internal node
		node = getFirstLeaf(node).getParent();

		// walk tree until we find an arc start
```

--------------------------------

### GET /ode/step-start

Source: https://www.hipparchus.org/hipparchus-ode/xref/org/hipparchus/ode/ODEIntegrator.html

Retrieves the state at the start of the current integration step.

```APIDOC
## GET /ode/step-start

### Description
Returns the state at step start time t_i. This method is intended to be called during integration.

### Method
GET

### Endpoint
/ode/step-start

### Response
#### Success Response (200)
- **state** (ODEStateAndDerivative) - The state at the beginning of the current step.
```

--------------------------------

### Setting Up and Executing Integration

Source: https://www.hipparchus.org/hipparchus-coverage/jacoco-aggregate/hipparchus-core/org.hipparchus.analysis.integration/BaseAbstractUnivariateIntegrator.java.html

Methods for initializing solver state and performing the integration process.

```java
protected void setup(final int maxEval,  
                           final UnivariateFunction f,  
                           final double lower, final double upper)  
        throws MathIllegalArgumentException, NullArgumentException {  
  
        // Checks.  
        MathUtils.checkNotNull(f);  
        UnivariateSolverUtils.verifyInterval(lower, upper);  
  
        // Reset.  
        min = lower;  
        max = upper;  
        function = f;  
        evaluations = evaluations.withMaximalCount(maxEval);  
        iterations.reset();  
    }  
  
    /** {@inheritDoc} */  
    @Override  
    public double integrate(final int maxEval, final UnivariateFunction f,  
                            final double lower, final double upper)  
        throws MathIllegalArgumentException, MathIllegalStateException, NullArgumentException {  
  
        // Initialization.  
        setup(maxEval, f, lower, upper);  
  
        // Perform computation.  
        return doIntegrate();  
  
    }  
  
    /**  
     * Method for implementing actual integration algorithms in derived  
     * classes.  
     *  
     * @return the root.  
     * @throws MathIllegalStateException if the maximal number of evaluations  
     * is exceeded.  
     * @throws MathIllegalStateException if the maximum iteration count is exceeded  
     * or the integrator detects convergence problems otherwise  
     */  
    protected abstract double doIntegrate()  
        throws MathIllegalStateException;
```

--------------------------------

### Get initial integration time

Source: https://www.hipparchus.org/hipparchus-ode/apidocs/org/hipparchus/ode/DenseOutputModel.html

Retrieves the starting time of the integration.

```java
public double getInitialTime()
```

--------------------------------

### Integration Setup and Execution

Source: https://www.hipparchus.org/hipparchus-core/jacoco/org.hipparchus.analysis.integration/BaseAbstractFieldUnivariateIntegrator.java.html

Methods for initializing the integration process and performing the computation.

```java
    /**  
     * Prepare for computation.  
     * Subclasses must call this method if they override any of the  
     * {@code solve} methods.  
     *  
     * @param maxEval Maximum number of evaluations.  
     * @param f the integrand function  
     * @param lower the min bound for the interval  
     * @param upper the upper bound for the interval  
     * @throws NullArgumentException if {@code f} is {@code null}.  
     * @throws MathIllegalArgumentException if {@code min >= max}.  
     */  
    protected void setup(final int maxEval,  
                         final CalculusFieldUnivariateFunction<T> f,  
                         final T lower, final T upper)  
        throws MathIllegalArgumentException, NullArgumentException {  
  
        // Checks.  
        MathUtils.checkNotNull(f);  
        UnivariateSolverUtils.verifyInterval(lower.getReal(), upper.getReal());  
  
        // Reset.  
        min = lower;  
        max = upper;  
        function = f;  
        evaluations = evaluations.withMaximalCount(maxEval);  
        iterations.reset();  
    }  
  
    /** {@inheritDoc} */  
    @Override  
    public T integrate(final int maxEval, final CalculusFieldUnivariateFunction<T> f,  
                       final T lower, final T upper)  
        throws MathIllegalArgumentException, MathIllegalStateException, NullArgumentException {  
  
        // Initialization.  
        setup(maxEval, f, lower, upper);  
  
        // Perform computation.  
        return doIntegrate();  
  
    }  
  
    /**  
     * Method for implementing actual integration algorithms in derived  
     * classes.  
     *  
     * @return the root.  
     * @throws MathIllegalStateException if the maximal number of evaluations  
     * is exceeded.  
     * @throws MathIllegalStateException if the maximum iteration count is exceeded  
     * or the integrator detects convergence problems otherwise  
     */  
    protected abstract T doIntegrate()  
        throws MathIllegalStateException;
```

--------------------------------

### Initialize QP Matrices and Vectors

Source: https://www.hipparchus.org/hipparchus-optim/jacoco/org.hipparchus.optim.nonlinear.vector.constrained/ADMMQPOptimizer.java.html

Sets up the quadratic term (H), gradient (q), and constraint matrices (A) and vectors (lb, ub) for the QP problem.

```java
RealMatrix H = function.getP();
RealVector q = function.getQ();
int me = 0;
int mi = 0;
int mb = 0;
if (eqConstraint != null) {
    me = eqConstraint.dimY();
}
if (iqConstraint != null) {
    mi = iqConstraint.dimY();
}
if (bqConstraint != null) {
    mb = bqConstraint.dimY();
}
RealVector lb = new ArrayRealVector(me + mi + mb);
RealVector ub = new ArrayRealVector(me + mi + mb);
RealMatrix A = new Array2DRowRealMatrix(me + mi + mb, n);
```

--------------------------------

### Get Left Abscissa of Interval

Source: https://www.hipparchus.org/hipparchus-core/jacoco/org.hipparchus.analysis.solvers/BracketedUnivariateSolver.java.html

Retrieves the x-coordinate of the start of the interval.

```java
public double getLeftAbscissa() {
    return leftAbscissa;
}
```

--------------------------------

### Get Start Index

Source: https://www.hipparchus.org/hipparchus-core/apidocs/org/hipparchus/util/ResizableDoubleArray.html

Returns the starting index of the first addressable element within the internal storage array. Addressable elements are in the range [getStartIndex(), getStartIndex() + getNumElements() - 1].

```java
protected int getStartIndex()
```

--------------------------------

### Solver Setup and Solve Methods

Source: https://www.hipparchus.org/hipparchus-core/apidocs/org/hipparchus/analysis/solvers/BaseAbstractUnivariateSolver.html

Methods for setting up the solver and performing the solve operation for univariate functions.

```APIDOC
## POST /websites/hipparchus/setup

### Description
Prepares the solver for computation by setting the function, interval, start value, and maximum evaluations.

### Method
POST

### Endpoint
/websites/hipparchus/setup

### Parameters
#### Request Body
- **maxEval** (int) - Maximum number of evaluations allowed.
- **f** (UnivariateFunction) - The function to solve.
- **min** (double) - The lower bound of the interval.
- **max** (double) - The upper bound of the interval.
- **startValue** (double) - The initial guess for the root.

### Request Example
```json
{
  "maxEval": 100,
  "f": "yourUnivariateFunctionImplementation",
  "min": 0.0,
  "max": 1.0,
  "startValue": 0.5
}
```

### Response
#### Success Response (200)
Indicates successful setup. No specific return value is documented for this setup method.

## POST /websites/hipparchus/solve

### Description
Solves for a zero of the univariate function within the specified interval, starting from an initial guess.

### Method
POST

### Endpoint
/websites/hipparchus/solve

### Parameters
#### Request Body
- **maxEval** (int) - Maximum number of evaluations.
- **f** (UnivariateFunction) - Function to solve.
- **min** (double) - Lower bound for the interval.
- **max** (double) - Upper bound for the interval.
- **startValue** (double) - Start value to use.

### Request Example
```json
{
  "maxEval": 100,
  "f": "yourUnivariateFunctionImplementation",
  "min": 0.0,
  "max": 1.0,
  "startValue": 0.5
}
```

### Response
#### Success Response (200)
- **root** (double) - A value where the function is zero.

#### Response Example
```json
{
  "root": 0.7071067811865476
}
```

#### Error Response
- **MathIllegalArgumentException**: If the arguments do not satisfy the requirements specified by the solver.
- **MathIllegalStateException**: If the allowed number of evaluations is exceeded.
```

--------------------------------

### Get All Optima - Hipparchus

Source: https://www.hipparchus.org/hipparchus-coverage/jacoco-aggregate/hipparchus-optim/org.hipparchus.optim/BaseMultiStartMultivariateOptimizer.java.html

Retrieves all optima found during the last optimization call, including those from failed starts. The array is sorted by objective value. Call this after `optimize` to get all results.

```java
/**  
  * Gets all the optima found during the last call to {@code optimize}.  
  * The optimizer stores all the optima found during a set of  
  * restarts. The {@code optimize} method returns the best point only.  
  * This method returns all the points found at the end of each starts,  
  * including the best one already returned by the {@code optimize} method.  
  * <br>  
  * The returned array as one element for each start as specified  
  * in the constructor. It is ordered with the results from the  
  * runs that did converge first, sorted from best to worst  
  * objective value (i.e in ascending order if minimizing and in  
  * descending order if maximizing), followed by {@code null} elements  
  * corresponding to the runs that did not converge. This means all  
  * elements will be {@code null} if the {@code optimize} method did throw  
  * an exception.  
  * This also means that if the first element is not {@code null}, it is  
  * the best point found across all starts.  
  * <br>  
  * The behaviour is undefined if this method is called before  
  * {@code optimize}; it will likely throw {@code NullPointerException}.  
  *  
  * @return an array containing the optima sorted from best to worst.  
  */  
  public abstract P[] getOptima();
```

--------------------------------

### BaseAbstractUnivariateSolver - Setup Method

Source: https://www.hipparchus.org/hipparchus-core/apidocs/org/hipparchus/exception/class-use/NullArgumentException.html

Prepares the solver for computation.

```APIDOC
## BaseAbstractUnivariateSolver.setup

### Description
Prepare for computation.

### Method
protected void

### Endpoint
N/A (Protected method within a solver class)

### Parameters
#### Path Parameters
None

#### Query Parameters
None

#### Request Body
None

### Request Example
None

### Response
#### Success Response (200)
None

#### Response Example
None
```

--------------------------------

### Setting up and Integrating Variational Equations

Source: https://www.hipparchus.org/hipparchus-coverage/jacoco-aggregate/hipparchus-ode/org.hipparchus.ode/VariationalEquation.java.html

This example demonstrates the proper scheduling for setting up, integrating, and extracting Jacobian matrices using the VariationalEquation class. Ensure setUpInitialState is called before integration to avoid errors.

```java
 ODEJacobiansProvider jode 			= new MyODE(...);
 			ExpandableODE 			expandable = new Expandable(jode);
 			VariationalEquation 			ve 			= new VariationalEquation(expandable, jode);
 			
 			 ODEState initWithoutDerivatives = new ODEState(t0, y0);
 			ve.setInitialMainStateJacobian(dYdY0); 
 			ve.setInitialParameterJacobian(name, dYdP); 
 			ODEState initWithDerivatives = ve.setUpInitialState(initWithoutDerivatives);
 			
 			 ODEStateAndDerivative finalState = integrator.integrate(expandable, initWithDerivatives, finalT);
 			
 			 dYdY0 = ve.extractMainSetJacobian(finalState);
 			 dYdP 	= ve.extractParameterJacobian(finalState, name);
```

--------------------------------

### Get Submatrix by Row and Column Indices

Source: https://www.hipparchus.org/apidocs/org/hipparchus/linear/BlockRealMatrix.html

Gets a submatrix defined by the given start and end row and column indices (inclusive, 0-based). Throws MathIllegalArgumentException if indices are invalid.

```java
public BlockRealMatrix getSubMatrix(int startRow,
                                    int endRow,
                                    int startColumn,
                                    int endColumn)
                             throws MathIllegalArgumentException
```

--------------------------------

### Get subvector

Source: https://www.hipparchus.org/hipparchus-core/apidocs/org/hipparchus/linear/SparseFieldVector.html

Extracts a contiguous subvector starting at the given index with the specified length.

```java
public FieldVector<T> getSubVector​(int index,
                                   int n)
                            throws MathIllegalArgumentException
```

--------------------------------

### POST /integrator/setup

Source: https://www.hipparchus.org/hipparchus-core/apidocs/org/hipparchus/analysis/integration/BaseAbstractUnivariateIntegrator.html

Prepares the integrator for a new computation task.

```APIDOC
## POST /integrator/setup

### Description
Prepares the integrator for computation by setting the maximum evaluations, the function to integrate, and the integration bounds.

### Method
POST

### Parameters
#### Request Body
- **maxEval** (int) - Required - Maximum number of evaluations.
- **f** (UnivariateFunction) - Required - The integrand function.
- **lower** (double) - Required - The min bound for the interval.
- **upper** (double) - Required - The upper bound for the interval.
```

--------------------------------

### GET /optima

Source: https://www.hipparchus.org/hipparchus-optim/jacoco/org.hipparchus.optim.univariate/MultiStartUnivariateOptimizer.java.html

Retrieves an array of all points found at the end of each start, sorted by objective value.

```APIDOC
## GET /optima

### Description
Returns all the points found at the end of each start, including the best one. The array is ordered by convergence and objective value (best to worst), with null elements for non-converged runs.

### Method
GET

### Response
#### Success Response (200)
- **optima** (UnivariatePointValuePair[]) - An array containing the optima found across all starts.

#### Error Handling
- **MathIllegalStateException** - Thrown if the optimize method has not been called yet.
```

--------------------------------

### Initialize BOBYQA setup

Source: https://www.hipparchus.org/hipparchus-optim/jacoco/org.hipparchus.optim.nonlinear.scalar.noderiv/BOBYQAOptimizer.java.html

Validates problem constraints and initializes the required matrices and vectors for the BOBYQA optimization process.

```java
private void setup(double[] lowerBound,  
						   double[] upperBound) {  

		double[] init = getStartPoint();  
		final int dimension = init.length;  

		// Check problem dimension.  
		if (dimension < MINIMUM_PROBLEM_DIMENSION) {  
			throw new MathIllegalArgumentException(LocalizedCoreFormats.NUMBER_TOO_SMALL,  
													dimension, MINIMUM_PROBLEM_DIMENSION);  
		}  
		// Check number of interpolation points.  
		final int[] nPointsInterval = { dimension + 2, (dimension + 2) * (dimension + 1) / 2 };  
		if (numberOfInterpolationPoints < nPointsInterval[0] ||  
			numberOfInterpolationPoints > nPointsInterval[1]) {  
			throw new MathIllegalArgumentException(LocalizedCoreFormats.NUMBER_OF_INTERPOLATION_POINTS,  
												numberOfInterpolationPoints,  
												nPointsInterval[0],  
												nPointsInterval[1]);  
		}  

		// Initialize bound differences.  
		boundDifference = new double[dimension];  

		double requiredMinDiff = 2 * initialTrustRegionRadius;  
		double minDiff = Double.POSITIVE_INFINITY;  
		for (int i = 0; i < dimension; i++) {  
			boundDifference[i] = upperBound[i] - lowerBound[i];  
			minDiff = FastMath.min(minDiff, boundDifference[i]);  
		}  
		if (minDiff < requiredMinDiff) {  
			initialTrustRegionRadius = minDiff / 3.0;  
		}  

		// Initialize the data structures used by the "bobyqa" method.  
		bMatrix = new Array2DRowRealMatrix(dimension + numberOfInterpolationPoints,  
											dimension);  
		zMatrix = new Array2DRowRealMatrix(numberOfInterpolationPoints,  
											numberOfInterpolationPoints - dimension - 1);  
		interpolationPoints = new Array2DRowRealMatrix(numberOfInterpolationPoints,  
														dimension);  
		originShift = new ArrayRealVector(dimension);  
		fAtInterpolationPoints = new ArrayRealVector(numberOfInterpolationPoints);  
		trustRegionCenterOffset = new ArrayRealVector(dimension);  
		gradientAtTrustRegionCenter = new ArrayRealVector(dimension);  
		lowerDifference = new ArrayRealVector(dimension);  
		upperDifference = new ArrayRealVector(dimension);  
		modelSecondDerivativesParameters = new ArrayRealVector(numberOfInterpolationPoints);  
		newPoint = new ArrayRealVector(dimension);  
		alternativeNewPoint = new ArrayRealVector(dimension);  
		trialStepPoint = new ArrayRealVector(dimension);  
		lagrangeValuesAtNewPoint = new ArrayRealVector(dimension + numberOfInterpolationPoints);  
		modelSecondDerivativesValues = new ArrayRealVector(dimension * (dimension + 1) / 2);  
	}
```

--------------------------------

### start Method

Source: https://www.hipparchus.org/apidocs/org/hipparchus/linear/FieldMatrixChangingVisitor.html

Initializes the visitor before traversing the matrix.

```APIDOC
## Method: start

### Description
Start visiting a matrix. This method is called once before any entry of the matrix is visited.

### Method Signature
```java
void start(int rows, int columns, int startRow, int endRow, int startColumn, int endColumn)
```

### Parameters
* **rows** (int) - Required - number of rows of the matrix
* **columns** (int) - Required - number of columns of the matrix
* **startRow** (int) - Required - Initial row index
* **endRow** (int) - Required - Final row index (inclusive)
* **startColumn** (int) - Required - Initial column index
* **endColumn** (int) - Required - Final column index (inclusive)
```

--------------------------------

### GET /optimizer/properties

Source: https://www.hipparchus.org/hipparchus-optim/apidocs/org/hipparchus/optim/univariate/UnivariateOptimizer.html

Retrieves configuration properties of the optimizer such as goal type, bounds, and start value.

```APIDOC
## GET /optimizer/properties

### Description
Retrieves the current configuration and state of the optimizer.

### Method
GET

### Response
#### Success Response (200)
- **getGoalType** (GoalType) - The optimization type.
- **getMin** (double) - The lower bounds.
- **getMax** (double) - The upper bounds.
- **getStartValue** (double) - The initial guess.
```

--------------------------------

### Get Optima

Source: https://www.hipparchus.org/hipparchus-optim/apidocs/org/hipparchus/optim/univariate/MultiStartUnivariateOptimizer.html

Returns an array of all optima found across all start attempts, ordered by quality.

```java
public UnivariatePointValuePair[] getOptima()
```

--------------------------------

### Solver Setup and Execution

Source: https://www.hipparchus.org/hipparchus-core/jacoco/org.hipparchus.analysis.solvers/BaseAbstractUnivariateSolver.java.html

Methods for initializing solver parameters and executing the solving process.

```java
    protected void setup(int maxEval,
                         F f,
                         double min, double max,
                         double startValue)
        throws NullArgumentException {
        // Checks.
        MathUtils.checkNotNull(f);

        // Reset.
        searchMin = min;
        searchMax = max;
        searchStart = startValue;
        function = f;
        evaluations = evaluations.withMaximalCount(maxEval);
    }

    @Override
    public double solve(int maxEval, F f, double min, double max, double startValue)
        throws MathIllegalArgumentException, MathIllegalStateException {
        // Initialization.
        setup(maxEval, f, min, max, startValue);

        // Perform computation.
        return doSolve();
    }

    @Override
    public double solve(int maxEval, F f, double min, double max) {
        return solve(maxEval, f, min, max, min + 0.5 * (max - min));
    }

    @Override
    public double solve(int maxEval, F f, double startValue)
        throws MathIllegalArgumentException, MathIllegalStateException {
        return solve(maxEval, f, Double.NaN, Double.NaN, startValue);
    }

    protected abstract double doSolve()
        throws MathIllegalArgumentException, MathIllegalStateException;
```

--------------------------------

### GET /optima

Source: https://www.hipparchus.org/hipparchus-coverage/jacoco-aggregate/hipparchus-optim/org.hipparchus.optim.univariate/MultiStartUnivariateOptimizer.java.html

Retrieves an array of all points found at the end of each start in the multi-start optimization process.

```APIDOC
## GET /optima

### Description
Returns all the points found at the end of each start, including the best one. The array is ordered by convergence and objective value.

### Method
GET

### Endpoint
/optima

### Response
#### Success Response (200)
- **optima** (UnivariatePointValuePair[]) - An array containing the optima found across all starts.

#### Error Handling
- **MathIllegalStateException** - Thrown if the optimize method has not been called yet.
```

--------------------------------

### Setting up and Integrating with VariationalEquation

Source: https://www.hipparchus.org/apidocs/org/hipparchus/ode/VariationalEquation.html

Demonstrates the correct sequence of calls for setting up equations, initializing the state with derivatives, performing integration, and extracting Jacobian matrices. Ensure `setUpInitialState` is called before integration.

```java
   // set up equations
   ODEJacobiansProvider jode       = new MyODE(...);
   ExpandableODE        expandable = new Expandable(jode);
   VariationalEquation  ve         = new VariationalEquation(expandable, jode);

   // set up initial state
   ODEState initWithoutDerivatives = new ODEState(t0, y0);
   ve.setInitialMainStateJacobian(dYdY0); // only needed if the default identity matrix is not suitable
   ve.setInitialParameterJacobian(name, dYdP); // only needed if the default zero matrix is not suitable
   ODEState initWithDerivatives = ve.setUpInitialState(initWithoutDerivatives);

   // perform integration on the expanded equations with the expanded initial state
   ODEStateAndDerivative finalState = integrator.integrate(expandable, initWithDerivatives, finalT);

   // extract Jacobian matrices
   dYdY0 = ve.extractMainSetJacobian(finalState);
   dYdP  = ve.extractParameterJacobian(finalState, name);
 
```

--------------------------------

### Setup Integration

Source: https://www.hipparchus.org/apidocs/org/hipparchus/analysis/integration/BaseAbstractUnivariateIntegrator.html

Prepares the integrator for a computation by setting maximum evaluations, the function, and integration bounds.

```APIDOC
## Setup Integration

### Description
Initializes the integrator with the necessary parameters for performing an integration. This method should be called by subclasses that override the `solve` methods.

### Method
`setup(int maxEval, UnivariateFunction f, double lower, double upper)`

### Parameters
- **maxEval** (int) - Required - The maximum number of function evaluations allowed.
- **f** (UnivariateFunction) - Required - The function to be integrated.
- **lower** (double) - Required - The lower bound of the integration interval.
- **upper** (double) - Required - The upper bound of the integration interval.

### Throws
- `MathIllegalArgumentException` - If parameters are invalid.
- `NullArgumentException` - If the function `f` is null.
```

--------------------------------

### GET /geometry/firstIntersection

Source: https://www.hipparchus.org/hipparchus-coverage/jacoco-aggregate/hipparchus-geometry/org.hipparchus.geometry.euclidean.threed/PolyhedronsSet.java.html

Calculates the first sub-hyperplane intersected by a semi-infinite line starting from a given point.

```APIDOC
## GET /geometry/firstIntersection

### Description
Retrieves the first sub-hyperplane crossed by a semi-infinite line after a specified point. If no intersection occurs, it returns null.

### Method
GET

### Endpoint
/geometry/firstIntersection

### Parameters
#### Query Parameters
- **point** (Vector3D) - Required - The starting point of the line segment.
- **line** (Line) - Required - The line to consider for intersection.

### Response
#### Success Response (200)
- **sub-hyperplane** (SubHyperplane) - The first sub-hyperplane intersected by the line.
```

--------------------------------

### Main Entry Point

Source: https://www.hipparchus.org/hipparchus-samples/apidocs/org/hipparchus/samples/geometry/GeometryExample.html

Standard main method for executing the geometry example application.

```java
public static void main​(String argv)
```

--------------------------------

### Get Column Vector

Source: https://www.hipparchus.org/hipparchus-core/jacoco/org.hipparchus.linear/RealMatrix.java.html

Retrieves a column from the matrix as a RealVector. Column indices start at 0.

```Java
RealVector getColumnVector(int column)
    throws MathIllegalArgumentException;
```

--------------------------------

### Initialize BOBYQA setup

Source: https://www.hipparchus.org/hipparchus-coverage/jacoco-aggregate/hipparchus-optim/org.hipparchus.optim.nonlinear.scalar.noderiv/BOBYQAOptimizer.java.html

Performs validity checks on problem dimensions and interpolation points, and initializes the required matrices and vectors for the BOBYQA algorithm.

```java
private void setup(double[] lowerBound,  
						   double[] upperBound) {  

		double[] init = getStartPoint();  
		final int dimension = init.length;  

		// Check problem dimension.  
		if (dimension < MINIMUM_PROBLEM_DIMENSION) {  
			throw new MathIllegalArgumentException(LocalizedCoreFormats.NUMBER_TOO_SMALL,  
												dimension, MINIMUM_PROBLEM_DIMENSION);  
		}  
		// Check number of interpolation points.  
		final int[] nPointsInterval = { dimension + 2, (dimension + 2) * (dimension + 1) / 2 };  
		if (numberOfInterpolationPoints < nPointsInterval[0] ||  
			numberOfInterpolationPoints > nPointsInterval[1]) {  
			throw new MathIllegalArgumentException(LocalizedCoreFormats.NUMBER_OF_INTERPOLATION_POINTS,  
											 numberOfInterpolationPoints,  
											 nPointsInterval[0],  
											 nPointsInterval[1]);  
		}  

		// Initialize bound differences.  
		boundDifference = new double[dimension];  

		double requiredMinDiff = 2 * initialTrustRegionRadius;  
		double minDiff = Double.POSITIVE_INFINITY;  
		for (int i = 0; i < dimension; i++) {  
			boundDifference[i] = upperBound[i] - lowerBound[i];  
			minDiff = FastMath.min(minDiff, boundDifference[i]);  
		}  
		if (minDiff < requiredMinDiff) {  
			initialTrustRegionRadius = minDiff / 3.0;  
		}  

		// Initialize the data structures used by the "bobyqa" method.  
		bMatrix = new Array2DRowRealMatrix(dimension + numberOfInterpolationPoints,  
											dimension);  
		zMatrix = new Array2DRowRealMatrix(numberOfInterpolationPoints,  
											numberOfInterpolationPoints - dimension - 1);  
		interpolationPoints = new Array2DRowRealMatrix(numberOfInterpolationPoints,  
														dimension);  
		originShift = new ArrayRealVector(dimension);  
		fAtInterpolationPoints = new ArrayRealVector(numberOfInterpolationPoints);  
		trustRegionCenterOffset = new ArrayRealVector(dimension);  
		gradientAtTrustRegionCenter = new ArrayRealVector(dimension);  
		lowerDifference = new ArrayRealVector(dimension);  
		upperDifference = new ArrayRealVector(dimension);  
		modelSecondDerivativesParameters = new ArrayRealVector(numberOfInterpolationPoints);  
		newPoint = new ArrayRealVector(dimension);  
		alternativeNewPoint = new ArrayRealVector(dimension);  
		trialStepPoint = new ArrayRealVector(dimension);  
		lagrangeValuesAtNewPoint = new ArrayRealVector(dimension + numberOfInterpolationPoints);  
		modelSecondDerivativesValues = new ArrayRealVector(dimension * (dimension + 1) / 2);  
	}
```

--------------------------------

### Get Row Vector

Source: https://www.hipparchus.org/hipparchus-core/jacoco/org.hipparchus.linear/RealMatrix.java.html

Retrieves a row from the matrix as a RealVector. Row indices start at 0.

```Java
RealVector getRowVector(int row)
    throws MathIllegalArgumentException;
```

--------------------------------

### Integration Setup and Execution

Source: https://www.hipparchus.org/hipparchus-core/jacoco/org.hipparchus.analysis.integration/BaseAbstractUnivariateIntegrator.java.html

Configures the solver parameters and provides the template for integration execution.

```java
protected void setup(final int maxEval,
                           final UnivariateFunction f,
                           final double lower, final double upper)
        throws MathIllegalArgumentException, NullArgumentException {

        // Checks.
        MathUtils.checkNotNull(f);
        UnivariateSolverUtils.verifyInterval(lower, upper);

        // Reset.
        min = lower;
        max = upper;
        function = f;
        evaluations = evaluations.withMaximalCount(maxEval);
        iterations.reset();
    }

    /** {@inheritDoc} */
    @Override
    public double integrate(final int maxEval, final UnivariateFunction f,
                            final double lower, final double upper)
        throws MathIllegalArgumentException, MathIllegalStateException, NullArgumentException {

        // Initialization.
        setup(maxEval, f, lower, upper);

        // Perform computation.
        return doIntegrate();

    }
```

--------------------------------

### First Intersection

Source: https://www.hipparchus.org/apidocs/org/hipparchus/geometry/euclidean/threed/PolyhedronsSet.html

Gets the first sub-hyperplane crossed by a semi-infinite line starting from a given point.

```APIDOC
## firstIntersection

### Description
Get the first sub-hyperplane crossed by a semi-infinite line.

### Method
`public SubHyperplane<Euclidean3D, Vector3D, Plane, SubPlane> firstIntersection(Vector3D point, Line line)`

### Parameters
- **point** (Vector3D) - start point of the part of the line considered
- **line** (Line) - line to consider (contains point)

### Returns
The first sub-hyperplane crossed by the line after the given point, or null if the line does not intersect any sub-hyperplane.
```

--------------------------------

### Configure Start Point

Source: https://www.hipparchus.org/apidocs/org/hipparchus/fitting/SimpleCurveFitter.html

Sets the initial guess for the parameters, returning a new instance.

```java
public SimpleCurveFitter withStartPoint​(double[] newStart)
```

--------------------------------

### Get Plus Inside Hyperplane

Source: https://www.hipparchus.org/hipparchus-coverage/jacoco-aggregate/hipparchus-geometry/org.hipparchus.geometry.partitioning/BoundaryAttribute.java.html

Retrieves the internal hyperplane. No specific setup or constraints are mentioned.

```Java
public I getPlusInside() {
    return plusInside;
}
```

--------------------------------

### ExampleUtils.ExampleFrame Constructor

Source: https://www.hipparchus.org/hipparchus-samples/apidocs/org/hipparchus/samples/ExampleUtils.ExampleFrame.html

Initializes a new instance of the ExampleFrame class.

```APIDOC
## ExampleUtils.ExampleFrame()

### Description
Empty constructor.

### Method
CONSTRUCTOR

### Endpoint
N/A

### Parameters
None

### Request Example
```json
{}
```

### Response
#### Success Response (200)
An instance of ExampleFrame.

#### Response Example
```json
{
  "message": "ExampleFrame instance created"
}
```
```

--------------------------------

### Initialize Step End Handler

Source: https://www.hipparchus.org/apidocs/org/hipparchus/ode/events/FieldODEStepEndHandler.html

Method called once at the start of integration to perform internal setup.

```java
default void init​(FieldODEStateAndDerivative<T> initialState,
                  T finalTime)
```

--------------------------------

### ExampleUtils.showExampleFrame

Source: https://www.hipparchus.org/apidocs/org/hipparchus/samples/class-use/ExampleUtils.ExampleFrame.html

Displays a specified example frame to the user interface.

```APIDOC
## Method: showExampleFrame

### Description
Displays an instance of ExampleUtils.ExampleFrame to the user interface.

### Parameters
#### Path Parameters
- **frame** (ExampleUtils.ExampleFrame) - Required - The frame instance to be displayed.

### Response
- **void** - This method does not return a value.
```

--------------------------------

### Get Column

Source: https://www.hipparchus.org/hipparchus-coverage/jacoco-aggregate/hipparchus-core/org.hipparchus.linear/RealMatrix.java.html

Retrieves the entries of a specified column as a double array. Column indices start at 0.

```APIDOC
## GET COLUMN

### Description
Get the entries at the given column index as an array. Column indices start at 0.

### Method
double[]

### Endpoint
N/A (Method within a class)

### Parameters
#### Path Parameters
None

#### Query Parameters
None

#### Request Body
- **column** (int) - Required - The index of the column to be fetched.

### Request Example
```json
{
  "column": 0
}
```

### Response
#### Success Response (200)
- **array** (double[]) - The array of entries in the column.

### Response Example
```json
{
  "array": [1.0, 2.0, 3.0]
}
```

### Error Handling
- **MathIllegalArgumentException**: if the specified column index is not valid.
```

--------------------------------

### Integration Start Method

Source: https://www.hipparchus.org/hipparchus-ode/jacoco/org.hipparchus.ode/MultistepIntegrator.java.html

Initiates the ODE integration process using the configured starter integrator.

```APIDOC
## PROTECTED void start(final ExpandableODE equations, final ODEState initialState, final double finalTime)

### Description
Starts the integration process. This method computes one step using the underlying starter integrator and initializes the Nordsieck vector. The starter integrator's primary role is to establish initial conditions; it does not propagate time or handle events itself. User events and step handlers are cleared before use, and a specific NordsieckInitializer step handler is added.

### Method
PROTECTED void

### Endpoint
start

### Parameters
#### Path Parameters
- **equations** (ExpandableODE) - Required - The complete set of differential equations to integrate.
- **initialState** (ODEState) - Required - The initial state, including time, primary, and secondary state vectors.
- **finalTime** (double) - Required - The target time for the integration. Can be less than initialState.getTime() for backward integration.

### Throws
- MathIllegalArgumentException: If array dimensions do not match equation settings, or if the integration step is too small.
- MathIllegalStateException: If the number of function evaluations is exceeded.
- MathIllegalArgumentException: If the location of an event cannot be bracketed.

### Response
#### Success Response (200)
This method does not return a value upon successful completion, but it initializes the integration state.

#### Response Example
(No direct response body, exceptions indicate errors.)
```

--------------------------------

### Get Column as Array

Source: https://www.hipparchus.org/hipparchus-core/jacoco/org.hipparchus.linear/RealMatrix.java.html

Retrieves a column from the matrix as a double array. Column indices start at 0.

```Java
double[] getColumn(int column) throws MathIllegalArgumentException;
```

--------------------------------

### Example constructor usage

Source: https://www.hipparchus.org/hipparchus-core/apidocs/org/hipparchus/linear/BlockRealMatrix.html

Demonstrates the internal logic equivalent to calling the raw data constructor.

```java
matrix = new BlockRealMatrix(rawData.length, rawData[0].length,
                                   toBlocksLayout(rawData), false);
```

--------------------------------

### Get Row as Array

Source: https://www.hipparchus.org/hipparchus-core/jacoco/org.hipparchus.linear/RealMatrix.java.html

Retrieves a row from the matrix as a double array. Row indices start at 0.

```Java
double[] getRow(int row) throws MathIllegalArgumentException;
```

--------------------------------

### POST /start

Source: https://www.hipparchus.org/hipparchus-ode/apidocs/org/hipparchus/ode/MultistepIntegrator.html

Starts the integration process by initializing the Nordsieck vector.

```APIDOC
## POST start

### Description
Start the integration. This method computes one step using the underlying starter integrator and initializes the Nordsieck vector.

### Parameters
- **equations** (ExpandableODE) - Required - Complete set of differential equations
- **initialState** (ODEState) - Required - Initial state
- **finalTime** (double) - Required - Target time for the integration

### Throws
- MathIllegalArgumentException - if arrays dimension do not match or integration step is too small
- MathIllegalStateException - if the number of functions evaluations is exceeded
```

--------------------------------

### Get Column Vector

Source: https://www.hipparchus.org/hipparchus-core/apidocs/org/hipparchus/linear/BlockRealMatrix.html

Retrieves the entries of a specified column as a vector. Column indices start at 0.

```APIDOC
## GET /websites/hipparchus/getColumnVector

### Description
Get the entries at the given column index as a vector. Column indices start at 0.

### Method
GET

### Endpoint
/websites/hipparchus/getColumnVector

### Parameters
#### Path Parameters
- **column** (int) - Required - The column index to be fetched.

### Returns
- **RealVector** - A column vector containing the entries of the specified column.

### Throws
- **MathIllegalArgumentException** - if the specified column index is invalid.
```