# PolymathPlus Examples

## Purpose

This file gives representative, validated examples for students, engineers, and AI agents that need to convert natural-language problems into valid PolymathPlus source.

Use these examples for modeling patterns and prompt grounding. Use the detailed specs for exact syntax:

- `spec_leq.md`
- `spec_nle.md`
- `spec_deq.md`
- `spec_reg.md`
- `spec_nlp.md`

This file is intentionally not exhaustive. For less common subtypes such as REG `mlinfit`, REG `integrate`, and NLP `fit:`, use the detailed specs.

Each example is one singular program type. Do not combine examples into a composite source file.

## LEQ: Linear Algebraic System

Natural-language problem:

Solve a two-equation linear balance system where `2*A + B = 10` and `A - B = 1`.

Program type: `LEQ`

```polymathplus
2*A + B = 10
A - B = 1
```

Modeling notes:

- This is LEQ because all unknowns are linear.
- There are no products of unknowns, derivatives, tables, or optimization statements.

## NLE0: Expression Evaluation

Natural-language problem:

Evaluate an engineering expression for pressure drop from known flow and resistance values.

Program type: `NLE`

```polymathplus
flow = 3.2
resistance = 4.5
pressure_drop = resistance*flow^2
```

Modeling notes:

- This is expression evaluation, not root solving.
- No initial guesses or bounds are needed because there is no nonlinear root variable.

## NLE1: Single Nonlinear Root

Natural-language problem:

Find `x` such that `x*log(x/2) + 4*x^2 - 1620 = 0` on the interval from `0.1` to `100`.

Program type: `NLE`

```polymathplus
f(x) = x*log(x/2) + 4*x^2 - 1620
x(min) = 0.1
x(max) = 100
```

Modeling notes:

- This is NLE1 because there is one nonlinear root variable.
- Bounds are required for the root variable.

## NLES: Simultaneous Nonlinear Roots

Natural-language problem:

Solve the nonlinear system `x^2 + y^2 = 25` and `x - y = 1`, using initial guesses near `x = 4` and `y = 3`.

Program type: `NLE`

```polymathplus
f(x) = x^2 + y^2 - 25
f(y) = x - y - 1

x(0) = 4
y(0) = 3
```

Modeling notes:

- This is NLES because there are simultaneous nonlinear root equations.
- Initial guesses are required.
- Use one `f(variable)` root line per nonlinear variable.

## DEQ: Ordinary Differential Equation

Natural-language problem:

A tank concentration `A` decays with first-order rate constant `k = 0.3`. The initial concentration is `10`. Simulate from `t = 0` to `t = 5`.

Program type: `DEQ`

```polymathplus
d(A)/d(t) = -k*A
k = 0.3
A(0) = 10
t(0) = 0
t(f) = 5
```

Modeling notes:

- This is DEQ because it contains a derivative.
- Initial and final values are required.

## REG: Polynomial Regression

Natural-language problem:

Fit a second-order polynomial to measured `x, y` data.

Program type: `REG`

```polymathplus
[
x y
1 2.1
2 4.2
3 8.1
4 16.3
5 25.2
]
polyfit x y 2
```

Modeling notes:

- This is REG because it uses a data table and a regression command.
- Use REG for ordinary table-based fitting before considering NLP `fit:`.

## REG: Nonlinear Regression

Natural-language problem:

Fit the model `y = a*x/(b + x)` to measured data, starting from `a = 2` and `b = 1`.

Program type: `REG`

```polymathplus
[
x y
0.5   1.255
0.387 1.25
0.24  1.189
0.136 1.124
0.04  0.783
0.011 0.402
]
nlinfit y = a*x/(b + x)
m(a)=2
m(b)=1
```

Modeling notes:

- This is REG `nlinfit`.
- Initial guesses are required for fitted parameters.

## NLP: Unconstrained Optimization

Natural-language problem:

Find `x` and `y` that minimize the squared distance from the point `(2, 3)`.

Program type: `NLP`

```polymathplus
min: (x - 2)^2 + (y - 3)^2
x| 1 0 10
y| 1 0 10
```

Modeling notes:

- This is NLP because it minimizes an objective.
- Guess/bounds lines are required for model variables.

## NLP: Constrained Optimization

Natural-language problem:

Minimize distance from `(1, 1)` subject to `x + y = 1` and inside the circle `x^2 + y^2 <= 1.5`.

Program type: `NLP`

```polymathplus
min: (x - 1)^2 + (y - 1)^2
x + y - 1 = 0
1.5 - (x^2 + y^2) > 0
x| 0.5 -2 2
y| 0.5 -2 2
```

Modeling notes:

- Equality constraints use residual form: `residual = 0`.
- Inequality constraints use residual form: `residual > 0`.
- Do not use `>=`, `<=`, or nonzero inequality right-hand sides.

## Unsupported Example: PDE Request

Natural-language problem:

Solve a heat equation PDE in one spatial dimension and time.

Program type: unsupported directly

Recommended agent response:

```txt
PolymathPlus does not directly support PDE source programs. It supports LEQ, NLE, DEQ, REG, and NLP. If you discretize the PDE into algebraic equations or ODEs, I can help build a supported LEQ, NLE, or DEQ formulation. Please specify the discretization or confirm that you want help choosing one.
```

Modeling notes:

- Do not invent PDE syntax.
- A supported source program can be generated only after a valid reformulation is selected.

## Missing Input Example: NLES Without Guesses

Natural-language problem:

Solve `x^2 + y^2 = 25` and `x - y = 1`.

Program type: `NLE`, but blocked until initial guesses are provided.

Recommended agent response:

```txt
This is a supported NLES problem, but PolymathPlus requires an initial guess for each nonlinear variable. I should not invent those as known data. Provide starting guesses for x and y, or confirm that I should use heuristic guesses based on the scale of the equations.
```

Modeling notes:

- Do not generate final NLES source without required guesses.
- Heuristic guesses may be offered only as assumptions.