A formula and a table of numbers answer different questions. A formula holds every solution at once, shows how the answer depends on the constants and the parameters, and says what happens as x grows without limit. Numbers answer one initial value problem, for one set of parameters, over one finite stretch — but they answer it for almost any equation. The two check each other: a formula is tested against numbers, and a program against equations whose solutions are known. And some questions only analysis can answer, because numbers never warn that a solution has stopped existing, or that it is not the only one.

A map and a GPS track. The map shows every road and where each one leads; the track records one journey exactly, even across open country with no road at all. Each checks the other, and neither replaces the other.

$y'=y^{2}$, $y(0)=1$ has the solution $\tfrac{1}{1-x}$, which becomes infinite at $x=1$ and does not exist beyond it. Euler’s method with $h=0.25$ never notices: its values at $x=1$ and at $x=1.25$ are the ordinary numbers $3.65$ and $6.99$.

An analytic solution is a formula: through its constants it gives every solution, it shows how the answer depends on the parameters, and it describes the behaviour for all $x$, including as $x\to\infty$. A numerical solution is a table for one initial value and one set of parameters on a bounded interval, with an error that can be estimated but not removed. Analysis says whether there is exactly one solution to compute and for how long it exists; numbers then compute it where no formula can. A program is tested by its observed order: the errors $E(h)$ on an equation with a known solution should fall by $2^{p}$ each time is halved.

• Test a program on an equation whose solution is known: if its errors fall by a factor $2^{p}$ each time $h$ is halved, the method is working at its order $p$.
• Check the existence and uniqueness conditions before computing: a method runs straight past a blow-up and follows one of many solutions without a warning.
• A formula can be wrong too — a slip in the algebra — and a quick numerical run exposes it.
• The whole family of solutions, the dependence on a parameter and the behaviour for all time belong to the formula; one accurate curve for an equation with no formula belongs to the numbers.

The right-hand side vanishes at y = 0. So the first Euler step from 0 adds h times 0, and by induction every value is 0, whatever the step. The zero function is a solution, but so is x³, and so is every function that waits at zero and then leaves along a cubic. The method reports one solution with complete confidence; only the failure of the uniqueness condition, which analysis checks, reveals the others.

The initial value.

If a value is zero, the slope there is zero and so is the next value.

By induction, whatever the step $h$.

Yet $x^{3}$ solves the same problem through the same point, and the method never finds it.

Run the method with h and with h/2 on an equation whose solution is known. The ratio of the two errors is about 2 to the power of the order.

Great work! You now know how to:

• say what a formula shows that numbers do not, and the reverse
• measure a method’s order from its errors
• prove that a method can follow one of many solutions without a warning
• use each kind of answer to check the other