A recently-formulated residue-squaring method for perturbation problems is subjected to an exacting test in its application to the problem of diagonalising the Hamiltonian of the nonlinear oscillator with quartic anharmonicity. Unlike other methods, this new iterative diagonalisation method enables several eigenvalues to be calculated simultaneously with little more labour than for a single eigenvalue. Values obtained for the four lowest even-parity levels of the anharmonic oscillator from just two or three iterations are shown to agree well with earlier accurate calculations. An approximate analytical formula for the energy levels is also presented.