**Schrödinger equation in one dimension is assumed solvable by the shooting method.****The first tentative "shot" (initiated in the origin) "misses" the asymptotic boundary conditions of course.****Our proposal: The "Newton's" wavefunction corrections are evaluated simultaneously.****The quadratic rate of convergence of energies is achieved.****Numerical (anharmonic oscillator) tests added.**

**Published in Phys. Lett. A 230 (1997) 283 - 287.**