Two new analysis methods for solar seismology are developed. Called the continuous orthonormalization (CON) and adjoint methods, their use enables both solar eigenfrequencies and eigenfrequency sensitivities (partial derivatives with respect to solar model parameters) to be computed more accurately and efficiently than with existing methods. The CON method integrates an eighth-order nonlinear system of ordinary differential equations (ODEs) which defines the linear adiabatic nonradial oscillation modes of the Sun. (The Cowling approximation is not used.) All normal modes of oscillation are treated identically, regardless of their type (pressure, gravity or fundamental) or their predominant location inside the Sun. The adjoint method integrates a related eighth-order linear inhomogeneous system of ODEs. From the resultant solution, an eigenfrequency's partial derivatives with respect to an extensive set of solar model parameters may be computed simultaneously. Extensive numerical tests confirm the validity of the two new methods. Eigenfrequencies obtained via the CON method have seven significant digits and match within 1% the eigenfrequencies obtained via finite difference or mesh approaches. (Exact agreement is neither expected nor attainable because differently defined solar models are analyzed. The CON method analyzes models which are functionally specified on a continuum of radial points; the other methods analyze models defined on discrete sets of radial points.) Eigenfrequency sensitivities obtained via the adjoint method match within 2% the results obtained by explicitly perturbing the solar model parameters and recomputing the eigenfrequencies. The usefulness and power of the two new methods are demonstrated by applying them to the solution of an elementary solar inversion problem. A sample solar model's f-mode frequencies (obtained via the CON method) are iteratively driven into agreement with an observed set of f-mode frequencies. Adjoint sensitivity results are used to alter solar model parameters within hundreds of radial bins. The frequency movement is large, comparable to the frequency separation between adjacent degree f-modes. Model changes are also large; the density near the base of the convection zone is roughly doubled, while slightly further out it is halved.