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This paper presents and analyzes new algorithms for computing the numerical values of derivatives, of arbitrary order, and of eigenvalues and eigenvectors of ${\bf A}(\rho){\bf x}(\rho) = \lambda(\rho){\bf B}(\rho){\bf x}(\rho)$ at a point $\rho=\rho_0$ at which the eigenvalues considered are multiple. Here ${\bf A}(\rho)$ and ${\bf B}(\rho)$ are hermitian matrices which depend analytically on a single real variable $\rho,$ and ${\bf B}(\rho_0)$ is positive definite. The algorithms are valid under more general conditions than previous algorithms. Numerical results support the theoretical analysis and show that the algorithms are also useful when eigenvalues are merely very close rather than coincident.