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I am looking for a reference to study logarithm of an invertible triangular matrix. What is a good algorithm? Are there any good reference which studies this topic both theoretically and from an algorithm view point? I am also looking for structure of the logarithm of an upper/lower triangular invertible matrix.

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You may want to take a look at "Functions of Matrices: Theory and Computation" by Higham. Chapter 11 is specifically devoted to the matrix logarithm. In particular, the chapter contains a thorough comparison of four different numerical algorithms.

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If your matrix is triangular, you can read off the eigenvalues, and it is not hard to find the eigenvector/generalized eigenvectors. Once you have those, it is easy to find the logarithm.

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