 Solution

ID: past-exam-of-the-year-1-of-the-computer-science-course-of-the-university-of-oxford/2016/functional-programming-design-and-analysis-of-algorithms/4/d/solution

Generate successive matrix powers and take cumulative sums using matrix addition:

seriesMat :: Num a => Matrix a -> [Matrix a]
seriesMat m = scanl1 addMat (iterate (multMat m) m)

For the stipulated square matrix, iterate (multMat m) m produces m, m*m, m*(m*m), and so on. Associativity of exact matrix multiplication identifies these as

M, M^{2}, M^{3}, \dots

. The left scan scanl1 addMat then returns

[M, M + M^{2}, M + M^{2} + M^{3}, \dots] .

All dimensions remain compatible. The infinite outer list is available a finite prefix at a time under lazy evaluation; no convergence of the matrix geometric series is needed. With floating-point entries, the expression specifies this particular multiplication and addition order, since machine arithmetic need not obey the exact algebraic laws.

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