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In this paper, a new algorithm based on the Tschirnhaus transforms is developed to reduce the computation complexity of the 10-order line spectrum pairs (LSP) frequencies. The first step of the proposed algorithm is to derive a quartic equation from the 1st derivative of the given 5-degree LSP polynomial. Then the extremes of the 5-degree LSP polynomial can be found by applying the Tschirnhaus transform to the above quartic equation. By the use of these extremes as the initial approximations, one can easily solve the roots of the 5-degree LSP polynomial via the Newton's method and get the accurate LSP frequencies. One of the main advantages of the proposed algorithm is the rapid root determination of a quartic equation without complex number operations and resulting in considerable computational saving. Compared to other methods, the proposed algorithm can determine the precise LSP frequencies with the lowest computational complexity.