Approximating the Matrix Sign Function Using a Novel Iterative Method

Table 1

Iteration functions for finding the sign (in the scalar form) from the Padé family (4).

	𝑛=0	𝑛=1	𝑛=2	𝑛=3	𝑛=4
𝑚=0	𝑥	2𝑥1+𝑥^2	8𝑥3+6𝑥^2−𝑥^4	16𝑥5+15𝑥^2−5𝑥^4+𝑥^6	128𝑥35+140𝑥^2−70𝑥^4+28𝑥^6−5𝑥^8
𝑚=1	𝑥2(3−𝑥^2)	𝑥(3+𝑥^2)1+3𝑥^2	4𝑥(1+𝑥^2)1+6𝑥^2+𝑥^4	−8𝑥3+5𝑥^2−5−45𝑥^2−15𝑥^4+𝑥^6	64𝑥3+7𝑥^235+420𝑥^2+210𝑥^4−28𝑥^6+3𝑥^8
𝑚=2	𝑥8(15−10𝑥^2+3𝑥^4)	𝑥415+10𝑥^2−𝑥^41+5𝑥^2	𝑥(5+10𝑥^2+𝑥^4)1+10𝑥^2+5𝑥^4	6𝑥+20𝑥^3+6𝑥^51+15𝑥^2+15𝑥^4+𝑥^6	−16𝑥3+7𝑥^22+𝑥^2−7−140𝑥^2−210𝑥^4−28𝑥^6+𝑥^8
𝑚=3	−𝑥−35+35𝑥^2−21𝑥^4+5𝑥^616	𝑥35+35𝑥^2−7𝑥^4+𝑥^68+56𝑥^2	35𝑥+105𝑥^3+21𝑥^5−𝑥^76+84𝑥^2+70𝑥^4	𝑥7+35𝑥^2+21𝑥^4+𝑥^61+7𝑥^23+5𝑥^2+𝑥^4	8𝑥+7𝑥^3+7𝑥^5+𝑥^71+28𝑥^2+70𝑥^4+28𝑥^6+𝑥^8
𝑚=4	𝑥315−420𝑥^2+378𝑥^4−180𝑥^6+35𝑥^8128	315𝑥+420𝑥^3−126𝑥^5+36𝑥^7−5𝑥^964+576𝑥^2	105𝑥+420𝑥^3+126𝑥^5−12𝑥^7+𝑥^916+288𝑥^2+336𝑥^4	63𝑥+420𝑥^3+378𝑥^5+36𝑥^7−𝑥^981+3𝑥^29+7𝑥^23+𝑥^2	𝑥3+𝑥^23+27𝑥^2+33𝑥^4+𝑥^61+36𝑥^2+126𝑥^4+84𝑥^6+9𝑥^8