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J. Eur. Opt. Society-Rapid Publ. 22, 31( 2026)
Figure 3. Accuracy assessment of the block – wise recursion compared to the component-wise recursion( Andersen’ s method [ 7 ]) for calculating Zernike basis functions:( a) 120 test positions from a circular region with five radii( 1.00, 0.96, 0.88, 0.72, 0.40) and 24 uniformly distributed azimuth angles;( b) 120 difference curves( color – coded) with respect to the polynomial orders, each curve representing the maximum absolute differences between two sets of Zernike components along the orders from 0 to 99 for a position in( a);( c) the logarithmic representation( log10) of the corresponding curves in( b), of which 17 curves are distinguished from the others by their difference value of over 3 10 �15 and whose corresponding positions all lie on the unit circle.
The comparison results show an absolute difference of less than 6 10 �14, with the maximum absolute difference exceeding 3 10 �15 at 17 positions on the unit circle from order 50 onwards. This preliminary result confirmed our expectation that the block-wise recursive method can be interpreted as equivalent to the component-wise recursive method, although the former is proposed on the basis of one Pascal triangle and the latter was developed from the TTRR relationship in orthogonal polynomials.
Naturally, the question arises as to the above 17 positions on the unit circle: which method is closer to the truth? Recursive methods( component-wise and block-wise