Differentiating Eq. (2), ($3N$) equations could be obtained and these may be expressed as a simple matrix $QR=P$, where $R$ refers to the sequence of the tilts, pistons, and reference errors of the Zernike coefficients. $R$ could be expressed as Display Formula
$R=(a0,0,a0,1,a0,2,a1,0,a0,1,a0,2,\u2026,aNs,0,aNs,1,aNs,2)T,$(3)
where $Q$ refers to a large-scale square matrix. The elements in the $i$’th row and $j$’th column are Display Formula$qij={(2\delta mn\u22121)(\u2211x2)mn,1\u2264m\u2264n\u2264NS,i=3m\u22122,j=3n\u22122(2\delta mn\u22121)(\u2211xy)mn,1\u2264m\u2264n\u2264NS,i=3m\u22122,j=3n\u22121(2\delta mn\u22121)(\u2211x)mn,1\u2264m\u2264n\u2264NS,i=3m\u22122,j=3n(2\delta mn\u22121)(\u2211y2)mn,1\u2264m\u2264n\u2264NS,i=3m\u22121,j=3n\u22121(2\delta mn\u22121)(\u2211y)mn,1\u2264m\u2264n\u2264NS,i=3m\u22121,j=3n(2\delta mn\u22121)(NP)mn,1\u2264m\u2264n\u2264NS,i=3m,j=3nqji,i>j,$(4)
where $P$ is a sequence with ($3\u2009\u2009N+k+1$) elements and could be expressed as Display Formula$pi={\u2211x(Wm*\u2212Wm),1\u2264m\u2264NS,i=3m\u22122\u2211y(Wm*\u2212Wm),1\u2264m\u2264NS,i=3m\u22121\u2211(Wm*\u2212Wm),1\u2264m\u2264NS,i=3m.$(5)