Levin collocation for a nonstationary oscillatory integral

Input $a<b$, $\omega \in \mathbb{R}\setminus \{0\}$, real $g\in C^1[a,b]$ with $\mathrm{\mid }{g}^{\mathrm{\prime }}\mathrm{\mid }\ge c>0$, complex $f\in C[a,b]$, an integer $n\ge 2$, and complex-valued $C^1$ functions ${\psi }_1,\dots ,{\psi }_n:[a,b]\to \mathbb{C}$ linearly independent over $\mathbb{C}$. Also input distinct collocation nodes $a=x_1<\cdots <x_n=b$. Define $L_{\omega }\psi ={\psi }^{\mathrm{\prime }}+i\omega g^{\mathrm{\prime }}\psi$, form $A_{jk}=L_{\omega }{\psi }_k(x_j)$ and $y_j=f(x_j)$. If $A$ is singular, return failure status and no quadrature value. Otherwise solve $Ac=y$ in exact arithmetic, put $p={\sum }_{k=1}^n{c}_k{\psi }_k$, and return $Q_L=p(b)e^{i\omega g(b)}-p(a)e^{i\omega g(a)}$ with $A,c$, and ${\kappa }_2(A)=\mathrm{\parallel }A{\mathrm{\parallel }}_2\mathrm{\parallel }{A}^{-1}{\mathrm{\parallel }}_2$, where vector norms are Euclidean and $\mathrm{\parallel }A{\mathrm{\parallel }}_2={\sup }_{\mathrm{\parallel }v{\mathrm{\parallel }}_2=1}\mathrm{\parallel }Av{\mathrm{\parallel }}_2$ is the induced matrix norm. This is a finite collocation procedure; residual and conditioning claims for the nonsingular branch are stated separately on this page. Rounded factorization error is outside this exact-arithmetic definition.