Different modulation methods can be adopted in interferometry and the desired information can be obtained with the interferograms. But sometimes the very nature of the experimental setup does not allow us to introduce any modulation method. Then we are forced to demodulate the interferograms with close fringes. Regularized phase-tracking (RPT) method can demodulate close fringes but with low precision. This paper presents a novel algorithm,called RPT approached & golden section algorithm (RPT&GS),to demodulate close fringes based on the RPT method. Starting from the result of classical RPT,we use the golden section algorithm to search for the accurate phase of single close fringe. Computer simulation shows the RPT&GS contains all the advantages of RPT besides high demodulation precision. Comparison experiments show good agreement with ZYGO GPI interferometer on real close fringes,and the average peak-valley (PV) value accuracy is better than λ/20.