The Bayesian Reconstruction (BR) Method

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//
//    Bayesian Spectral Reconstruction and Extended MEM
//
//    Code: Dr. Alexander Karl Rothkopf               DATE: July 14th 2021
//
//    Method: Dr. Yannis Burnier, Dr. Alexander Karl Rothkopf DATE: 2013
//    ========================================================================
//
//    THIS PROGRAM IS AN IMPLEMENTATION OF TWO BAYESIAN APPROACHES
//    FOR THE NUMERICAL INVERSION OF THE TYPICALLY ILL DEFINED TASK
//
//    DATA[t] = Int_-WMIN^WMAX [ K[w,t] * SPEC[w] ]dw
//
//    WHERE THE GOAL IS TO EXTRACT THE SPECTRAL FUNCTION SPEC[w] FROM
//    THE PROVIDED LATTICE DATA DATA[t] USING A PREVIOUSLY SPECIFIED
//    KERNEL FUNCTION K[w,t].
//
//    References:
//
//    Y. Burnier and A. Rothkopf,
//    "Bayesian Approach to Spectral Function Reconstruction for
//     Euclidean Quantum Field Theories"
//    arXiv:1307.6106 [hep-lat]
//    Phys. Rev. Lett. 111, 182003 (2013)
//
//    A. Rothkopf
//    "Improved Maximum Entropy Analysis with an Extended Search Space"
//    arXiv:1110.6285 [physics.comp-ph]
//    J.Comput.Phys. 238 (2013) 106-114
//
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//    This program uses adapted code from two libraries:
//
//    1. The GNU Scientific Library:
//
//        SINGULAR VALUE DECOMPOSITION
//        EIGENSYSTEMS OF SYMMETRIC MATRICES
//        MATRIX INVERSION BASED ON QR DECOMPOSITION
//        SOLVER FOR Ax=b BASED ON QR DECOMPOSITION
//
//    2. The ALGLIB Library (MPFR part 2.6.):
//
//        LBFGS MINIMIZER
//        LEVENBERG MARQUARD + LBFGS PRECONDITIONER MINIMIZER
//        EIGENVALUES FOR NON_SYMMETRIC MATRICES
//
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