[Master Index]

# Index for PublicToolbox/regutools in BrainStorm 2.0 (Alpha)

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### Contents

     Regularization Tools.
     Version 2.0  16-March-93.
     Copyright (c) 1993 by Per Christian Hansen and UNI-C.

     Demonstration.
       regudemo  - Tutorial introduction to Regularization Tools.

     Test examples.
       baart     - Fredholm integral equation of the first kind.
       deriv2    - Computation of the second derivative.
       foxgood   - Severely ill-posed problem.
       heat      - Inverse heat equation.
       ilaplace  - Inverse Laplace transformation.
       parallax  - Stellar parallax problem with 28 fixed observations.
       phillips  - Philips' "famous" test problem.
       shaw      - One-dimensional image restoration problem.
       spikes    - Test problem with a "spiky" solution.
       ursell    - Integral equation with no square integrable solution.
       wing      - Test problem with a discontinuous solution.

     Regularization routines.
       cgls      - Computes the least squares solution based on k steps
                   of the conjugate gradient algorithm.
       discrep   - Minimizes the solution (semi-)norm subject to an upper
                   bound on the residual norm (discrepancy principle).
       dsvd      - Computes a damped SVD/GSVD solution.
       lsqi      - Minimizes the residual norm subject to an upper bound
                   on the (semi-)norm of the solution.
       lsqr      - Computes the least squares solution based on k steps
                   of the LSQR algorithm (Lanczos bidiagonalization).
       maxent    - Computes the maximum entropy regularized solution.
       mtsvd     - Computes the modified TSVD solution.
       nu        - Computes the solution based on k steps of Brakhage's
                   iterative nu-method.
       pcgls     - Same as cgls, but for general-form regularization.
       plsqr     - Same as lsqr, but for general-form regularization.
       pnu       - Same as nu, but for general-form regularization.
       tgsvd     - Computes the truncated GSVD solution.
       tikhonov  - Computes the Tikhonov regularized solution.
       tsvd      - Computes the truncated SVD solution.
       ttls      - Computes the truncated TLS solution.

     Analysis routines.
       fil_fac   - Computes filter factors for some regularization methods.
       gcv       - Plots the GCV function and computes its minimum.
       l_corner  - Locates the L-shaped corner of the L-curve.
       l_curve   - Computes the L-curve, plots it, and computes its corner.
       lagrange  - Plots the Lagrange function ||Ax-b||^2 + lambda^2*||Lx||^,
                   and its derivative.
       picard    - Plots the (generalized) singular values,
                   the Fourier coefficient for the right-hand side, and a
                   (smoothed curve of) the solution Fourier-coefficients.
       plot_lc   - Plots an L-curve.
       quasiopt  - Plots the quasi-optimality function and computes its minimum.

     Routines for transforming a problem in general form into one in
     standard form, and back again.
       gen_form  - Transforms a standard-form solution back into the
                   general-form setting.
       std_form  - Transforms a general-form problem into one in
                   standard form.

     Utility routines.
       bidiag    - Bidiagonalization of a matrix by Householder transformations.
       bsvd      - Computes the singular values, or the compact SVD,
                   of a bidiagonal matrix stored in compact form.
       csd       - Computes the CS decomposition.
       csvd      - Computes the compact SVD of an m-by-n matrix.
       get_l     - Produces a p-by-n matrix which is the discrete
                   approximation to the d'th order derivative operator.
       gsvd      - Computes the generalized SVD of a matrix pair.
       lanc_b    - Performs k steps of the Lanczos bidiagonalization
                   process with/without reorthogonalization.

     Auxiliary routines required by some of the above routines.
       app_hh_l  - Applies a Householder transformation from the left.
       gen_hh    - Generates a Householder transformation.
       heb_new   - Newton-Raphson iteration with Hebden's rational
                   approximation, used in lsqi.
       heb_new2  - Ditto, used in discrep.
       lsolve    - Inversion with A-weighted generalized inverse of L.
       ltsolve   - Inversion with transposed A-weighted inverse of L.
       mgs       - Modified Gram-Schmidt orthonormalization.
       newton    - Newton-Raphson iteration, used in discrep.
       pinit     - Initialization for treating general-form problems.
       pythag    - Computes sqrt(a^2 + b^2).
       spleval   - Computes points on a spline or spline curve.

     The routine l_corner requires the following routines from the Spline
     Toolbox:
       fnder, ppbrk, ppcut, ppmak, sp2pp, spbrk, spmak.
     If the Spline Toolbox is not available, then dummy functions with these
     names should reside in the same directory as Regularization Tools.

## Matlab files in this Directory

    app_hh              APP_HH Apply a Householder transformation.
    baart               BAART Test problem: Fredholm integral equation of the first kind.
    bidiag              BIDIAG Bidiagonalization of an m-times-n matrix with m >= n.
    bsvd                BSVD SVD of a bidiagonal matrix stored in "compact form".
    cgls                CGLS Conjugate gradient algorithm applied implicitly to the normal equations.
    Contents            Regularization Tools.
    csd                 CSD CS decomposition.
    csvd                CSVD Compact singular value decomposition.
    deriv2              DERIV2 Test problem: computation of the second derivative.
    discrep             DISCREP Discrepancy principle criterion for choosing the reg. parameter.
    dsvd                DSVD Damped SVD regularization.
    fil_fac             FIL_FAC Filter factors for some regularization methods.
    fnder               FNDER Dummy function for Regularization Tools
    foxgood             FOXGOOD Severely ill-posed testproblem.
    gcv                 GCV Plot the GCV function and find its minimum.
    gen_form            GEN_FORM Transform a standard-form problem back to the general-form setting.
    gen_hh              GEN_HH Generate a Householder transformation.
    get_l               GET_L Compute discrete derivative operators.
    gsvd                GSVD Generalized SVD of a matrix pair.
    heat                HEAT Test problem: inverse heat equation.
    heb_new             HEB_NEW Newton iteration with Hebden model (utility routine for LSQI).
    ilaplace            ILAPLACE Test problem: inverse Laplace transformation.
    lagrange            LAGRANGE Plot the Lagrange function for Tikhonov regularization.
    lanc_b              LANC_B Lanczos bidiagonalization.
    lsolve              LSOLVE Utility routine for "preconditioned" iterative methods.
    lsqi                LSQI Least squares minimizaiton with a quadratic inequality constraint.
    lsqr                LSQR Solution of least squares problems by Lanczos bidiagonalization.
    ltsolve             LTSOLVE Utility routine for "preconditioned" iterative methods.
    l_corner            L_CORNER Locate the "corner" of the L-curve.
    l_curve             L_CURVE Plot the L-curve and find its "corner".
    maxent              MAXENT Maximum entropy regularization.
    mgs                 MGS Modified Gram-Schmidt orthonormalization.
    mtsvd               MTSVD Modified truncated SVD regularization.
    newton              NEWTON Newton iteration (utility routine for DISCREP).
    nu                  NU Brakhage's nu-method.
    parallax            PARALLAX Stellar parallax problem with 28 fixed, real observations.
    pcgls               PCGLS "Preconditioned" conjugate gradients appl. implicitly to normal equations.
    phillips            PHILLIPS Phillips' "famous" test problem.
    picard              PICARD Visual inspection of the Picard condition.
    pinit               PINIT Utility init.-procedure for "preconditioned" iterative methods.
    plot_lc             PLOT_LC Plot the L-curve.
    plsqr               PLSQR "Preconditioned" version of the LSQR Lanczos bidiagonalization algorithm.
    pnu                 PNU "Preconditioned" version of Brakhage's nu-method.
    ppbrk               PPBRK Dummy function for Regularization Tools
    ppcut               PPCUT Dummy function for Regularization Tools
    ppmak               PPMAK Dummy function for Regularization Tools
    ppual               PPUAL Dummy function for Regularization Tools
    pythag              PYTHAG Computes sqrt( y^2 + z^2 ).
    quasiopt            QUASIOPT Quasi-optimality criterion for choosing the regularization parameter.
    regudemo            REGUDEMO Tutorial script for Regularization Tools.
    shaw                SHAW Test problem: one-dimensional iimage restoration model.
    sorted              SORTED Dummy function for Regularization Tools
    sp2pp               SP2PP Dummy function for Regularization Tools
    spbrk               SPBRK Dummy function for Regularization Tools
    spikes              SPIKES Test problem with a "spiky" solution.
    spleval             SPLEVAL Evaluation of a spline or spline curve.
    spmak               SPMAK Dummy function for Regularization Tools
    sprpp               SPRPP Dummy function for Regularization Tools
    std_form            STD_FORM Transform a general-form reg. problem into one in standard form.
    tgsvd               TGSVD Truncated GSVD regularization.
    tikhonov            TIKHONOV Tikhonov regularization.
    tsvd                TSVD Truncated SVD regularization.
    ttls                TTLS Truncated TLS regularization.
    ursell              URSELL Test problem: integral equation wiht no square integrable solution.
    wing                WING Test problem with a discontinuous solution.