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| vigra/eigensystem.hxx |
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    00001 /************************************************************************/
    00002 /*                                                                      */
    00003 /*                  Copyright 2004 by Ullrich Koethe                    */
    00004 /*       Cognitive Systems Group, University of Hamburg, Germany        */
    00005 /*                                                                      */
    00006 /*    This file is part of the VIGRA computer vision library.           */
    00007 /*    ( Version 1.4.0, Dec 21 2005 )                                    */
    00008 /*    The VIGRA Website is                                              */
    00009 /*        http://kogs-www.informatik.uni-hamburg.de/~koethe/vigra/      */
    00010 /*    Please direct questions, bug reports, and contributions to        */
    00011 /*        koethe@informatik.uni-hamburg.de          or                  */
    00012 /*        vigra@kogs1.informatik.uni-hamburg.de                         */
    00013 /*                                                                      */
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    00022 /*                                                                      */
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    00025 /*    Software.                                                         */
    00026 /*                                                                      */
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    00035 /*                                                                      */
    00036 /************************************************************************/
    00037
    00038
    00039 #ifndef VIGRA_EIGENSYSTEM_HXX
    00040 #define VIGRA_EIGENSYSTEM_HXX
    00041
    00042 #include <algorithm>
    00043 #include <complex>
    00044 #include "vigra/matrix.hxx"
    00045
    00046
    00047 namespace vigra
    00048 {
    00049
    00050 namespace linalg
    00051 {
    00052
    00053 namespace detail
    00054 {
    00055
    00056 // code adapted from JAMA
    00057 // a and b will be overwritten
    00058 template <class T, class C1, class C2>
    00059 void
    00060 housholderTridiagonalization(MultiArrayView<2, T, C1> &a, MultiArrayView<2, T, C2> &b)
    00061 {
    00062     const unsigned int n = rowCount(a);
    00063     vigra_precondition(n == columnCount(a),
    00064         "housholderTridiagonalization(): matrix must be square.");
    00065     vigra_precondition(n == rowCount(b) && 2 <= columnCount(b),
    00066         "housholderTridiagonalization(): matrix size mismatch.");
    00067
    00068     MultiArrayView<1, T, C2> d = b.bindOuter(0);
    00069     MultiArrayView<1, T, C2> e = b.bindOuter(1);
    00070
    00071     for(unsigned int j = 0; j < n; ++j)
    00072     {
    00073         d(j) = a(n-1, j);
    00074     }
    00075
    00076     // Householder reduction to tridiagonalMatrix form.
    00077
    00078     for(int i = n-1; i > 0; --i)
    00079     {
    00080         // Scale to avoid under/overflow.
    00081
    00082         T scale = 0.0;
    00083         T h = 0.0;
    00084         for(int k = 0; k < i; ++k)
    00085         {
    00086             scale = scale + abs(d(k));
    00087         }
    00088         if(scale == 0.0)
    00089         {
    00090             e(i) = d(i-1);
    00091             for(int j = 0; j < i; ++j)
    00092             {
    00093                 d(j) = a(i-1, j);
    00094                 a(i, j) = 0.0;
    00095                 a(j, i) = 0.0;
    00096             }
    00097         }
    00098         else
    00099         {
    00100             // Generate Householder vector.
    00101
    00102             for(int k = 0; k < i; ++k)
    00103             {
    00104                 d(k) /= scale;
    00105                 h += sq(d(k));
    00106             }
    00107             T f = d(i-1);
    00108             T g = VIGRA_CSTD::sqrt(h);
    00109             if(f > 0) {
    00110                 g = -g;
    00111             }
    00112             e(i) = scale * g;
    00113             h -= f * g;
    00114             d(i-1) = f - g;
    00115             for(int j = 0; j < i; ++j)
    00116             {
    00117                e(j) = 0.0;
    00118             }
    00119
    00120             // Apply similarity transformation to remaining columns.
    00121
    00122             for(int j = 0; j < i; ++j)
    00123             {
    00124                f = d(j);
    00125                a(j, i) = f;
    00126                g = e(j) + a(j, j) * f;
    00127                for(int k = j+1; k <= i-1; ++k)
    00128                {
    00129                    g += a(k, j) * d(k);
    00130                    e(k) += a(k, j) * f;
    00131                }
    00132                e(j) = g;
    00133             }
    00134             f = 0.0;
    00135             for(int j = 0; j < i; ++j)
    00136             {
    00137                 e(j) /= h;
    00138                 f += e(j) * d(j);
    00139             }
    00140             T hh = f / (h + h);
    00141             for(int j = 0; j < i; ++j)
    00142             {
    00143                 e(j) -= hh * d(j);
    00144             }
    00145             for(int j = 0; j < i; ++j)
    00146             {
    00147                 f = d(j);
    00148                 g = e(j);
    00149                 for(int k = j; k <= i-1; ++k)
    00150                 {
    00151                     a(k, j) -= (f * e(k) + g * d(k));
    00152                 }
    00153                 d(j) = a(i-1, j);
    00154                 a(i, j) = 0.0;
    00155             }
    00156         }
    00157         d(i) = h;
    00158     }
    00159
    00160     // Accumulate transformations.
    00161
    00162     for(unsigned int i = 0; i < n-1; ++i)
    00163     {
    00164         a(n-1, i) = a(i, i);
    00165         a(i, i) = 1.0;
    00166         T h = d(i+1);
    00167         if(h != 0.0)
    00168         {
    00169             for(unsigned int k = 0; k <= i; ++k)
    00170             {
    00171                 d(k) = a(k, i+1) / h;
    00172             }
    00173             for(unsigned int j = 0; j <= i; ++j)
    00174             {
    00175                 T g = 0.0;
    00176                 for(unsigned int k = 0; k <= i; ++k)
    00177                 {
    00178                     g += a(k, i+1) * a(k, j);
    00179                 }
    00180                 for(unsigned int k = 0; k <= i; ++k)
    00181                 {
    00182                     a(k, j) -= g * d(k);
    00183                 }
    00184             }
    00185         }
    00186         for(unsigned int k = 0; k <= i; ++k)
    00187         {
    00188             a(k, i+1) = 0.0;
    00189         }
    00190     }
    00191     for(unsigned int j = 0; j < n; ++j)
    00192     {
    00193         d(j) = a(n-1, j);
    00194         a(n-1, j) = 0.0;
    00195     }
    00196     a(n-1, n-1) = 1.0;
    00197     e(0) = 0.0;
    00198 }
    00199
    00200 // code adapted from JAMA
    00201 // de and z will be overwritten
    00202 template <class T, class C1, class C2>
    00203 bool
    00204 tridiagonalMatrixEigensystem(MultiArrayView<2, T, C1> &de, MultiArrayView<2, T, C2> &z)
    00205 {
    00206     const unsigned int n = rowCount(z);
    00207     vigra_precondition(n == columnCount(z),
    00208         "tridiagonalMatrixEigensystem(): matrix must be square.");
    00209     vigra_precondition(n == rowCount(de) && 2 <= columnCount(de),
    00210         "tridiagonalMatrixEigensystem(): matrix size mismatch.");
    00211
    00212     MultiArrayView<1, T, C2> d = de.bindOuter(0);
    00213     MultiArrayView<1, T, C2> e = de.bindOuter(1);
    00214
    00215     for(unsigned int i = 1; i < n; i++) {
    00216        e(i-1) = e(i);
    00217     }
    00218     e(n-1) = 0.0;
    00219
    00220     T f = 0.0;
    00221     T tst1 = 0.0;
    00222     T eps = VIGRA_CSTD::pow(2.0,-52.0);
    00223     for(unsigned int l = 0; l < n; ++l)
    00224     {
    00225         // Find small subdiagonalMatrix element
    00226
    00227         tst1 = std::max(tst1, abs(d(l)) + abs(e(l)));
    00228         unsigned int m = l;
    00229
    00230         // Original while-loop from Java code
    00231         while(m < n)
    00232         {
    00233             if(abs(e(m)) <= eps*tst1)
    00234                 break;
    00235             ++m;
    00236         }
    00237
    00238         // If m == l, d(l) is an eigenvalue,
    00239         // otherwise, iterate.
    00240
    00241         if(m > l)
    00242         {
    00243             int iter = 0;
    00244             do
    00245             {
    00246                 if(++iter > 50)
    00247                    return false; // too many iterations
    00248
    00249                 // Compute implicit shift
    00250
    00251                 T g = d(l);
    00252                 T p = (d(l+1) - g) / (2.0 * e(l));
    00253                 T r = hypot(p,1.0);
    00254                 if(p < 0)
    00255                 {
    00256                     r = -r;
    00257                 }
    00258                 d(l) = e(l) / (p + r);
    00259                 d(l+1) = e(l) * (p + r);
    00260                 T dl1 = d(l+1);
    00261                 T h = g - d(l);
    00262                 for(unsigned int i = l+2; i < n; ++i)
    00263                 {
    00264                    d(i) -= h;
    00265                 }
    00266                 f = f + h;
    00267
    00268                 // Implicit QL transformation.
    00269
    00270                 p = d(m);
    00271                 T c = 1.0;
    00272                 T c2 = c;
    00273                 T c3 = c;
    00274                 T el1 = e(l+1);
    00275                 T s = 0.0;
    00276                 T s2 = 0.0;
    00277                 for(int i = m-1; i >= (int)l; --i)
    00278                 {
    00279                     c3 = c2;
    00280                     c2 = c;
    00281                     s2 = s;
    00282                     g = c * e(i);
    00283                     h = c * p;
    00284                     r = hypot(p,e(i));
    00285                     e(i+1) = s * r;
    00286                     s = e(i) / r;
    00287                     c = p / r;
    00288                     p = c * d(i) - s * g;
    00289                     d(i+1) = h + s * (c * g + s * d(i));
    00290
    00291                     // Accumulate transformation.
    00292
    00293                     for(unsigned int k = 0; k < n; ++k)
    00294                     {
    00295                          h = z(k, i+1);
    00296                          z(k, i+1) = s * z(k, i) + c * h;
    00297                         z(k, i) = c * z(k, i) - s * h;
    00298                     }
    00299                 }
    00300                 p = -s * s2 * c3 * el1 * e(l) / dl1;
    00301                 e(l) = s * p;
    00302                 d(l) = c * p;
    00303
    00304                 // Check for convergence.
    00305
    00306             } while(abs(e(l)) > eps*tst1);
    00307         }
    00308         d(l) = d(l) + f;
    00309         e(l) = 0.0;
    00310     }
    00311
    00312     // Sort eigenvalues and corresponding vectors.
    00313
    00314     for(unsigned int i = 0; i < n-1; ++i)
    00315     {
    00316         int k = i;
    00317         T p = abs(d(i));
    00318         for(unsigned int j = i+1; j < n; ++j)
    00319         {
    00320             T p1 = abs(d(j));
    00321             if(p < p1)
    00322             {
    00323                 k = j;
    00324                 p = p1;
    00325             }
    00326         }
    00327         if(k != i)
    00328         {
    00329             std::swap(d(k), d(i));
    00330             for(unsigned int j = 0; j < n; ++j)
    00331             {
    00332                 std::swap(z(j, i), z(j, k));
    00333             }
    00334         }
    00335     }
    00336     return true;
    00337 }
    00338
    00339 // Nonsymmetric reduction to Hessenberg form.
    00340
    00341 template <class T, class C1, class C2>
    00342 void nonsymmetricHessenbergReduction(MultiArrayView<2, T, C1> & H, MultiArrayView<2, T, C2> & V)
    00343 {
    00344     //  This is derived from the Algol procedures orthes and ortran,
    00345     //  by Martin and Wilkinson, Handbook for Auto. Comp.,
    00346     //  Vol.ii-Linear Algebra, and the corresponding
    00347     //  Fortran subroutines in EISPACK.
    00348
    00349     int n = rowCount(H);
    00350     int low = 0;
    00351     int high = n-1;
    00352     ArrayVector<T> ort(n);
    00353
    00354     for(int m = low+1; m <= high-1; ++m)
    00355     {
    00356         // Scale column.
    00357
    00358         T scale = 0.0;
    00359         for(int i = m; i <= high; ++i)
    00360         {
    00361             scale = scale + abs(H(i, m-1));
    00362         }
    00363         if(scale != 0.0)
    00364         {
    00365
    00366             // Compute Householder transformation.
    00367
    00368             T h = 0.0;
    00369             for(int i = high; i >= m; --i)
    00370             {
    00371                 ort[i] = H(i, m-1)/scale;
    00372                 h += sq(ort[i]);
    00373             }
    00374             T g = VIGRA_CSTD::sqrt(h);
    00375             if(ort[m] > 0)
    00376             {
    00377                 g = -g;
    00378             }
    00379             h = h - ort[m] * g;
    00380             ort[m] = ort[m] - g;
    00381
    00382             // Apply Householder similarity transformation
    00383             // H = (I-u*u'/h)*H*(I-u*u')/h)
    00384
    00385             for(int j = m; j < n; ++j)
    00386             {
    00387                 T f = 0.0;
    00388                 for(int i = high; i >= m; --i)
    00389                 {
    00390                     f += ort[i]*H(i, j);
    00391                 }
    00392                 f = f/h;
    00393                 for(int i = m; i <= high; ++i)
    00394                 {
    00395                     H(i, j) -= f*ort[i];
    00396                 }
    00397             }
    00398
    00399             for(int i = 0; i <= high; ++i)
    00400             {
    00401                 T f = 0.0;
    00402                 for(int j = high; j >= m; --j)
    00403                 {
    00404                     f += ort[j]*H(i, j);
    00405                 }
    00406                 f = f/h;
    00407                 for(int j = m; j <= high; ++j)
    00408                 {
    00409                     H(i, j) -= f*ort[j];
    00410                 }
    00411             }
    00412             ort[m] = scale*ort[m];
    00413             H(m, m-1) = scale*g;
    00414         }
    00415     }
    00416
    00417     // Accumulate transformations (Algol's ortran).
    00418
    00419     for(int i = 0; i < n; ++i)
    00420     {
    00421         for(int j = 0; j < n; ++j)
    00422         {
    00423             V(i, j) = (i == j ? 1.0 : 0.0);
    00424         }
    00425     }
    00426
    00427     for(int m = high-1; m >= low+1; --m)
    00428     {
    00429         if(H(m, m-1) != 0.0)
    00430         {
    00431             for(int i = m+1; i <= high; ++i)
    00432             {
    00433                 ort[i] = H(i, m-1);
    00434             }
    00435             for(int j = m; j <= high; ++j)
    00436             {
    00437                 T g = 0.0;
    00438                 for(int i = m; i <= high; ++i)
    00439                 {
    00440                     g += ort[i] * V(i, j);
    00441                 }
    00442                 // Double division avoids possible underflow
    00443                 g = (g / ort[m]) / H(m, m-1);
    00444                 for(int i = m; i <= high; ++i)
    00445                 {
    00446                     V(i, j) += g * ort[i];
    00447                 }
    00448             }
    00449         }
    00450     }
    00451 }
    00452
    00453
    00454 // Complex scalar division.
    00455
    00456 template <class T>
    00457 void cdiv(T xr, T xi, T yr, T yi, T & cdivr, T & cdivi)
    00458 {
    00459     T r,d;
    00460     if(abs(yr) > abs(yi))
    00461     {
    00462         r = yi/yr;
    00463         d = yr + r*yi;
    00464         cdivr = (xr + r*xi)/d;
    00465         cdivi = (xi - r*xr)/d;
    00466     }
    00467     else
    00468     {
    00469         r = yr/yi;
    00470         d = yi + r*yr;
    00471         cdivr = (r*xr + xi)/d;
    00472         cdivi = (r*xi - xr)/d;
    00473     }
    00474 }
    00475
    00476 template <class T, class C>
    00477 int hessenbergQrDecompositionHelper(MultiArrayView<2, T, C> & H, int n, int l, double eps,
    00478              T & p, T & q, T & r, T & s, T & w, T & x, T & y, T & z)
    00479 {
    00480     int m = n-2;
    00481     while(m >= l)
    00482     {
    00483         z = H(m, m);
    00484         r = x - z;
    00485         s = y - z;
    00486         p = (r * s - w) / H(m+1, m) + H(m, m+1);
    00487         q = H(m+1, m+1) - z - r - s;
    00488         r = H(m+2, m+1);
    00489         s = abs(p) + abs(q) + abs(r);
    00490         p = p / s;
    00491         q = q / s;
    00492         r = r / s;
    00493         if(m == l)
    00494         {
    00495             break;
    00496         }
    00497         if(abs(H(m, m-1)) * (abs(q) + abs(r)) <
    00498             eps * (abs(p) * (abs(H(m-1, m-1)) + abs(z) +
    00499             abs(H(m+1, m+1)))))
    00500         {
    00501                 break;
    00502         }
    00503         --m;
    00504     }
    00505     return m;
    00506 }
    00507
    00508
    00509
    00510 // Nonsymmetric reduction from Hessenberg to real Schur form.
    00511
    00512 template <class T, class C1, class C2, class C3>
    00513 bool hessenbergQrDecomposition(MultiArrayView<2, T, C1> & H, MultiArrayView<2, T, C2> & V,
    00514                                      MultiArrayView<2, T, C3> & de)
    00515 {
    00516
    00517     //  This is derived from the Algol procedure hqr2,
    00518     //  by Martin and Wilkinson, Handbook for Auto. Comp.,
    00519     //  Vol.ii-Linear Algebra, and the corresponding
    00520     //  Fortran subroutine in EISPACK.
    00521
    00522     // Initialize
    00523     MultiArrayView<1, T, C3> d = de.bindOuter(0);
    00524     MultiArrayView<1, T, C3> e = de.bindOuter(1);
    00525
    00526     int nn = rowCount(H);
    00527     int n = nn-1;
    00528     int low = 0;
    00529     int high = nn-1;
    00530     T eps = VIGRA_CSTD::pow(2.0, sizeof(T) == sizeof(float)
    00531                                      ? -23.0
    00532                                      : -52.0);
    00533     T exshift = 0.0;
    00534     T p=0,q=0,r=0,s=0,z=0,t,w,x,y;
    00535     T norm = vigra::norm(H);
    00536
    00537     // Outer loop over eigenvalue index
    00538     int iter = 0;
    00539     while(n >= low)
    00540     {
    00541
    00542         // Look for single small sub-diagonal element
    00543         int l = n;
    00544         while (l > low)
    00545         {
    00546             s = abs(H(l-1, l-1)) + abs(H(l, l));
    00547             if(s == 0.0)
    00548             {
    00549                 s = norm;
    00550             }
    00551             if(abs(H(l, l-1)) < eps * s)
    00552             {
    00553                 break;
    00554             }
    00555             --l;
    00556         }
    00557
    00558         // Check for convergence
    00559         // One root found
    00560         if(l == n)
    00561         {
    00562             H(n, n) = H(n, n) + exshift;
    00563             d(n) = H(n, n);
    00564             e(n) = 0.0;
    00565             --n;
    00566             iter = 0;
    00567
    00568         // Two roots found
    00569
    00570         }
    00571         else if(l == n-1)
    00572         {
    00573             w = H(n, n-1) * H(n-1, n);
    00574             p = (H(n-1, n-1) - H(n, n)) / 2.0;
    00575             q = p * p + w;
    00576             z = VIGRA_CSTD::sqrt(abs(q));
    00577             H(n, n) = H(n, n) + exshift;
    00578             H(n-1, n-1) = H(n-1, n-1) + exshift;
    00579             x = H(n, n);
    00580
    00581             // Real pair
    00582
    00583             if(q >= 0)
    00584             {
    00585                 if(p >= 0)
    00586                 {
    00587                     z = p + z;
    00588                 }
    00589                 else
    00590                 {
    00591                     z = p - z;
    00592                 }
    00593                 d(n-1) = x + z;
    00594                 d(n) = d(n-1);
    00595                 if(z != 0.0)
    00596                 {
    00597                     d(n) = x - w / z;
    00598                 }
    00599                 e(n-1) = 0.0;
    00600                 e(n) = 0.0;
    00601                 x = H(n, n-1);
    00602                 s = abs(x) + abs(z);
    00603                 p = x / s;
    00604                 q = z / s;
    00605                 r = VIGRA_CSTD::sqrt(p * p+q * q);
    00606                 p = p / r;
    00607                 q = q / r;
    00608
    00609                 // Row modification
    00610
    00611                 for(int j = n-1; j < nn; ++j)
    00612                 {
    00613                     z = H(n-1, j);
    00614                     H(n-1, j) = q * z + p * H(n, j);
    00615                     H(n, j) = q * H(n, j) - p * z;
    00616                 }
    00617
    00618                 // Column modification
    00619
    00620                 for(int i = 0; i <= n; ++i)
    00621                 {
    00622                     z = H(i, n-1);
    00623                     H(i, n-1) = q * z + p * H(i, n);
    00624                     H(i, n) = q * H(i, n) - p * z;
    00625                 }
    00626
    00627                 // Accumulate transformations
    00628
    00629                 for(int i = low; i <= high; ++i)
    00630                 {
    00631                     z = V(i, n-1);
    00632                     V(i, n-1) = q * z + p * V(i, n);
    00633                     V(i, n) = q * V(i, n) - p * z;
    00634                 }
    00635
    00636             // Complex pair
    00637
    00638             }
    00639             else
    00640             {
    00641                 d(n-1) = x + p;
    00642                 d(n) = x + p;
    00643                 e(n-1) = z;
    00644                 e(n) = -z;
    00645             }
    00646             n = n - 2;
    00647             iter = 0;
    00648
    00649         // No convergence yet
    00650
    00651         }
    00652         else
    00653         {
    00654
    00655             // Form shift
    00656
    00657             x = H(n, n);
    00658             y = 0.0;
    00659             w = 0.0;
    00660             if(l < n)
    00661             {
    00662                 y = H(n-1, n-1);
    00663                 w = H(n, n-1) * H(n-1, n);
    00664             }
    00665
    00666             // Wilkinson's original ad hoc shift
    00667
    00668             if(iter == 10)
    00669             {
    00670                 exshift += x;
    00671                 for(int i = low; i <= n; ++i)
    00672                 {
    00673                     H(i, i) -= x;
    00674                 }
    00675                 s = abs(H(n, n-1)) + abs(H(n-1, n-2));
    00676                 x = y = 0.75 * s;
    00677                 w = -0.4375 * s * s;
    00678             }
    00679
    00680             // MATLAB's new ad hoc shift
    00681
    00682             if(iter == 30)
    00683             {
    00684                  s = (y - x) / 2.0;
    00685                  s = s * s + w;
    00686                  if(s > 0)
    00687                  {
    00688                       s = VIGRA_CSTD::sqrt(s);
    00689                       if(y < x)
    00690                       {
    00691                           s = -s;
    00692                       }
    00693                       s = x - w / ((y - x) / 2.0 + s);
    00694                       for(int i = low; i <= n; ++i)
    00695                       {
    00696                           H(i, i) -= s;
    00697                       }
    00698                       exshift += s;
    00699                       x = y = w = 0.964;
    00700                  }
    00701             }
    00702
    00703             iter = iter + 1;
    00704             if(iter > 60)
    00705                 return false;
    00706
    00707             // Look for two consecutive small sub-diagonal elements
    00708             int m = hessenbergQrDecompositionHelper(H, n, l, eps, p, q, r, s, w, x, y, z);
    00709             for(int i = m+2; i <= n; ++i)
    00710             {
    00711                 H(i, i-2) = 0.0;
    00712                 if(i > m+2)
    00713                 {
    00714                     H(i, i-3) = 0.0;
    00715                 }
    00716             }
    00717
    00718             // Double QR step involving rows l:n and columns m:n
    00719
    00720             for(int k = m; k <= n-1; ++k)
    00721             {
    00722                 int notlast = (k != n-1);
    00723                 if(k != m) {
    00724                     p = H(k, k-1);
    00725                     q = H(k+1, k-1);
    00726                     r = (notlast ? H(k+2, k-1) : 0.0);
    00727                     x = abs(p) + abs(q) + abs(r);
    00728                     if(x != 0.0)
    00729                     {
    00730                         p = p / x;
    00731                         q = q / x;
    00732                         r = r / x;
    00733                     }
    00734                 }
    00735                 if(x == 0.0)
    00736                 {
    00737                     break;
    00738                 }
    00739                 s = VIGRA_CSTD::sqrt(p * p + q * q + r * r);
    00740                 if(p < 0)
    00741                 {
    00742                     s = -s;
    00743                 }
    00744                 if(s != 0)
    00745                 {
    00746                     if(k != m)
    00747                     {
    00748                         H(k, k-1) = -s * x;
    00749                     }
    00750                     else if(l != m)
    00751                     {
    00752                         H(k, k-1) = -H(k, k-1);
    00753                     }
    00754                     p = p + s;
    00755                     x = p / s;
    00756                     y = q / s;
    00757                     z = r / s;
    00758                     q = q / p;
    00759                     r = r / p;
    00760
    00761                     // Row modification
    00762
    00763                     for(int j = k; j < nn; ++j)
    00764                     {
    00765                         p = H(k, j) + q * H(k+1, j);
    00766                         if(notlast)
    00767                         {
    00768                             p = p + r * H(k+2, j);
    00769                             H(k+2, j) = H(k+2, j) - p * z;
    00770                         }
    00771                         H(k, j) = H(k, j) - p * x;
    00772                         H(k+1, j) = H(k+1, j) - p * y;
    00773                     }
    00774
    00775                     // Column modification
    00776
    00777                     for(int i = 0; i <= std::min(n,k+3); ++i)
    00778                     {
    00779                         p = x * H(i, k) + y * H(i, k+1);
    00780                         if(notlast)
    00781                         {
    00782                             p = p + z * H(i, k+2);
    00783                             H(i, k+2) = H(i, k+2) - p * r;
    00784                         }
    00785                         H(i, k) = H(i, k) - p;
    00786                         H(i, k+1) = H(i, k+1) - p * q;
    00787                     }
    00788
    00789                     // Accumulate transformations
    00790
    00791                     for(int i = low; i <= high; ++i)
    00792                     {
    00793                         p = x * V(i, k) + y * V(i, k+1);
    00794                         if(notlast)
    00795                         {
    00796                             p = p + z * V(i, k+2);
    00797                             V(i, k+2) = V(i, k+2) - p * r;
    00798                         }
    00799                         V(i, k) = V(i, k) - p;
    00800                         V(i, k+1) = V(i, k+1) - p * q;
    00801                     }
    00802                 }  // (s != 0)
    00803             }  // k loop
    00804         }  // check convergence
    00805     }  // while (n >= low)
    00806
    00807     // Backsubstitute to find vectors of upper triangular form
    00808
    00809     if(norm == 0.0)
    00810     {
    00811         return false;
    00812     }
    00813
    00814     for(n = nn-1; n >= 0; --n)
    00815     {
    00816         p = d(n);
    00817         q = e(n);
    00818
    00819         // Real vector
    00820
    00821         if(q == 0)
    00822         {
    00823             int l = n;
    00824             H(n, n) = 1.0;
    00825             for(int i = n-1; i >= 0; --i)
    00826             {
    00827                 w = H(i, i) - p;
    00828                 r = 0.0;
    00829                 for(int j = l; j <= n; ++j)
    00830                 {
    00831                     r = r + H(i, j) * H(j, n);
    00832                 }
    00833                 if(e(i) < 0.0)
    00834                 {
    00835                     z = w;
    00836                     s = r;
    00837                 }
    00838                 else
    00839                 {
    00840                     l = i;
    00841                     if(e(i) == 0.0)
    00842                     {
    00843                         if(w != 0.0)
    00844                         {
    00845                             H(i, n) = -r / w;
    00846                         }
    00847                         else
    00848                         {
    00849                             H(i, n) = -r / (eps * norm);
    00850                         }
    00851
    00852                     // Solve real equations
    00853
    00854                     }
    00855                     else
    00856                     {
    00857                         x = H(i, i+1);
    00858                         y = H(i+1, i);
    00859                         q = (d(i) - p) * (d(i) - p) + e(i) * e(i);
    00860                         t = (x * s - z * r) / q;
    00861                         H(i, n) = t;
    00862                         if(abs(x) > abs(z))
    00863                         {
    00864                             H(i+1, n) = (-r - w * t) / x;
    00865                         }
    00866                         else
    00867                         {
    00868                             H(i+1, n) = (-s - y * t) / z;
    00869                         }
    00870                     }
    00871
    00872                     // Overflow control
    00873
    00874                     t = abs(H(i, n));
    00875                     if((eps * t) * t > 1)
    00876                     {
    00877                         for(int j = i; j <= n; ++j)
    00878                         {
    00879                             H(j, n) = H(j, n) / t;
    00880                         }
    00881                     }
    00882                 }
    00883             }
    00884
    00885         // Complex vector
    00886
    00887         }
    00888         else if(q < 0)
    00889         {
    00890             int l = n-1;
    00891
    00892             // Last vector component imaginary so matrix is triangular
    00893
    00894             if(abs(H(n, n-1)) > abs(H(n-1, n)))
    00895             {
    00896                 H(n-1, n-1) = q / H(n, n-1);
    00897                 H(n-1, n) = -(H(n, n) - p) / H(n, n-1);
    00898             }
    00899             else
    00900             {
    00901                 cdiv(0.0,-H(n-1, n),H(n-1, n-1)-p,q, H(n-1, n-1), H(n-1, n));
    00902             }
    00903             H(n, n-1) = 0.0;
    00904             H(n, n) = 1.0;
    00905             for(int i = n-2; i >= 0; --i)
    00906             {
    00907                 T ra,sa,vr,vi;
    00908                 ra = 0.0;
    00909                 sa = 0.0;
    00910                 for(int j = l; j <= n; ++j)
    00911                 {
    00912                     ra = ra + H(j, n-1) * H(i, j);
    00913                     sa = sa + H(j, n) * H(i, j);
    00914                 }
    00915                 w = H(i, i) - p;
    00916
    00917                 if(e(i) < 0.0)
    00918                 {
    00919                     z = w;
    00920                     r = ra;
    00921                     s = sa;
    00922                 }
    00923                 else
    00924                 {
    00925                     l = i;
    00926                     if(e(i) == 0)
    00927                     {
    00928                         cdiv(-ra,-sa,w,q, H(i, n-1), H(i, n));
    00929                     }
    00930                     else
    00931                     {
    00932                         // Solve complex equations
    00933
    00934                         x = H(i, i+1);
    00935                         y = H(i+1, i);
    00936                         vr = (d(i) - p) * (d(i) - p) + e(i) * e(i) - q * q;
    00937                         vi = (d(i) - p) * 2.0 * q;
    00938                         if((vr == 0.0) && (vi == 0.0))
    00939                         {
    00940                             vr = eps * norm * (abs(w) + abs(q) +
    00941                             abs(x) + abs(y) + abs(z));
    00942                         }
    00943                         cdiv(x*r-z*ra+q*sa,x*s-z*sa-q*ra,vr,vi, H(i, n-1), H(i, n));
    00944                         if(abs(x) > (abs(z) + abs(q)))
    00945                         {
    00946                             H(i+1, n-1) = (-ra - w * H(i, n-1) + q * H(i, n)) / x;
    00947                             H(i+1, n) = (-sa - w * H(i, n) - q * H(i, n-1)) / x;
    00948                         }
    00949                         else
    00950                         {
    00951                             cdiv(-r-y*H(i, n-1),-s-y*H(i, n),z,q, H(i+1, n-1), H(i+1, n));
    00952                         }
    00953                     }
    00954
    00955                     // Overflow control
    00956
    00957                     t = std::max(abs(H(i, n-1)),abs(H(i, n)));
    00958                     if((eps * t) * t > 1)
    00959                     {
    00960                         for(int j = i; j <= n; ++j)
    00961                         {
    00962                             H(j, n-1) = H(j, n-1) / t;
    00963                             H(j, n) = H(j, n) / t;
    00964                         }
    00965                     }
    00966                 }
    00967             }
    00968         }
    00969     }
    00970
    00971     // Back transformation to get eigenvectors of original matrix
    00972
    00973     for(int j = nn-1; j >= low; --j)
    00974     {
    00975         for(int i = low; i <= high; ++i)
    00976         {
    00977             z = 0.0;
    00978             for(int k = low; k <= std::min(j,high); ++k)
    00979             {
    00980                 z = z + V(i, k) * H(k, j);
    00981             }
    00982             V(i, j) = z;
    00983         }
    00984     }
    00985     return true;
    00986 }
    00987
    00988 } // namespace detail
    00989
    00990 /** \addtogroup LinearAlgebraFunctions Matrix functions
    00991  */
    00992 //@{
    00993     /** Compute the eigensystem of a symmetric matrix.
    00994
    00995         \a a is a real symmetric matrix, \a ew is a single-column matrix
    00996         holding the eigenvalues, and \a ev is a matrix of the same size as
    00997         \a a whose columns are the corresponding eigenvectors. Eigenvalues
    00998         will be sorted from largest to smallest magnitude.
    00999         The algorithm returns <tt>false</tt> when it doesn't
    01000         converge. It can be applied in-place, i.e. <tt>&a == &ev</tt> is allowed.
    01001         The code of this function was adapted from JAMA.
    01002
    01003     <b>\#include</b> "<a href="eigensystem_8hxx-source.html">vigra/eigensystem.hxx</a>" or<br>
    01004     <b>\#include</b> "<a href="linear__algebra_8hxx-source.html">vigra/linear_algebra.hxx</a>"<br>
    01005         Namespaces: vigra and vigra::linalg
    01006      */
    01007 template <class T, class C1, class C2, class C3>
    01008 bool
    01009 symmetricEigensystem(MultiArrayView<2, T, C1> const & a,
    01010             MultiArrayView<2, T, C2> & ew, MultiArrayView<2, T, C3> & ev)
    01011 {
    01012     vigra_precondition(isSymmetric(a),
    01013         "symmetricEigensystem(): symmetric input matrix required.");
    01014     unsigned int acols = columnCount(a);
    01015     vigra_precondition(1 == columnCount(ew) && acols == rowCount(ew) &&
    01016                        acols == columnCount(ev) && acols == rowCount(ev),
    01017         "symmetricEigensystem(): matrix shape mismatch.");
    01018
    01019     ev.copy(a); // does nothing if &ev == &a
    01020     Matrix<T> de(acols, 2);
    01021     detail::housholderTridiagonalization(ev, de);
    01022     if(!detail::tridiagonalMatrixEigensystem(de, ev))
    01023         return false;
    01024
    01025     ew.copy(columnVector(de, 0));
    01026     return true;
    01027 }
    01028
    01029     /** Compute the eigensystem of a square, but
    01030         not necessarily symmetric matrix.
    01031
    01032         \a a is a real square matrix, \a ew is a single-column matrix
    01033         holding the possibly complex eigenvalues, and \a ev is a matrix of
    01034         the same size as \a a whose columns are the corresponding eigenvectors.
    01035         Eigenvalues will be sorted from largest to smallest magnitude.
    01036         The algorithm returns <tt>false</tt> when it doesn't
    01037         converge. It can be applied in-place, i.e. <tt>&a == &ev</tt> is allowed.
    01038         The code of this function was adapted from JAMA.
    01039
    01040     <b>\#include</b> "<a href="eigensystem_8hxx-source.html">vigra/eigensystem.hxx</a>" or<br>
    01041     <b>\#include</b> "<a href="linear__algebra_8hxx-source.html">vigra/linear_algebra.hxx</a>"<br>
    01042         Namespaces: vigra and vigra::linalg
    01043      */
    01044 template <class T, class C1, class C2, class C3>
    01045 bool
    01046 nonsymmetricEigensystem(MultiArrayView<2, T, C1> const & a,
    01047          MultiArrayView<2, std::complex<T>, C2> & ew, MultiArrayView<2, T, C3> & ev)
    01048 {
    01049     unsigned int acols = columnCount(a);
    01050     vigra_precondition(acols == rowCount(a),
    01051         "nonsymmetricEigensystem(): square input matrix required.");
    01052     vigra_precondition(1 == columnCount(ew) && acols == rowCount(ew) &&
    01053                        acols == columnCount(ev) && acols == rowCount(ev),
    01054         "nonsymmetricEigensystem(): matrix shape mismatch.");
    01055
    01056     Matrix<T> H(a);
    01057     Matrix<T> de(acols, 2);
    01058     detail::nonsymmetricHessenbergReduction(H, ev);
    01059     if(!detail::hessenbergQrDecomposition(H, ev, de))
    01060         return false;
    01061
    01062     for(unsigned int i=0; i < acols; ++i)
    01063     {
    01064         ew(i,0) = std::complex<T>(de(i, 0), de(i, 1));
    01065     }
    01066     return true;
    01067 }
    01068
    01069     /** Compute the roots of a polynomial using the eigenvalue method.
    01070
    01071         \a poly is a real polynomial (compatible to \ref vigra::PolynomialView),
    01072         and \a roots a complex valued vector (compatible to <tt>std::vector</tt>
    01073         with a <tt>value_type</tt> compatible to the type <tt>POLYNOMIAL::Complex</tt>) to which
    01074         the roots are appended. The function calls \ref nonsymmetricEigensystem() with the standard
    01075         companion matrix yielding the roots as eigenvalues. It returns <tt>false</tt> if
    01076         it fails to converge.
    01077
    01078         <b>\#include</b> "<a href="eigensystem_8hxx-source.html">vigra/eigensystem.hxx</a>" or<br>
    01079         <b>\#include</b> "<a href="linear__algebra_8hxx-source.html">vigra/linear_algebra.hxx</a>"<br>
    01080         Namespaces: vigra and vigra::linalg
    01081
    01082         \see polynomialRoots(), vigra::Polynomial
    01083      */
    01084 template <class POLYNOMIAL, class VECTOR>
    01085 bool polynomialRootsEigenvalueMethod(POLYNOMIAL const & poly, VECTOR & roots, bool polishRoots)
    01086 {
    01087     typedef typename POLYNOMIAL::value_type T;
    01088     typedef typename POLYNOMIAL::Real    Real;
    01089     typedef typename POLYNOMIAL::Complex Complex;
    01090     typedef Matrix<T> TMatrix;
    01091     typedef Matrix<Complex> ComplexMatrix;
    01092
    01093     int const degree = poly.order();
    01094     double const eps = poly.epsilon();
    01095
    01096     TMatrix inMatrix(degree, degree);
    01097     for(int i = 0; i < degree; ++i)
    01098         inMatrix(0, i) = -poly[degree - i - 1] / poly[degree];
    01099     for(int i = 0; i < degree - 1; ++i)
    01100         inMatrix(i + 1, i) = NumericTraits<T>::one();
    01101     ComplexMatrix ew(degree, 1);
    01102     TMatrix ev(degree, degree);
    01103     bool success = nonsymmetricEigensystem(inMatrix, ew, ev);
    01104     if(!success)
    01105         return false;
    01106     for(int i = 0; i < degree; ++i)
    01107     {
    01108         if(polishRoots)
    01109             vigra::detail::laguerre1Root(poly, ew(i,0), 1);
    01110         roots.push_back(vigra::detail::deleteBelowEpsilon(ew(i,0), eps));
    01111     }
    01112     std::sort(roots.begin(), roots.end(), vigra::detail::PolynomialRootCompare<Real>(eps));
    01113     return true;
    01114 }
    01115
    01116 template <class POLYNOMIAL, class VECTOR>
    01117 bool polynomialRootsEigenvalueMethod(POLYNOMIAL const & poly, VECTOR & roots)
    01118 {
    01119     return polynomialRootsEigenvalueMethod(poly, roots, true);
    01120 }
    01121
    01122     /** Compute the real roots of a real polynomial using the eigenvalue method.
    01123
    01124         \a poly is a real polynomial (compatible to \ref vigra::PolynomialView),
    01125         and \a roots a real valued vector (compatible to <tt>std::vector</tt>
    01126         with a <tt>value_type</tt> compatible to the type <tt>POLYNOMIAL::Real</tt>) to which
    01127         the roots are appended. The function calls \ref polynomialRootsEigenvalueMethod() and
    01128         throws away all complex roots. It returns <tt>false</tt> if it fails to converge.
    01129
    01130         <b>\#include</b> "<a href="eigensystem_8hxx-source.html">vigra/eigensystem.hxx</a>" or<br>
    01131         <b>\#include</b> "<a href="linear__algebra_8hxx-source.html">vigra/linear_algebra.hxx</a>"<br>
    01132         Namespaces: vigra and vigra::linalg
    01133
    01134         \see polynomialRealRoots(), vigra::Polynomial
    01135      */
    01136 template <class POLYNOMIAL, class VECTOR>
    01137 bool polynomialRealRootsEigenvalueMethod(POLYNOMIAL const & p, VECTOR & roots, bool polishRoots)
    01138 {
    01139     typedef typename NumericTraits<typename VECTOR::value_type>::ComplexPromote Complex;
    01140     ArrayVector<Complex> croots;
    01141     if(!polynomialRootsEigenvalueMethod(p, croots))
    01142         return false;
    01143     for(unsigned int i = 0; i < croots.size(); ++i)
    01144         if(croots[i].imag() == 0.0)
    01145             roots.push_back(croots[i].real());
    01146     return true;
    01147 }
    01148
    01149 template <class POLYNOMIAL, class VECTOR>
    01150 bool polynomialRealRootsEigenvalueMethod(POLYNOMIAL const & p, VECTOR & roots)
    01151 {
    01152     return polynomialRealRootsEigenvalueMethod(p, roots, true);
    01153 }
    01154
    01155
    01156 //@}
    01157
    01158 } // namespace linalg
    01159
    01160 using linalg::symmetricEigensystem;
    01161 using linalg::nonsymmetricEigensystem;
    01162 using linalg::polynomialRootsEigenvalueMethod;
    01163 using linalg::polynomialRealRootsEigenvalueMethod;
    01164
    01165 } // namespace vigra
    01166
    01167 #endif // VIGRA_EIGENSYSTEM_HXX

© Ullrich Köthe (koethe@informatik.uni-hamburg.de) Cognitive Systems Group, University of Hamburg, Germany | html generated using doxygen and Python VIGRA 1.4.0 (21 Dec 2005)