EigenvalueDecomposition< Real > Class Template Reference

`#include <jama-eig.h>`

Collaboration diagram for EigenvalueDecomposition< Real >:

## Public Member Functions

EigenvalueDecomposition (const MatrixBase< Real > &A)

~EigenvalueDecomposition ()

void GetV (MatrixBase< Real > *V_out)

void GetRealEigenvalues (VectorBase< Real > *r_out)

void GetImagEigenvalues (VectorBase< Real > *i_out)

## Private Member Functions

Real & H (int r, int c)

Real & V (int r, int c)

void Hqr2 ()

void Tred2 ()

void Tql2 ()

void Orthes ()

## Static Private Member Functions

static void cdiv (Real xr, Real xi, Real yr, Real yi, Real *cdivr, Real *cdivi)

int n_

Real * d_

Real * e_

Real * V_

Real * H_

Real * ort_

## Detailed Description

### template<typename Real> class kaldi::EigenvalueDecomposition< Real >

Definition at line 39 of file jama-eig.h.

## ◆ EigenvalueDecomposition()

 EigenvalueDecomposition ( const MatrixBase< Real > & A )

Definition at line 876 of file jama-eig.h.

876  {
877  KALDI_ASSERT(A.NumCols() == A.NumRows() && A.NumCols() >= 1);
878  n_ = A.NumRows();
879  V_ = new Real[n_*n_];
880  d_ = new Real[n_];
881  e_ = new Real[n_];
882  H_ = NULL;
883  ort_ = NULL;
884  if (A.IsSymmetric(0.0)) {
885
886  for (int i = 0; i < n_; i++)
887  for (int j = 0; j < n_; j++)
888  V(i, j) = A(i, j); // Note that V(i, j) is a member function; A(i, j) is an operator
889  // of the matrix A.
890  // Tridiagonalize.
891  Tred2();
892
893  // Diagonalize.
894  Tql2();
895  } else {
896  H_ = new Real[n_*n_];
897  ort_ = new Real[n_];
898  for (int i = 0; i < n_; i++)
899  for (int j = 0; j < n_; j++)
900  H(i, j) = A(i, j); // as before: H is member function, A(i, j) is operator of matrix.
901
902  // Reduce to Hessenberg form.
903  Orthes();
904
905  // Reduce Hessenberg to real Schur form.
906  Hqr2();
907  }
908 }
Real & V(int r, int c)
Definition: jama-eig.h:70
#define KALDI_ASSERT(cond)
Definition: kaldi-error.h:185
Real & H(int r, int c)
Definition: jama-eig.h:69

## ◆ ~EigenvalueDecomposition()

 ~EigenvalueDecomposition ( )

Definition at line 911 of file jama-eig.h.

911  {
912  delete [] d_;
913  delete [] e_;
914  delete [] V_;
915  delete [] H_;
916  delete [] ort_;
917 }

## ◆ cdiv()

 static void cdiv ( Real xr, Real xi, Real yr, Real yi, Real * cdivr, Real * cdivi )
inlinestaticprivate

Definition at line 73 of file jama-eig.h.

References rnnlm::d, and EigenvalueDecomposition< Real >::Hqr2().

Referenced by EigenvalueDecomposition< Real >::Hqr2().

73  {
74  Real r, d;
75  if (std::abs(yr) > std::abs(yi)) {
76  r = yi/yr;
77  d = yr + r*yi;
78  *cdivr = (xr + r*xi)/d;
79  *cdivi = (xi - r*xr)/d;
80  } else {
81  r = yr/yi;
82  d = yi + r*yr;
83  *cdivr = (r*xr + xi)/d;
84  *cdivi = (r*xi - xr)/d;
85  }
86  }

## ◆ GetImagEigenvalues()

 void GetImagEigenvalues ( VectorBase< Real > * i_out )
inline

Definition at line 61 of file jama-eig.h.

Referenced by MatrixBase< float >::Eig().

61  {
62  // returns imaginary part of eigenvalues.
63  KALDI_ASSERT(i_out->Dim() == static_cast<MatrixIndexT>(n_));
64  for (int i = 0; i < n_; i++)
65  (*i_out)(i) = e_[i];
66  }
int32 MatrixIndexT
Definition: matrix-common.h:98
#define KALDI_ASSERT(cond)
Definition: kaldi-error.h:185

## ◆ GetRealEigenvalues()

 void GetRealEigenvalues ( VectorBase< Real > * r_out )
inline

Definition at line 55 of file jama-eig.h.

Referenced by MatrixBase< float >::Eig().

55  {
56  // returns real part of eigenvalues.
57  KALDI_ASSERT(r_out->Dim() == static_cast<MatrixIndexT>(n_));
58  for (int i = 0; i < n_; i++)
59  (*r_out)(i) = d_[i];
60  }
int32 MatrixIndexT
Definition: matrix-common.h:98
#define KALDI_ASSERT(cond)
Definition: kaldi-error.h:185

## ◆ GetV()

 void GetV ( MatrixBase< Real > * V_out )
inline

Definition at line 47 of file jama-eig.h.

Referenced by MatrixBase< float >::Eig().

47  { // V is what we call P externally; it's the matrix of
48  // eigenvectors.
49  KALDI_ASSERT(V_out->NumRows() == static_cast<MatrixIndexT>(n_)
50  && V_out->NumCols() == static_cast<MatrixIndexT>(n_));
51  for (int i = 0; i < n_; i++)
52  for (int j = 0; j < n_; j++)
53  (*V_out)(i, j) = V(i, j); // V(i, j) is member function.
54  }
int32 MatrixIndexT
Definition: matrix-common.h:98
Real & V(int r, int c)
Definition: jama-eig.h:70
#define KALDI_ASSERT(cond)
Definition: kaldi-error.h:185

## ◆ H()

 Real& H ( int r, int c )
inlineprivate

Definition at line 69 of file jama-eig.h.

69 { return H_[r*n_ + c]; }

## ◆ Hqr2()

 void Hqr2 ( )
private

Definition at line 436 of file jama-eig.h.

436  {
437  // This is derived from the Algol procedure hqr2,
438  // by Martin and Wilkinson, Handbook for Auto. Comp.,
439  // Vol.ii-Linear Algebra, and the corresponding
440  // Fortran subroutine in EISPACK.
441
442  int nn = n_;
443  int n = nn-1;
444  int low = 0;
445  int high = nn-1;
446  Real eps = std::numeric_limits<Real>::epsilon();
447  Real exshift = 0.0;
448  Real p = 0, q = 0, r = 0, s = 0, z=0, t, w, x, y;
449
450  // Store roots isolated by balanc and compute matrix norm
451
452  Real norm = 0.0;
453  for (int i = 0; i < nn; i++) {
454  if (i < low || i > high) {
455  d_[i] = H(i, i);
456  e_[i] = 0.0;
457  }
458  for (int j = std::max(i-1, 0); j < nn; j++) {
459  norm = norm + std::abs(H(i, j));
460  }
461  }
462
463  // Outer loop over eigenvalue index
464
465  int iter = 0;
466  while (n >= low) {
467
468  // Look for single small sub-diagonal element
469
470  int l = n;
471  while (l > low) {
472  s = std::abs(H(l-1, l-1)) + std::abs(H(l, l));
473  if (s == 0.0) {
474  s = norm;
475  }
476  if (std::abs(H(l, l-1)) < eps * s) {
477  break;
478  }
479  l--;
480  }
481
482  // Check for convergence
483  // One root found
484
485  if (l == n) {
486  H(n, n) = H(n, n) + exshift;
487  d_[n] = H(n, n);
488  e_[n] = 0.0;
489  n--;
490  iter = 0;
491
492  // Two roots found
493
494  } else if (l == n-1) {
495  w = H(n, n-1) * H(n-1, n);
496  p = (H(n-1, n-1) - H(n, n)) / 2.0;
497  q = p * p + w;
498  z = std::sqrt(std::abs(q));
499  H(n, n) = H(n, n) + exshift;
500  H(n-1, n-1) = H(n-1, n-1) + exshift;
501  x = H(n, n);
502
503  // Real pair
504
505  if (q >= 0) {
506  if (p >= 0) {
507  z = p + z;
508  } else {
509  z = p - z;
510  }
511  d_[n-1] = x + z;
512  d_[n] = d_[n-1];
513  if (z != 0.0) {
514  d_[n] = x - w / z;
515  }
516  e_[n-1] = 0.0;
517  e_[n] = 0.0;
518  x = H(n, n-1);
519  s = std::abs(x) + std::abs(z);
520  p = x / s;
521  q = z / s;
522  r = std::sqrt(p * p+q * q);
523  p = p / r;
524  q = q / r;
525
526  // Row modification
527
528  for (int j = n-1; j < nn; j++) {
529  z = H(n-1, j);
530  H(n-1, j) = q * z + p * H(n, j);
531  H(n, j) = q * H(n, j) - p * z;
532  }
533
534  // Column modification
535
536  for (int i = 0; i <= n; i++) {
537  z = H(i, n-1);
538  H(i, n-1) = q * z + p * H(i, n);
539  H(i, n) = q * H(i, n) - p * z;
540  }
541
542  // Accumulate transformations
543
544  for (int i = low; i <= high; i++) {
545  z = V(i, n-1);
546  V(i, n-1) = q * z + p * V(i, n);
547  V(i, n) = q * V(i, n) - p * z;
548  }
549
550  // Complex pair
551
552  } else {
553  d_[n-1] = x + p;
554  d_[n] = x + p;
555  e_[n-1] = z;
556  e_[n] = -z;
557  }
558  n = n - 2;
559  iter = 0;
560
561  // No convergence yet
562
563  } else {
564
565  // Form shift
566
567  x = H(n, n);
568  y = 0.0;
569  w = 0.0;
570  if (l < n) {
571  y = H(n-1, n-1);
572  w = H(n, n-1) * H(n-1, n);
573  }
574
575  // Wilkinson's original ad hoc shift
576
577  if (iter == 10) {
578  exshift += x;
579  for (int i = low; i <= n; i++) {
580  H(i, i) -= x;
581  }
582  s = std::abs(H(n, n-1)) + std::abs(H(n-1, n-2));
583  x = y = 0.75 * s;
584  w = -0.4375 * s * s;
585  }
586
587  // MATLAB's new ad hoc shift
588
589  if (iter == 30) {
590  s = (y - x) / 2.0;
591  s = s * s + w;
592  if (s > 0) {
593  s = std::sqrt(s);
594  if (y < x) {
595  s = -s;
596  }
597  s = x - w / ((y - x) / 2.0 + s);
598  for (int i = low; i <= n; i++) {
599  H(i, i) -= s;
600  }
601  exshift += s;
602  x = y = w = 0.964;
603  }
604  }
605
606  iter = iter + 1; // (Could check iteration count here.)
607
608  // Look for two consecutive small sub-diagonal elements
609
610  int m = n-2;
611  while (m >= l) {
612  z = H(m, m);
613  r = x - z;
614  s = y - z;
615  p = (r * s - w) / H(m+1, m) + H(m, m+1);
616  q = H(m+1, m+1) - z - r - s;
617  r = H(m+2, m+1);
618  s = std::abs(p) + std::abs(q) + std::abs(r);
619  p = p / s;
620  q = q / s;
621  r = r / s;
622  if (m == l) {
623  break;
624  }
625  if (std::abs(H(m, m-1)) * (std::abs(q) + std::abs(r)) <
626  eps * (std::abs(p) * (std::abs(H(m-1, m-1)) + std::abs(z) +
627  std::abs(H(m+1, m+1))))) {
628  break;
629  }
630  m--;
631  }
632
633  for (int i = m+2; i <= n; i++) {
634  H(i, i-2) = 0.0;
635  if (i > m+2) {
636  H(i, i-3) = 0.0;
637  }
638  }
639
640  // Double QR step involving rows l:n and columns m:n
641
642  for (int k = m; k <= n-1; k++) {
643  bool notlast = (k != n-1);
644  if (k != m) {
645  p = H(k, k-1);
646  q = H(k+1, k-1);
647  r = (notlast ? H(k+2, k-1) : 0.0);
648  x = std::abs(p) + std::abs(q) + std::abs(r);
649  if (x != 0.0) {
650  p = p / x;
651  q = q / x;
652  r = r / x;
653  }
654  }
655  if (x == 0.0) {
656  break;
657  }
658  s = std::sqrt(p * p + q * q + r * r);
659  if (p < 0) {
660  s = -s;
661  }
662  if (s != 0) {
663  if (k != m) {
664  H(k, k-1) = -s * x;
665  } else if (l != m) {
666  H(k, k-1) = -H(k, k-1);
667  }
668  p = p + s;
669  x = p / s;
670  y = q / s;
671  z = r / s;
672  q = q / p;
673  r = r / p;
674
675  // Row modification
676
677  for (int j = k; j < nn; j++) {
678  p = H(k, j) + q * H(k+1, j);
679  if (notlast) {
680  p = p + r * H(k+2, j);
681  H(k+2, j) = H(k+2, j) - p * z;
682  }
683  H(k, j) = H(k, j) - p * x;
684  H(k+1, j) = H(k+1, j) - p * y;
685  }
686
687  // Column modification
688
689  for (int i = 0; i <= std::min(n, k+3); i++) {
690  p = x * H(i, k) + y * H(i, k+1);
691  if (notlast) {
692  p = p + z * H(i, k+2);
693  H(i, k+2) = H(i, k+2) - p * r;
694  }
695  H(i, k) = H(i, k) - p;
696  H(i, k+1) = H(i, k+1) - p * q;
697  }
698
699  // Accumulate transformations
700
701  for (int i = low; i <= high; i++) {
702  p = x * V(i, k) + y * V(i, k+1);
703  if (notlast) {
704  p = p + z * V(i, k+2);
705  V(i, k+2) = V(i, k+2) - p * r;
706  }
707  V(i, k) = V(i, k) - p;
708  V(i, k+1) = V(i, k+1) - p * q;
709  }
710  } // (s != 0)
711  } // k loop
712  } // check convergence
713  } // while (n >= low)
714
715  // Backsubstitute to find vectors of upper triangular form
716
717  if (norm == 0.0) {
718  return;
719  }
720
721  for (n = nn-1; n >= 0; n--) {
722  p = d_[n];
723  q = e_[n];
724
725  // Real vector
726
727  if (q == 0) {
728  int l = n;
729  H(n, n) = 1.0;
730  for (int i = n-1; i >= 0; i--) {
731  w = H(i, i) - p;
732  r = 0.0;
733  for (int j = l; j <= n; j++) {
734  r = r + H(i, j) * H(j, n);
735  }
736  if (e_[i] < 0.0) {
737  z = w;
738  s = r;
739  } else {
740  l = i;
741  if (e_[i] == 0.0) {
742  if (w != 0.0) {
743  H(i, n) = -r / w;
744  } else {
745  H(i, n) = -r / (eps * norm);
746  }
747
748  // Solve real equations
749
750  } else {
751  x = H(i, i+1);
752  y = H(i+1, i);
753  q = (d_[i] - p) * (d_[i] - p) +e_[i] *e_[i];
754  t = (x * s - z * r) / q;
755  H(i, n) = t;
756  if (std::abs(x) > std::abs(z)) {
757  H(i+1, n) = (-r - w * t) / x;
758  } else {
759  H(i+1, n) = (-s - y * t) / z;
760  }
761  }
762
763  // Overflow control
764
765  t = std::abs(H(i, n));
766  if ((eps * t) * t > 1) {
767  for (int j = i; j <= n; j++) {
768  H(j, n) = H(j, n) / t;
769  }
770  }
771  }
772  }
773
774  // Complex vector
775
776  } else if (q < 0) {
777  int l = n-1;
778
779  // Last vector component imaginary so matrix is triangular
780
781  if (std::abs(H(n, n-1)) > std::abs(H(n-1, n))) {
782  H(n-1, n-1) = q / H(n, n-1);
783  H(n-1, n) = -(H(n, n) - p) / H(n, n-1);
784  } else {
785  Real cdivr, cdivi;
786  cdiv(0.0, -H(n-1, n), H(n-1, n-1)-p, q, &cdivr, &cdivi);
787  H(n-1, n-1) = cdivr;
788  H(n-1, n) = cdivi;
789  }
790  H(n, n-1) = 0.0;
791  H(n, n) = 1.0;
792  for (int i = n-2; i >= 0; i--) {
793  Real ra, sa, vr, vi;
794  ra = 0.0;
795  sa = 0.0;
796  for (int j = l; j <= n; j++) {
797  ra = ra + H(i, j) * H(j, n-1);
798  sa = sa + H(i, j) * H(j, n);
799  }
800  w = H(i, i) - p;
801
802  if (e_[i] < 0.0) {
803  z = w;
804  r = ra;
805  s = sa;
806  } else {
807  l = i;
808  if (e_[i] == 0) {
809  Real cdivr, cdivi;
810  cdiv(-ra, -sa, w, q, &cdivr, &cdivi);
811  H(i, n-1) = cdivr;
812  H(i, n) = cdivi;
813  } else {
814  Real cdivr, cdivi;
815  // Solve complex equations
816
817  x = H(i, i+1);
818  y = H(i+1, i);
819  vr = (d_[i] - p) * (d_[i] - p) +e_[i] *e_[i] - q * q;
820  vi = (d_[i] - p) * 2.0 * q;
821  if (vr == 0.0 && vi == 0.0) {
822  vr = eps * norm * (std::abs(w) + std::abs(q) +
823  std::abs(x) + std::abs(y) + std::abs(z));
824  }
825  cdiv(x*r-z*ra+q*sa, x*s-z*sa-q*ra, vr, vi, &cdivr, &cdivi);
826  H(i, n-1) = cdivr;
827  H(i, n) = cdivi;
828  if (std::abs(x) > (std::abs(z) + std::abs(q))) {
829  H(i+1, n-1) = (-ra - w * H(i, n-1) + q * H(i, n)) / x;
830  H(i+1, n) = (-sa - w * H(i, n) - q * H(i, n-1)) / x;
831  } else {
832  cdiv(-r-y*H(i, n-1), -s-y*H(i, n), z, q, &cdivr, &cdivi);
833  H(i+1, n-1) = cdivr;
834  H(i+1, n) = cdivi;
835  }
836  }
837
838  // Overflow control
839
840  t = std::max(std::abs(H(i, n-1)), std::abs(H(i, n)));
841  if ((eps * t) * t > 1) {
842  for (int j = i; j <= n; j++) {
843  H(j, n-1) = H(j, n-1) / t;
844  H(j, n) = H(j, n) / t;
845  }
846  }
847  }
848  }
849  }
850  }
851
852  // Vectors of isolated roots
853
854  for (int i = 0; i < nn; i++) {
855  if (i < low || i > high) {
856  for (int j = i; j < nn; j++) {
857  V(i, j) = H(i, j);
858  }
859  }
860  }
861
862  // Back transformation to get eigenvectors of original matrix
863
864  for (int j = nn-1; j >= low; j--) {
865  for (int i = low; i <= high; i++) {
866  z = 0.0;
867  for (int k = low; k <= std::min(j, high); k++) {
868  z = z + V(i, k) * H(k, j);
869  }
870  V(i, j) = z;
871  }
872  }
873 }
static void cdiv(Real xr, Real xi, Real yr, Real yi, Real *cdivr, Real *cdivi)
Definition: jama-eig.h:73
struct rnnlm::@11::@12 n
Real & V(int r, int c)
Definition: jama-eig.h:70
Real & H(int r, int c)
Definition: jama-eig.h:69

## ◆ Orthes()

 void Orthes ( )
private

Definition at line 345 of file jama-eig.h.

345  {
346
347  // This is derived from the Algol procedures orthes and ortran,
348  // by Martin and Wilkinson, Handbook for Auto. Comp.,
349  // Vol.ii-Linear Algebra, and the corresponding
350  // Fortran subroutines in EISPACK.
351
352  int low = 0;
353  int high = n_-1;
354
355  for (int m = low+1; m <= high-1; m++) {
356
357  // Scale column.
358
359  Real scale = 0.0;
360  for (int i = m; i <= high; i++) {
361  scale = scale + std::abs(H(i, m-1));
362  }
363  if (scale != 0.0) {
364
365  // Compute Householder transformation.
366
367  Real h = 0.0;
368  for (int i = high; i >= m; i--) {
369  ort_[i] = H(i, m-1)/scale;
370  h += ort_[i] * ort_[i];
371  }
372  Real g = std::sqrt(h);
373  if (ort_[m] > 0) {
374  g = -g;
375  }
376  h = h - ort_[m] * g;
377  ort_[m] = ort_[m] - g;
378
379  // Apply Householder similarity transformation
380  // H = (I-u*u'/h)*H*(I-u*u')/h)
381
382  for (int j = m; j < n_; j++) {
383  Real f = 0.0;
384  for (int i = high; i >= m; i--) {
385  f += ort_[i]*H(i, j);
386  }
387  f = f/h;
388  for (int i = m; i <= high; i++) {
389  H(i, j) -= f*ort_[i];
390  }
391  }
392
393  for (int i = 0; i <= high; i++) {
394  Real f = 0.0;
395  for (int j = high; j >= m; j--) {
396  f += ort_[j]*H(i, j);
397  }
398  f = f/h;
399  for (int j = m; j <= high; j++) {
400  H(i, j) -= f*ort_[j];
401  }
402  }
403  ort_[m] = scale*ort_[m];
404  H(m, m-1) = scale*g;
405  }
406  }
407
408  // Accumulate transformations (Algol's ortran).
409
410  for (int i = 0; i < n_; i++) {
411  for (int j = 0; j < n_; j++) {
412  V(i, j) = (i == j ? 1.0 : 0.0);
413  }
414  }
415
416  for (int m = high-1; m >= low+1; m--) {
417  if (H(m, m-1) != 0.0) {
418  for (int i = m+1; i <= high; i++) {
419  ort_[i] = H(i, m-1);
420  }
421  for (int j = m; j <= high; j++) {
422  Real g = 0.0;
423  for (int i = m; i <= high; i++) {
424  g += ort_[i] * V(i, j);
425  }
426  // Double division avoids possible underflow
427  g = (g / ort_[m]) / H(m, m-1);
428  for (int i = m; i <= high; i++) {
429  V(i, j) += g * ort_[i];
430  }
431  }
432  }
433  }
434 }
Real & V(int r, int c)
Definition: jama-eig.h:70
Real & H(int r, int c)
Definition: jama-eig.h:69

## ◆ Tql2()

 void Tql2 ( )
private

Definition at line 227 of file jama-eig.h.

227  {
228  // This is derived from the Algol procedures tql2, by
229  // Bowdler, Martin, Reinsch, and Wilkinson, Handbook for
230  // Auto. Comp., Vol.ii-Linear Algebra, and the corresponding
231  // Fortran subroutine in EISPACK.
232
233  for (int i = 1; i < n_; i++) {
234  e_[i-1] = e_[i];
235  }
236  e_[n_-1] = 0.0;
237
238  Real f = 0.0;
239  Real tst1 = 0.0;
240  Real eps = std::numeric_limits<Real>::epsilon();
241  for (int l = 0; l < n_; l++) {
242
243  // Find small subdiagonal element
244
245  tst1 = std::max(tst1, std::abs(d_[l]) + std::abs(e_[l]));
246  int m = l;
247  while (m < n_) {
248  if (std::abs(e_[m]) <= eps*tst1) {
249  break;
250  }
251  m++;
252  }
253
254  // If m == l, d_[l] is an eigenvalue,
255  // otherwise, iterate.
256
257  if (m > l) {
258  int iter = 0;
259  do {
260  iter = iter + 1; // (Could check iteration count here.)
261
262  // Compute implicit shift
263
264  Real g = d_[l];
265  Real p = (d_[l+1] - g) / (2.0 *e_[l]);
266  Real r = Hypot(p, static_cast<Real>(1.0)); // This is a Kaldi version of hypot that works with templates.
267  if (p < 0) {
268  r = -r;
269  }
270  d_[l] =e_[l] / (p + r);
271  d_[l+1] =e_[l] * (p + r);
272  Real dl1 = d_[l+1];
273  Real h = g - d_[l];
274  for (int i = l+2; i < n_; i++) {
275  d_[i] -= h;
276  }
277  f = f + h;
278
279  // Implicit QL transformation.
280
281  p = d_[m];
282  Real c = 1.0;
283  Real c2 = c;
284  Real c3 = c;
285  Real el1 =e_[l+1];
286  Real s = 0.0;
287  Real s2 = 0.0;
288  for (int i = m-1; i >= l; i--) {
289  c3 = c2;
290  c2 = c;
291  s2 = s;
292  g = c *e_[i];
293  h = c * p;
294  r = Hypot(p, e_[i]); // This is a Kaldi version of Hypot that works with templates.
295  e_[i+1] = s * r;
296  s =e_[i] / r;
297  c = p / r;
298  p = c * d_[i] - s * g;
299  d_[i+1] = h + s * (c * g + s * d_[i]);
300
301  // Accumulate transformation.
302
303  for (int k = 0; k < n_; k++) {
304  h = V(k, i+1);
305  V(k, i+1) = s * V(k, i) + c * h;
306  V(k, i) = c * V(k, i) - s * h;
307  }
308  }
309  p = -s * s2 * c3 * el1 *e_[l] / dl1;
310  e_[l] = s * p;
311  d_[l] = c * p;
312
313  // Check for convergence.
314
315  } while (std::abs(e_[l]) > eps*tst1);
316  }
317  d_[l] = d_[l] + f;
318  e_[l] = 0.0;
319  }
320
321  // Sort eigenvalues and corresponding vectors.
322
323  for (int i = 0; i < n_-1; i++) {
324  int k = i;
325  Real p = d_[i];
326  for (int j = i+1; j < n_; j++) {
327  if (d_[j] < p) {
328  k = j;
329  p = d_[j];
330  }
331  }
332  if (k != i) {
333  d_[k] = d_[i];
334  d_[i] = p;
335  for (int j = 0; j < n_; j++) {
336  p = V(j, i);
337  V(j, i) = V(j, k);
338  V(j, k) = p;
339  }
340  }
341  }
342 }
Real & V(int r, int c)
Definition: jama-eig.h:70
double Hypot(double x, double y)
Definition: kaldi-math.h:354

## ◆ Tred2()

 void Tred2 ( )
private

Definition at line 113 of file jama-eig.h.

113  {
114  // This is derived from the Algol procedures tred2 by
115  // Bowdler, Martin, Reinsch, and Wilkinson, Handbook for
116  // Auto. Comp., Vol.ii-Linear Algebra, and the corresponding
117  // Fortran subroutine in EISPACK.
118
119  for (int j = 0; j < n_; j++) {
120  d_[j] = V(n_-1, j);
121  }
122
123  // Householder reduction to tridiagonal form.
124
125  for (int i = n_-1; i > 0; i--) {
126
127  // Scale to avoid under/overflow.
128
129  Real scale = 0.0;
130  Real h = 0.0;
131  for (int k = 0; k < i; k++) {
132  scale = scale + std::abs(d_[k]);
133  }
134  if (scale == 0.0) {
135  e_[i] = d_[i-1];
136  for (int j = 0; j < i; j++) {
137  d_[j] = V(i-1, j);
138  V(i, j) = 0.0;
139  V(j, i) = 0.0;
140  }
141  } else {
142
143  // Generate Householder vector.
144
145  for (int k = 0; k < i; k++) {
146  d_[k] /= scale;
147  h += d_[k] * d_[k];
148  }
149  Real f = d_[i-1];
150  Real g = std::sqrt(h);
151  if (f > 0) {
152  g = -g;
153  }
154  e_[i] = scale * g;
155  h = h - f * g;
156  d_[i-1] = f - g;
157  for (int j = 0; j < i; j++) {
158  e_[j] = 0.0;
159  }
160
161  // Apply similarity transformation to remaining columns.
162
163  for (int j = 0; j < i; j++) {
164  f = d_[j];
165  V(j, i) = f;
166  g =e_[j] + V(j, j) * f;
167  for (int k = j+1; k <= i-1; k++) {
168  g += V(k, j) * d_[k];
169  e_[k] += V(k, j) * f;
170  }
171  e_[j] = g;
172  }
173  f = 0.0;
174  for (int j = 0; j < i; j++) {
175  e_[j] /= h;
176  f += e_[j] * d_[j];
177  }
178  Real hh = f / (h + h);
179  for (int j = 0; j < i; j++) {
180  e_[j] -= hh * d_[j];
181  }
182  for (int j = 0; j < i; j++) {
183  f = d_[j];
184  g = e_[j];
185  for (int k = j; k <= i-1; k++) {
186  V(k, j) -= (f * e_[k] + g * d_[k]);
187  }
188  d_[j] = V(i-1, j);
189  V(i, j) = 0.0;
190  }
191  }
192  d_[i] = h;
193  }
194
195  // Accumulate transformations.
196
197  for (int i = 0; i < n_-1; i++) {
198  V(n_-1, i) = V(i, i);
199  V(i, i) = 1.0;
200  Real h = d_[i+1];
201  if (h != 0.0) {
202  for (int k = 0; k <= i; k++) {
203  d_[k] = V(k, i+1) / h;
204  }
205  for (int j = 0; j <= i; j++) {
206  Real g = 0.0;
207  for (int k = 0; k <= i; k++) {
208  g += V(k, i+1) * V(k, j);
209  }
210  for (int k = 0; k <= i; k++) {
211  V(k, j) -= g * d_[k];
212  }
213  }
214  }
215  for (int k = 0; k <= i; k++) {
216  V(k, i+1) = 0.0;
217  }
218  }
219  for (int j = 0; j < n_; j++) {
220  d_[j] = V(n_-1, j);
221  V(n_-1, j) = 0.0;
222  }
223  V(n_-1, n_-1) = 1.0;
224  e_[0] = 0.0;
225 }
Real & V(int r, int c)
Definition: jama-eig.h:70

## ◆ V()

 Real& V ( int r, int c )
inlineprivate

Definition at line 70 of file jama-eig.h.

70 { return V_[r*n_ + c]; }

## ◆ d_

 Real* d_
private

## ◆ e_

 Real * e_
private

## ◆ H_

 Real* H_
private

Definition at line 96 of file jama-eig.h.

## ◆ ort_

 Real* ort_
private

Definition at line 97 of file jama-eig.h.

## ◆ V_

 Real* V_
private

Definition at line 95 of file jama-eig.h.

The documentation for this class was generated from the following file: