CPTEQR computes all eigenvalues and, optionally, eigenvectors of a


 symmetric positive definite tridiagonal matrix by first factoring the


 matrix using SPTTRF and then calling CBDSQR to compute the singular


 values of the bidiagonal factor.


 This routine computes the eigenvalues of the positive definite


 tridiagonal matrix to high relative accuracy.  This means that if the


 eigenvalues range over many orders of magnitude in size, then the


 small eigenvalues and corresponding eigenvectors will be computed


 more accurately than, for example, with the standard QR method.


 The eigenvectors of a full or band positive definite Hermitian matrix


 can also be found if CHETRD, CHPTRD, or CHBTRD has been used to


 reduce this matrix to tridiagonal form.  (The reduction to


 tridiagonal form, however, may preclude the possibility of obtaining


 high relative accuracy in the small eigenvalues of the original


 matrix, if these eigenvalues range over many orders of magnitude.)

COMPZ

          COMPZ is CHARACTER*1


          = 'N':  Compute eigenvalues only.


          = 'V':  Compute eigenvectors of original Hermitian


                  matrix also.  Array Z contains the unitary matrix


                  used to reduce the original matrix to tridiagonal


                  form.


          = 'I':  Compute eigenvectors of tridiagonal matrix also.

N

          N is INTEGER


          The order of the matrix.  N >= 0.

D

          D is REAL array, dimension (N)


          On entry, the n diagonal elements of the tridiagonal matrix.


          On normal exit, D contains the eigenvalues, in descending


          order.

E

          E is REAL array, dimension (N-1)


          On entry, the (n-1) subdiagonal elements of the tridiagonal


          matrix.


          On exit, E has been destroyed.

Z

          Z is COMPLEX array, dimension (LDZ, N)


          On entry, if COMPZ = 'V', the unitary matrix used in the


          reduction to tridiagonal form.


          On exit, if COMPZ = 'V', the orthonormal eigenvectors of the


          original Hermitian matrix;


          if COMPZ = 'I', the orthonormal eigenvectors of the


          tridiagonal matrix.


          If INFO > 0 on exit, Z contains the eigenvectors associated


          with only the stored eigenvalues.


          If  COMPZ = 'N', then Z is not referenced.

LDZ

          LDZ is INTEGER


          The leading dimension of the array Z.  LDZ >= 1, and if


          COMPZ = 'V' or 'I', LDZ >= max(1,N).

WORK

          WORK is REAL array, dimension (4*N)

INFO

          INFO is INTEGER


          = 0:  successful exit.


          < 0:  if INFO = -i, the i-th argument had an illegal value.


          > 0:  if INFO = i, and i is:


                <= N  the Cholesky factorization of the matrix could


                      not be performed because the leading principal


                      minor of order i was not positive.


                > N   the SVD algorithm failed to converge;


                      if INFO = N+i, i off-diagonal elements of the


                      bidiagonal factor did not converge to zero.

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

 DPTEQR computes all eigenvalues and, optionally, eigenvectors of a


 symmetric positive definite tridiagonal matrix by first factoring the


 matrix using DPTTRF, and then calling DBDSQR to compute the singular


 values of the bidiagonal factor.


 This routine computes the eigenvalues of the positive definite


 tridiagonal matrix to high relative accuracy.  This means that if the


 eigenvalues range over many orders of magnitude in size, then the


 small eigenvalues and corresponding eigenvectors will be computed


 more accurately than, for example, with the standard QR method.


 The eigenvectors of a full or band symmetric positive definite matrix


 can also be found if DSYTRD, DSPTRD, or DSBTRD has been used to


 reduce this matrix to tridiagonal form. (The reduction to tridiagonal


 form, however, may preclude the possibility of obtaining high


 relative accuracy in the small eigenvalues of the original matrix, if


 these eigenvalues range over many orders of magnitude.)

COMPZ

          COMPZ is CHARACTER*1


          = 'N':  Compute eigenvalues only.


          = 'V':  Compute eigenvectors of original symmetric


                  matrix also.  Array Z contains the orthogonal


                  matrix used to reduce the original matrix to


                  tridiagonal form.


          = 'I':  Compute eigenvectors of tridiagonal matrix also.

N

          N is INTEGER


          The order of the matrix.  N >= 0.

D

          D is DOUBLE PRECISION array, dimension (N)


          On entry, the n diagonal elements of the tridiagonal


          matrix.


          On normal exit, D contains the eigenvalues, in descending


          order.

E

          E is DOUBLE PRECISION array, dimension (N-1)


          On entry, the (n-1) subdiagonal elements of the tridiagonal


          matrix.


          On exit, E has been destroyed.

Z

          Z is DOUBLE PRECISION array, dimension (LDZ, N)


          On entry, if COMPZ = 'V', the orthogonal matrix used in the


          reduction to tridiagonal form.


          On exit, if COMPZ = 'V', the orthonormal eigenvectors of the


          original symmetric matrix;


          if COMPZ = 'I', the orthonormal eigenvectors of the


          tridiagonal matrix.


          If INFO > 0 on exit, Z contains the eigenvectors associated


          with only the stored eigenvalues.


          If  COMPZ = 'N', then Z is not referenced.

LDZ

          LDZ is INTEGER


          The leading dimension of the array Z.  LDZ >= 1, and if


          COMPZ = 'V' or 'I', LDZ >= max(1,N).

WORK

          WORK is DOUBLE PRECISION array, dimension (4*N)

INFO

          INFO is INTEGER


          = 0:  successful exit.


          < 0:  if INFO = -i, the i-th argument had an illegal value.


          > 0:  if INFO = i, and i is:


                <= N  the Cholesky factorization of the matrix could


                      not be performed because the leading principal


                      minor of order i was not positive.


                > N   the SVD algorithm failed to converge;


                      if INFO = N+i, i off-diagonal elements of the


                      bidiagonal factor did not converge to zero.

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

 SPTEQR computes all eigenvalues and, optionally, eigenvectors of a


 symmetric positive definite tridiagonal matrix by first factoring the


 matrix using SPTTRF, and then calling SBDSQR to compute the singular


 values of the bidiagonal factor.


 This routine computes the eigenvalues of the positive definite


 tridiagonal matrix to high relative accuracy.  This means that if the


 eigenvalues range over many orders of magnitude in size, then the


 small eigenvalues and corresponding eigenvectors will be computed


 more accurately than, for example, with the standard QR method.


 The eigenvectors of a full or band symmetric positive definite matrix


 can also be found if SSYTRD, SSPTRD, or SSBTRD has been used to


 reduce this matrix to tridiagonal form. (The reduction to tridiagonal


 form, however, may preclude the possibility of obtaining high


 relative accuracy in the small eigenvalues of the original matrix, if


 these eigenvalues range over many orders of magnitude.)

COMPZ

          COMPZ is CHARACTER*1


          = 'N':  Compute eigenvalues only.


          = 'V':  Compute eigenvectors of original symmetric


                  matrix also.  Array Z contains the orthogonal


                  matrix used to reduce the original matrix to


                  tridiagonal form.


          = 'I':  Compute eigenvectors of tridiagonal matrix also.

N

          N is INTEGER


          The order of the matrix.  N >= 0.

D

          D is REAL array, dimension (N)


          On entry, the n diagonal elements of the tridiagonal


          matrix.


          On normal exit, D contains the eigenvalues, in descending


          order.

E

          E is REAL array, dimension (N-1)


          On entry, the (n-1) subdiagonal elements of the tridiagonal


          matrix.


          On exit, E has been destroyed.

Z

          Z is REAL array, dimension (LDZ, N)


          On entry, if COMPZ = 'V', the orthogonal matrix used in the


          reduction to tridiagonal form.


          On exit, if COMPZ = 'V', the orthonormal eigenvectors of the


          original symmetric matrix;


          if COMPZ = 'I', the orthonormal eigenvectors of the


          tridiagonal matrix.


          If INFO > 0 on exit, Z contains the eigenvectors associated


          with only the stored eigenvalues.


          If  COMPZ = 'N', then Z is not referenced.

LDZ

          LDZ is INTEGER


          The leading dimension of the array Z.  LDZ >= 1, and if


          COMPZ = 'V' or 'I', LDZ >= max(1,N).

WORK

          WORK is REAL array, dimension (4*N)

INFO

          INFO is INTEGER


          = 0:  successful exit.


          < 0:  if INFO = -i, the i-th argument had an illegal value.


          > 0:  if INFO = i, and i is:


                <= N  the Cholesky factorization of the matrix could


                      not be performed because the leading principal


                      minor of order i was not positive.


                > N   the SVD algorithm failed to converge;


                      if INFO = N+i, i off-diagonal elements of the


                      bidiagonal factor did not converge to zero.

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

 ZPTEQR computes all eigenvalues and, optionally, eigenvectors of a


 symmetric positive definite tridiagonal matrix by first factoring the


 matrix using DPTTRF and then calling ZBDSQR to compute the singular


 values of the bidiagonal factor.


 This routine computes the eigenvalues of the positive definite


 tridiagonal matrix to high relative accuracy.  This means that if the


 eigenvalues range over many orders of magnitude in size, then the


 small eigenvalues and corresponding eigenvectors will be computed


 more accurately than, for example, with the standard QR method.


 The eigenvectors of a full or band positive definite Hermitian matrix


 can also be found if ZHETRD, ZHPTRD, or ZHBTRD has been used to


 reduce this matrix to tridiagonal form.  (The reduction to


 tridiagonal form, however, may preclude the possibility of obtaining


 high relative accuracy in the small eigenvalues of the original


 matrix, if these eigenvalues range over many orders of magnitude.)

COMPZ

          COMPZ is CHARACTER*1


          = 'N':  Compute eigenvalues only.


          = 'V':  Compute eigenvectors of original Hermitian


                  matrix also.  Array Z contains the unitary matrix


                  used to reduce the original matrix to tridiagonal


                  form.


          = 'I':  Compute eigenvectors of tridiagonal matrix also.

N

          N is INTEGER


          The order of the matrix.  N >= 0.

D

          D is DOUBLE PRECISION array, dimension (N)


          On entry, the n diagonal elements of the tridiagonal matrix.


          On normal exit, D contains the eigenvalues, in descending


          order.

E

          E is DOUBLE PRECISION array, dimension (N-1)


          On entry, the (n-1) subdiagonal elements of the tridiagonal


          matrix.


          On exit, E has been destroyed.

Z

          Z is COMPLEX*16 array, dimension (LDZ, N)


          On entry, if COMPZ = 'V', the unitary matrix used in the


          reduction to tridiagonal form.


          On exit, if COMPZ = 'V', the orthonormal eigenvectors of the


          original Hermitian matrix;


          if COMPZ = 'I', the orthonormal eigenvectors of the


          tridiagonal matrix.


          If INFO > 0 on exit, Z contains the eigenvectors associated


          with only the stored eigenvalues.


          If  COMPZ = 'N', then Z is not referenced.

LDZ

          LDZ is INTEGER


          The leading dimension of the array Z.  LDZ >= 1, and if


          COMPZ = 'V' or 'I', LDZ >= max(1,N).

WORK

          WORK is DOUBLE PRECISION array, dimension (4*N)

INFO

          INFO is INTEGER


          = 0:  successful exit.


          < 0:  if INFO = -i, the i-th argument had an illegal value.


          > 0:  if INFO = i, and i is:


                <= N  the Cholesky factorization of the matrix could


                      not be performed because the leading principal


                      minor of order i was not positive.


                > N   the SVD algorithm failed to converge;


                      if INFO = N+i, i off-diagonal elements of the


                      bidiagonal factor did not converge to zero.

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.