text:
MAGMA  2.3.0 Matrix Algebra for GPU and Multicore Architectures
sy/heevr: Solves using MRRR (driver)

## Functions

magma_int_t magma_cheevr (magma_vec_t jobz, magma_range_t range, magma_uplo_t uplo, magma_int_t n, magmaFloatComplex *A, magma_int_t lda, float vl, float vu, magma_int_t il, magma_int_t iu, float abstol, magma_int_t *m, float *w, magmaFloatComplex *Z, magma_int_t ldz, magma_int_t *isuppz, magmaFloatComplex *work, magma_int_t lwork, float *rwork, magma_int_t lrwork, magma_int_t *iwork, magma_int_t liwork, magma_int_t *info)
CHEEVR computes selected eigenvalues and, optionally, eigenvectors of a complex Hermitian matrix T. More...

magma_int_t magma_cheevr_gpu (magma_vec_t jobz, magma_range_t range, magma_uplo_t uplo, magma_int_t n, magmaFloatComplex_ptr dA, magma_int_t ldda, float vl, float vu, magma_int_t il, magma_int_t iu, float abstol, magma_int_t *m, float *w, magmaFloatComplex_ptr dZ, magma_int_t lddz, magma_int_t *isuppz, magmaFloatComplex *wA, magma_int_t ldwa, magmaFloatComplex *wZ, magma_int_t ldwz, magmaFloatComplex *work, magma_int_t lwork, float *rwork, magma_int_t lrwork, magma_int_t *iwork, magma_int_t liwork, magma_int_t *info)
CHEEVR computes selected eigenvalues and, optionally, eigenvectors of a complex Hermitian matrix T. More...

magma_int_t magma_zheevr (magma_vec_t jobz, magma_range_t range, magma_uplo_t uplo, magma_int_t n, magmaDoubleComplex *A, magma_int_t lda, double vl, double vu, magma_int_t il, magma_int_t iu, double abstol, magma_int_t *m, double *w, magmaDoubleComplex *Z, magma_int_t ldz, magma_int_t *isuppz, magmaDoubleComplex *work, magma_int_t lwork, double *rwork, magma_int_t lrwork, magma_int_t *iwork, magma_int_t liwork, magma_int_t *info)
ZHEEVR computes selected eigenvalues and, optionally, eigenvectors of a complex Hermitian matrix T. More...

magma_int_t magma_zheevr_gpu (magma_vec_t jobz, magma_range_t range, magma_uplo_t uplo, magma_int_t n, magmaDoubleComplex_ptr dA, magma_int_t ldda, double vl, double vu, magma_int_t il, magma_int_t iu, double abstol, magma_int_t *m, double *w, magmaDoubleComplex_ptr dZ, magma_int_t lddz, magma_int_t *isuppz, magmaDoubleComplex *wA, magma_int_t ldwa, magmaDoubleComplex *wZ, magma_int_t ldwz, magmaDoubleComplex *work, magma_int_t lwork, double *rwork, magma_int_t lrwork, magma_int_t *iwork, magma_int_t liwork, magma_int_t *info)
ZHEEVR computes selected eigenvalues and, optionally, eigenvectors of a complex Hermitian matrix T. More...

## Function Documentation

 magma_int_t magma_cheevr ( magma_vec_t jobz, magma_range_t range, magma_uplo_t uplo, magma_int_t n, magmaFloatComplex * A, magma_int_t lda, float vl, float vu, magma_int_t il, magma_int_t iu, float abstol, magma_int_t * m, float * w, magmaFloatComplex * Z, magma_int_t ldz, magma_int_t * isuppz, magmaFloatComplex * work, magma_int_t lwork, float * rwork, magma_int_t lrwork, magma_int_t * iwork, magma_int_t liwork, magma_int_t * info )

CHEEVR computes selected eigenvalues and, optionally, eigenvectors of a complex Hermitian matrix T.

Eigenvalues and eigenvectors can be selected by specifying either a range of values or a range of indices for the desired eigenvalues.

Whenever possible, CHEEVR calls CSTEGR to compute the eigenspectrum using Relatively Robust Representations. CSTEGR computes eigenvalues by the dqds algorithm, while orthogonal eigenvectors are computed from various "good" L D L^T representations (also known as Relatively Robust Representations). Gram-Schmidt orthogonalization is avoided as far as possible. More specifically, the various steps of the algorithm are as follows. For the i-th unreduced block of T,

1. Compute T - sigma_i = L_i D_i L_i^T, such that L_i D_i L_i^T is a relatively robust representation,
2. Compute the eigenvalues, lambda_j, of L_i D_i L_i^T to high relative accuracy by the dqds algorithm,
3. If there is a cluster of close eigenvalues, "choose" sigma_i close to the cluster, and go to step (a),
4. Given the approximate eigenvalue lambda_j of L_i D_i L_i^T, compute the corresponding eigenvector by forming a rank-revealing twisted factorization. The desired accuracy of the output can be specified by the input parameter ABSTOL.

For more details, see "A new O(n^2) algorithm for the symmetric tridiagonal eigenvalue/eigenvector problem", by Inderjit Dhillon, Computer Science Division Technical Report No. UCB//CSD-97-971, UC Berkeley, May 1997.

Note 1 : CHEEVR calls CSTEGR when the full spectrum is requested on machines which conform to the ieee-754 floating point standard. CHEEVR calls SSTEBZ and CSTEIN on non-ieee machines and when partial spectrum requests are made.

Normal execution of CSTEGR may create NaNs and infinities and hence may abort due to a floating point exception in environments which do not handle NaNs and infinities in the ieee standard default manner.

Parameters
 [in] jobz magma_vec_t = MagmaNoVec: Compute eigenvalues only; = MagmaVec: Compute eigenvalues and eigenvectors. [in] range magma_range_t = MagmaRangeAll: all eigenvalues will be found. = MagmaRangeV: all eigenvalues in the half-open interval (VL,VU] will be found. = MagmaRangeI: the IL-th through IU-th eigenvalues will be found. [in] uplo magma_uplo_t = MagmaUpper: Upper triangle of A is stored; = MagmaLower: Lower triangle of A is stored. [in] n INTEGER The order of the matrix A. N >= 0. [in,out] A COMPLEX array, dimension (LDA, N) On entry, the Hermitian matrix A. If UPLO = MagmaUpper, the leading N-by-N upper triangular part of A contains the upper triangular part of the matrix A. If UPLO = MagmaLower, the leading N-by-N lower triangular part of A contains the lower triangular part of the matrix A. On exit, the lower triangle (if UPLO=MagmaLower) or the upper triangle (if UPLO=MagmaUpper) of A, including the diagonal, is destroyed. [in] lda INTEGER The leading dimension of the array A. LDA >= max(1,N). [in] vl REAL [in] vu REAL If RANGE=MagmaRangeV, the lower and upper bounds of the interval to be searched for eigenvalues. VL < VU. Not referenced if RANGE = MagmaRangeAll or MagmaRangeI. [in] il INTEGER [in] iu INTEGER If RANGE=MagmaRangeI, the indices (in ascending order) of the smallest and largest eigenvalues to be returned. 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0. Not referenced if RANGE = MagmaRangeAll or MagmaRangeV. [in] abstol REAL The absolute error tolerance for the eigenvalues. An approximate eigenvalue is accepted as converged when it is determined to lie in an interval [a,b] of width less than or equal to  ABSTOL + EPS * max( |a|,|b| ),  where EPS is the machine precision. If ABSTOL is less than or equal to zero, then EPS*|T| will be used in its place, where |T| is the 1-norm of the tridiagonal matrix obtained by reducing A to tridiagonal form. See "Computing Small Singular Values of Bidiagonal Matrices with Guaranteed High Relative Accuracy," by Demmel and Kahan, LAPACK Working Note #3. If high relative accuracy is important, set ABSTOL to SLAMCH( 'Safe minimum' ). Doing so will guarantee that eigenvalues are computed to high relative accuracy when possible in future releases. The current code does not make any guarantees about high relative accuracy, but furutre releases will. See J. Barlow and J. Demmel, "Computing Accurate Eigensystems of Scaled Diagonally Dominant Matrices", LAPACK Working Note #7, for a discussion of which matrices define their eigenvalues to high relative accuracy. [out] m INTEGER The total number of eigenvalues found. 0 <= M <= N. If RANGE = MagmaRangeAll, M = N, and if RANGE = MagmaRangeI, M = IU-IL+1. [out] w REAL array, dimension (N) The first M elements contain the selected eigenvalues in ascending order. [out] Z COMPLEX array, dimension (LDZ, max(1,M)) If JOBZ = MagmaVec, then if INFO = 0, the first M columns of Z contain the orthonormal eigenvectors of the matrix A corresponding to the selected eigenvalues, with the i-th column of Z holding the eigenvector associated with W(i). If JOBZ = MagmaNoVec, then Z is not referenced. Note: the user must ensure that at least max(1,M) columns are supplied in the array Z; if RANGE = MagmaRangeV, the exact value of M is not known in advance and an upper bound must be used. [in] ldz INTEGER The leading dimension of the array Z. LDZ >= 1, and if JOBZ = MagmaVec, LDZ >= max(1,N). [out] isuppz INTEGER ARRAY, dimension ( 2*max(1,M) ) The support of the eigenvectors in Z, i.e., the indices indicating the nonzero elements in Z. The i-th eigenvector is nonzero only in elements ISUPPZ( 2*i-1 ) through ISUPPZ( 2*i ). Implemented only for RANGE = MagmaRangeAll or MagmaRangeI and IU - IL = N - 1 [out] work (workspace) COMPLEX array, dimension (LWORK) On exit, if INFO = 0, WORK[0] returns the optimal LWORK. [in] lwork INTEGER The length of the array WORK. LWORK >= max(1,2*N). For optimal efficiency, LWORK >= (NB+1)*N, where NB is the max of the blocksize for CHETRD and for CUNMTR as returned by ILAENV. If LWORK = -1, then a workspace query is assumed; the routine only calculates the optimal size of the WORK array, returns this value as the first entry of the WORK array, and no error message related to LWORK is issued by XERBLA. [out] rwork (workspace) REAL array, dimension (LRWORK) On exit, if INFO = 0, RWORK[0] returns the optimal (and minimal) LRWORK. [in] lrwork INTEGER The length of the array RWORK. LRWORK >= max(1,24*N). If LRWORK = -1, then a workspace query is assumed; the routine only calculates the optimal size of the RWORK array, returns this value as the first entry of the RWORK array, and no error message related to LRWORK is issued by XERBLA. [out] iwork (workspace) INTEGER array, dimension (LIWORK) On exit, if INFO = 0, IWORK[0] returns the optimal (and minimal) LIWORK. [in] liwork INTEGER The dimension of the array IWORK. LIWORK >= max(1,10*N). If LIWORK = -1, then a workspace query is assumed; the routine only calculates the optimal size of the IWORK array, returns this value as the first entry of the IWORK array, and no error message related to LIWORK is issued by XERBLA. [out] info INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value > 0: Internal error

## Further Details

Based on contributions by Inderjit Dhillon, IBM Almaden, USA Osni Marques, LBNL/NERSC, USA Ken Stanley, Computer Science Division, University of California at Berkeley, USA

 magma_int_t magma_cheevr_gpu ( magma_vec_t jobz, magma_range_t range, magma_uplo_t uplo, magma_int_t n, magmaFloatComplex_ptr dA, magma_int_t ldda, float vl, float vu, magma_int_t il, magma_int_t iu, float abstol, magma_int_t * m, float * w, magmaFloatComplex_ptr dZ, magma_int_t lddz, magma_int_t * isuppz, magmaFloatComplex * wA, magma_int_t ldwa, magmaFloatComplex * wZ, magma_int_t ldwz, magmaFloatComplex * work, magma_int_t lwork, float * rwork, magma_int_t lrwork, magma_int_t * iwork, magma_int_t liwork, magma_int_t * info )

CHEEVR computes selected eigenvalues and, optionally, eigenvectors of a complex Hermitian matrix T.

Eigenvalues and eigenvectors can be selected by specifying either a range of values or a range of indices for the desired eigenvalues.

Whenever possible, CHEEVR calls CSTEGR to compute the eigenspectrum using Relatively Robust Representations. CSTEGR computes eigenvalues by the dqds algorithm, while orthogonal eigenvectors are computed from various "good" L D L^T representations (also known as Relatively Robust Representations). Gram-Schmidt orthogonalization is avoided as far as possible. More specifically, the various steps of the algorithm are as follows. For the i-th unreduced block of T,

1. Compute T - sigma_i = L_i D_i L_i^T, such that L_i D_i L_i^T is a relatively robust representation,
2. Compute the eigenvalues, lambda_j, of L_i D_i L_i^T to high relative accuracy by the dqds algorithm,
3. If there is a cluster of close eigenvalues, "choose" sigma_i close to the cluster, and go to step (a),
4. Given the approximate eigenvalue lambda_j of L_i D_i L_i^T, compute the corresponding eigenvector by forming a rank-revealing twisted factorization. The desired accuracy of the output can be specified by the input parameter ABSTOL.

For more details, see "A new O(n^2) algorithm for the symmetric tridiagonal eigenvalue/eigenvector problem", by Inderjit Dhillon, Computer Science Division Technical Report No. UCB//CSD-97-971, UC Berkeley, May 1997.

Note 1 : CHEEVR calls CSTEGR when the full spectrum is requested on machines which conform to the ieee-754 floating point standard. CHEEVR calls SSTEBZ and CSTEIN on non-ieee machines and when partial spectrum requests are made.

Normal execution of CSTEGR may create NaNs and infinities and hence may abort due to a floating point exception in environments which do not handle NaNs and infinities in the ieee standard default manner.

Parameters
 [in] jobz magma_vec_t = MagmaNoVec: Compute eigenvalues only; = MagmaVec: Compute eigenvalues and eigenvectors. [in] range magma_range_t = MagmaRangeAll: all eigenvalues will be found. = MagmaRangeV: all eigenvalues in the half-open interval (VL,VU] will be found. = MagmaRangeI: the IL-th through IU-th eigenvalues will be found. [in] uplo magma_uplo_t = MagmaUpper: Upper triangle of A is stored; = MagmaLower: Lower triangle of A is stored. [in] n INTEGER The order of the matrix A. N >= 0. [in,out] dA COMPLEX array, dimension (LDDA, N) On entry, the Hermitian matrix A. If UPLO = MagmaUpper, the leading N-by-N upper triangular part of A contains the upper triangular part of the matrix A. If UPLO = MagmaLower, the leading N-by-N lower triangular part of A contains the lower triangular part of the matrix A. On exit, DA is destroyed. [in] ldda INTEGER The leading dimension of the array A. LDDA >= max(1,N). [in] vl REAL [in] vu REAL If RANGE=MagmaRangeV, the lower and upper bounds of the interval to be searched for eigenvalues. VL < VU. Not referenced if RANGE = MagmaRangeAll or MagmaRangeI. [in] il INTEGER [in] iu INTEGER If RANGE=MagmaRangeI, the indices (in ascending order) of the smallest and largest eigenvalues to be returned. 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0. Not referenced if RANGE = MagmaRangeAll or MagmaRangeV. [in] abstol REAL The absolute error tolerance for the eigenvalues. An approximate eigenvalue is accepted as converged when it is determined to lie in an interval [a,b] of width less than or equal to  ABSTOL + EPS * max( |a|,|b| ),  where EPS is the machine precision. If ABSTOL is less than or equal to zero, then EPS*|T| will be used in its place, where |T| is the 1-norm of the tridiagonal matrix obtained by reducing A to tridiagonal form. See "Computing Small Singular Values of Bidiagonal Matrices with Guaranteed High Relative Accuracy," by Demmel and Kahan, LAPACK Working Note #3. If high relative accuracy is important, set ABSTOL to SLAMCH( 'Safe minimum' ). Doing so will guarantee that eigenvalues are computed to high relative accuracy when possible in future releases. The current code does not make any guarantees about high relative accuracy, but furutre releases will. See J. Barlow and J. Demmel, "Computing Accurate Eigensystems of Scaled Diagonally Dominant Matrices", LAPACK Working Note #7, for a discussion of which matrices define their eigenvalues to high relative accuracy. [out] m INTEGER The total number of eigenvalues found. 0 <= M <= N. If RANGE = MagmaRangeAll, M = N, and if RANGE = MagmaRangeI, M = IU-IL+1. [out] w REAL array, dimension (N) The first M elements contain the selected eigenvalues in ascending order. [out] dZ COMPLEX array, dimension (LDDZ, max(1,M)) If JOBZ = MagmaVec, then if INFO = 0, the first M columns of Z contain the orthonormal eigenvectors of the matrix A corresponding to the selected eigenvalues, with the i-th column of Z holding the eigenvector associated with W(i). If JOBZ = MagmaNoVec, then Z is not referenced. Note: the user must ensure that at least max(1,M) columns are supplied in the array Z; if RANGE = MagmaRangeV, the exact value of M is not known in advance and an upper bound must be used. (workspace) If FAST_HEMV is defined DZ should be (LDDZ, max(1,N)) in both cases. [in] lddz INTEGER The leading dimension of the array Z. LDDZ >= 1, and if JOBZ = MagmaVec, LDDZ >= max(1,N). [out] isuppz INTEGER ARRAY, dimension ( 2*max(1,M) ) The support of the eigenvectors in Z, i.e., the indices indicating the nonzero elements in Z. The i-th eigenvector is nonzero only in elements ISUPPZ( 2*i-1 ) through ISUPPZ( 2*i ). Implemented only for RANGE = MagmaRangeAll or MagmaRangeI and IU - IL = N - 1 wA (workspace) COMPLEX array, dimension (LDWA, N) [in] ldwa INTEGER The leading dimension of the array wA. LDWA >= max(1,N). wZ (workspace) COMPLEX array, dimension (LDWZ, max(1,M)) [in] ldwz INTEGER The leading dimension of the array wZ. LDWZ >= 1, and if JOBZ = MagmaVec, LDWZ >= max(1,N). [out] work (workspace) COMPLEX array, dimension (LWORK) On exit, if INFO = 0, WORK[0] returns the optimal LWORK. [in] lwork INTEGER The length of the array WORK. LWORK >= (NB+1)*N, where NB is the max of the blocksize for CHETRD If LWORK = -1, then a workspace query is assumed; the routine only calculates the optimal size of the WORK array, returns this value as the first entry of the WORK array, and no error message related to LWORK is issued by XERBLA. [out] rwork (workspace) REAL array, dimension (LRWORK) On exit, if INFO = 0, RWORK[0] returns the optimal (and minimal) LRWORK. [in] lrwork INTEGER The length of the array RWORK. LRWORK >= max(1,24*N). If LRWORK = -1, then a workspace query is assumed; the routine only calculates the optimal size of the RWORK array, returns this value as the first entry of the RWORK array, and no error message related to LRWORK is issued by XERBLA. [out] iwork (workspace) INTEGER array, dimension (LIWORK) On exit, if INFO = 0, IWORK[0] returns the optimal (and minimal) LIWORK. [in] liwork INTEGER The dimension of the array IWORK. LIWORK >= max(1,10*N). If LIWORK = -1, then a workspace query is assumed; the routine only calculates the optimal size of the IWORK array, returns this value as the first entry of the IWORK array, and no error message related to LIWORK is issued by XERBLA. [out] info INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value > 0: Internal error

## Further Details

Based on contributions by Inderjit Dhillon, IBM Almaden, USA Osni Marques, LBNL/NERSC, USA Ken Stanley, Computer Science Division, University of California at Berkeley, USA

 magma_int_t magma_zheevr ( magma_vec_t jobz, magma_range_t range, magma_uplo_t uplo, magma_int_t n, magmaDoubleComplex * A, magma_int_t lda, double vl, double vu, magma_int_t il, magma_int_t iu, double abstol, magma_int_t * m, double * w, magmaDoubleComplex * Z, magma_int_t ldz, magma_int_t * isuppz, magmaDoubleComplex * work, magma_int_t lwork, double * rwork, magma_int_t lrwork, magma_int_t * iwork, magma_int_t liwork, magma_int_t * info )

ZHEEVR computes selected eigenvalues and, optionally, eigenvectors of a complex Hermitian matrix T.

Eigenvalues and eigenvectors can be selected by specifying either a range of values or a range of indices for the desired eigenvalues.

Whenever possible, ZHEEVR calls ZSTEGR to compute the eigenspectrum using Relatively Robust Representations. ZSTEGR computes eigenvalues by the dqds algorithm, while orthogonal eigenvectors are computed from various "good" L D L^T representations (also known as Relatively Robust Representations). Gram-Schmidt orthogonalization is avoided as far as possible. More specifically, the various steps of the algorithm are as follows. For the i-th unreduced block of T,

1. Compute T - sigma_i = L_i D_i L_i^T, such that L_i D_i L_i^T is a relatively robust representation,
2. Compute the eigenvalues, lambda_j, of L_i D_i L_i^T to high relative accuracy by the dqds algorithm,
3. If there is a cluster of close eigenvalues, "choose" sigma_i close to the cluster, and go to step (a),
4. Given the approximate eigenvalue lambda_j of L_i D_i L_i^T, compute the corresponding eigenvector by forming a rank-revealing twisted factorization. The desired accuracy of the output can be specified by the input parameter ABSTOL.

For more details, see "A new O(n^2) algorithm for the symmetric tridiagonal eigenvalue/eigenvector problem", by Inderjit Dhillon, Computer Science Division Technical Report No. UCB//CSD-97-971, UC Berkeley, May 1997.

Note 1 : ZHEEVR calls ZSTEGR when the full spectrum is requested on machines which conform to the ieee-754 floating point standard. ZHEEVR calls DSTEBZ and ZSTEIN on non-ieee machines and when partial spectrum requests are made.

Normal execution of ZSTEGR may create NaNs and infinities and hence may abort due to a floating point exception in environments which do not handle NaNs and infinities in the ieee standard default manner.

Parameters
 [in] jobz magma_vec_t = MagmaNoVec: Compute eigenvalues only; = MagmaVec: Compute eigenvalues and eigenvectors. [in] range magma_range_t = MagmaRangeAll: all eigenvalues will be found. = MagmaRangeV: all eigenvalues in the half-open interval (VL,VU] will be found. = MagmaRangeI: the IL-th through IU-th eigenvalues will be found. [in] uplo magma_uplo_t = MagmaUpper: Upper triangle of A is stored; = MagmaLower: Lower triangle of A is stored. [in] n INTEGER The order of the matrix A. N >= 0. [in,out] A COMPLEX_16 array, dimension (LDA, N) On entry, the Hermitian matrix A. If UPLO = MagmaUpper, the leading N-by-N upper triangular part of A contains the upper triangular part of the matrix A. If UPLO = MagmaLower, the leading N-by-N lower triangular part of A contains the lower triangular part of the matrix A. On exit, the lower triangle (if UPLO=MagmaLower) or the upper triangle (if UPLO=MagmaUpper) of A, including the diagonal, is destroyed. [in] lda INTEGER The leading dimension of the array A. LDA >= max(1,N). [in] vl DOUBLE PRECISION [in] vu DOUBLE PRECISION If RANGE=MagmaRangeV, the lower and upper bounds of the interval to be searched for eigenvalues. VL < VU. Not referenced if RANGE = MagmaRangeAll or MagmaRangeI. [in] il INTEGER [in] iu INTEGER If RANGE=MagmaRangeI, the indices (in ascending order) of the smallest and largest eigenvalues to be returned. 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0. Not referenced if RANGE = MagmaRangeAll or MagmaRangeV. [in] abstol DOUBLE PRECISION The absolute error tolerance for the eigenvalues. An approximate eigenvalue is accepted as converged when it is determined to lie in an interval [a,b] of width less than or equal to  ABSTOL + EPS * max( |a|,|b| ),  where EPS is the machine precision. If ABSTOL is less than or equal to zero, then EPS*|T| will be used in its place, where |T| is the 1-norm of the tridiagonal matrix obtained by reducing A to tridiagonal form. See "Computing Small Singular Values of Bidiagonal Matrices with Guaranteed High Relative Accuracy," by Demmel and Kahan, LAPACK Working Note #3. If high relative accuracy is important, set ABSTOL to DLAMCH( 'Safe minimum' ). Doing so will guarantee that eigenvalues are computed to high relative accuracy when possible in future releases. The current code does not make any guarantees about high relative accuracy, but furutre releases will. See J. Barlow and J. Demmel, "Computing Accurate Eigensystems of Scaled Diagonally Dominant Matrices", LAPACK Working Note #7, for a discussion of which matrices define their eigenvalues to high relative accuracy. [out] m INTEGER The total number of eigenvalues found. 0 <= M <= N. If RANGE = MagmaRangeAll, M = N, and if RANGE = MagmaRangeI, M = IU-IL+1. [out] w DOUBLE PRECISION array, dimension (N) The first M elements contain the selected eigenvalues in ascending order. [out] Z COMPLEX_16 array, dimension (LDZ, max(1,M)) If JOBZ = MagmaVec, then if INFO = 0, the first M columns of Z contain the orthonormal eigenvectors of the matrix A corresponding to the selected eigenvalues, with the i-th column of Z holding the eigenvector associated with W(i). If JOBZ = MagmaNoVec, then Z is not referenced. Note: the user must ensure that at least max(1,M) columns are supplied in the array Z; if RANGE = MagmaRangeV, the exact value of M is not known in advance and an upper bound must be used. [in] ldz INTEGER The leading dimension of the array Z. LDZ >= 1, and if JOBZ = MagmaVec, LDZ >= max(1,N). [out] isuppz INTEGER ARRAY, dimension ( 2*max(1,M) ) The support of the eigenvectors in Z, i.e., the indices indicating the nonzero elements in Z. The i-th eigenvector is nonzero only in elements ISUPPZ( 2*i-1 ) through ISUPPZ( 2*i ). Implemented only for RANGE = MagmaRangeAll or MagmaRangeI and IU - IL = N - 1 [out] work (workspace) COMPLEX_16 array, dimension (LWORK) On exit, if INFO = 0, WORK[0] returns the optimal LWORK. [in] lwork INTEGER The length of the array WORK. LWORK >= max(1,2*N). For optimal efficiency, LWORK >= (NB+1)*N, where NB is the max of the blocksize for ZHETRD and for ZUNMTR as returned by ILAENV. If LWORK = -1, then a workspace query is assumed; the routine only calculates the optimal size of the WORK array, returns this value as the first entry of the WORK array, and no error message related to LWORK is issued by XERBLA. [out] rwork (workspace) DOUBLE PRECISION array, dimension (LRWORK) On exit, if INFO = 0, RWORK[0] returns the optimal (and minimal) LRWORK. [in] lrwork INTEGER The length of the array RWORK. LRWORK >= max(1,24*N). If LRWORK = -1, then a workspace query is assumed; the routine only calculates the optimal size of the RWORK array, returns this value as the first entry of the RWORK array, and no error message related to LRWORK is issued by XERBLA. [out] iwork (workspace) INTEGER array, dimension (LIWORK) On exit, if INFO = 0, IWORK[0] returns the optimal (and minimal) LIWORK. [in] liwork INTEGER The dimension of the array IWORK. LIWORK >= max(1,10*N). If LIWORK = -1, then a workspace query is assumed; the routine only calculates the optimal size of the IWORK array, returns this value as the first entry of the IWORK array, and no error message related to LIWORK is issued by XERBLA. [out] info INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value > 0: Internal error

## Further Details

Based on contributions by Inderjit Dhillon, IBM Almaden, USA Osni Marques, LBNL/NERSC, USA Ken Stanley, Computer Science Division, University of California at Berkeley, USA

 magma_int_t magma_zheevr_gpu ( magma_vec_t jobz, magma_range_t range, magma_uplo_t uplo, magma_int_t n, magmaDoubleComplex_ptr dA, magma_int_t ldda, double vl, double vu, magma_int_t il, magma_int_t iu, double abstol, magma_int_t * m, double * w, magmaDoubleComplex_ptr dZ, magma_int_t lddz, magma_int_t * isuppz, magmaDoubleComplex * wA, magma_int_t ldwa, magmaDoubleComplex * wZ, magma_int_t ldwz, magmaDoubleComplex * work, magma_int_t lwork, double * rwork, magma_int_t lrwork, magma_int_t * iwork, magma_int_t liwork, magma_int_t * info )

ZHEEVR computes selected eigenvalues and, optionally, eigenvectors of a complex Hermitian matrix T.

Eigenvalues and eigenvectors can be selected by specifying either a range of values or a range of indices for the desired eigenvalues.

Whenever possible, ZHEEVR calls ZSTEGR to compute the eigenspectrum using Relatively Robust Representations. ZSTEGR computes eigenvalues by the dqds algorithm, while orthogonal eigenvectors are computed from various "good" L D L^T representations (also known as Relatively Robust Representations). Gram-Schmidt orthogonalization is avoided as far as possible. More specifically, the various steps of the algorithm are as follows. For the i-th unreduced block of T,

1. Compute T - sigma_i = L_i D_i L_i^T, such that L_i D_i L_i^T is a relatively robust representation,
2. Compute the eigenvalues, lambda_j, of L_i D_i L_i^T to high relative accuracy by the dqds algorithm,
3. If there is a cluster of close eigenvalues, "choose" sigma_i close to the cluster, and go to step (a),
4. Given the approximate eigenvalue lambda_j of L_i D_i L_i^T, compute the corresponding eigenvector by forming a rank-revealing twisted factorization. The desired accuracy of the output can be specified by the input parameter ABSTOL.

For more details, see "A new O(n^2) algorithm for the symmetric tridiagonal eigenvalue/eigenvector problem", by Inderjit Dhillon, Computer Science Division Technical Report No. UCB//CSD-97-971, UC Berkeley, May 1997.

Note 1 : ZHEEVR calls ZSTEGR when the full spectrum is requested on machines which conform to the ieee-754 floating point standard. ZHEEVR calls DSTEBZ and ZSTEIN on non-ieee machines and when partial spectrum requests are made.

Normal execution of ZSTEGR may create NaNs and infinities and hence may abort due to a floating point exception in environments which do not handle NaNs and infinities in the ieee standard default manner.

Parameters
 [in] jobz magma_vec_t = MagmaNoVec: Compute eigenvalues only; = MagmaVec: Compute eigenvalues and eigenvectors. [in] range magma_range_t = MagmaRangeAll: all eigenvalues will be found. = MagmaRangeV: all eigenvalues in the half-open interval (VL,VU] will be found. = MagmaRangeI: the IL-th through IU-th eigenvalues will be found. [in] uplo magma_uplo_t = MagmaUpper: Upper triangle of A is stored; = MagmaLower: Lower triangle of A is stored. [in] n INTEGER The order of the matrix A. N >= 0. [in,out] dA COMPLEX_16 array, dimension (LDDA, N) On entry, the Hermitian matrix A. If UPLO = MagmaUpper, the leading N-by-N upper triangular part of A contains the upper triangular part of the matrix A. If UPLO = MagmaLower, the leading N-by-N lower triangular part of A contains the lower triangular part of the matrix A. On exit, DA is destroyed. [in] ldda INTEGER The leading dimension of the array A. LDDA >= max(1,N). [in] vl DOUBLE PRECISION [in] vu DOUBLE PRECISION If RANGE=MagmaRangeV, the lower and upper bounds of the interval to be searched for eigenvalues. VL < VU. Not referenced if RANGE = MagmaRangeAll or MagmaRangeI. [in] il INTEGER [in] iu INTEGER If RANGE=MagmaRangeI, the indices (in ascending order) of the smallest and largest eigenvalues to be returned. 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0. Not referenced if RANGE = MagmaRangeAll or MagmaRangeV. [in] abstol DOUBLE PRECISION The absolute error tolerance for the eigenvalues. An approximate eigenvalue is accepted as converged when it is determined to lie in an interval [a,b] of width less than or equal to  ABSTOL + EPS * max( |a|,|b| ),  where EPS is the machine precision. If ABSTOL is less than or equal to zero, then EPS*|T| will be used in its place, where |T| is the 1-norm of the tridiagonal matrix obtained by reducing A to tridiagonal form. See "Computing Small Singular Values of Bidiagonal Matrices with Guaranteed High Relative Accuracy," by Demmel and Kahan, LAPACK Working Note #3. If high relative accuracy is important, set ABSTOL to DLAMCH( 'Safe minimum' ). Doing so will guarantee that eigenvalues are computed to high relative accuracy when possible in future releases. The current code does not make any guarantees about high relative accuracy, but furutre releases will. See J. Barlow and J. Demmel, "Computing Accurate Eigensystems of Scaled Diagonally Dominant Matrices", LAPACK Working Note #7, for a discussion of which matrices define their eigenvalues to high relative accuracy. [out] m INTEGER The total number of eigenvalues found. 0 <= M <= N. If RANGE = MagmaRangeAll, M = N, and if RANGE = MagmaRangeI, M = IU-IL+1. [out] w DOUBLE PRECISION array, dimension (N) The first M elements contain the selected eigenvalues in ascending order. [out] dZ COMPLEX_16 array, dimension (LDDZ, max(1,M)) If JOBZ = MagmaVec, then if INFO = 0, the first M columns of Z contain the orthonormal eigenvectors of the matrix A corresponding to the selected eigenvalues, with the i-th column of Z holding the eigenvector associated with W(i). If JOBZ = MagmaNoVec, then Z is not referenced. Note: the user must ensure that at least max(1,M) columns are supplied in the array Z; if RANGE = MagmaRangeV, the exact value of M is not known in advance and an upper bound must be used. (workspace) If FAST_HEMV is defined DZ should be (LDDZ, max(1,N)) in both cases. [in] lddz INTEGER The leading dimension of the array Z. LDDZ >= 1, and if JOBZ = MagmaVec, LDDZ >= max(1,N). [out] isuppz INTEGER ARRAY, dimension ( 2*max(1,M) ) The support of the eigenvectors in Z, i.e., the indices indicating the nonzero elements in Z. The i-th eigenvector is nonzero only in elements ISUPPZ( 2*i-1 ) through ISUPPZ( 2*i ). Implemented only for RANGE = MagmaRangeAll or MagmaRangeI and IU - IL = N - 1 wA (workspace) COMPLEX_16 array, dimension (LDWA, N) [in] ldwa INTEGER The leading dimension of the array wA. LDWA >= max(1,N). wZ (workspace) COMPLEX_16 array, dimension (LDWZ, max(1,M)) [in] ldwz INTEGER The leading dimension of the array wZ. LDWZ >= 1, and if JOBZ = MagmaVec, LDWZ >= max(1,N). [out] work (workspace) COMPLEX_16 array, dimension (LWORK) On exit, if INFO = 0, WORK[0] returns the optimal LWORK. [in] lwork INTEGER The length of the array WORK. LWORK >= (NB+1)*N, where NB is the max of the blocksize for ZHETRD If LWORK = -1, then a workspace query is assumed; the routine only calculates the optimal size of the WORK array, returns this value as the first entry of the WORK array, and no error message related to LWORK is issued by XERBLA. [out] rwork (workspace) DOUBLE PRECISION array, dimension (LRWORK) On exit, if INFO = 0, RWORK[0] returns the optimal (and minimal) LRWORK. [in] lrwork INTEGER The length of the array RWORK. LRWORK >= max(1,24*N). If LRWORK = -1, then a workspace query is assumed; the routine only calculates the optimal size of the RWORK array, returns this value as the first entry of the RWORK array, and no error message related to LRWORK is issued by XERBLA. [out] iwork (workspace) INTEGER array, dimension (LIWORK) On exit, if INFO = 0, IWORK[0] returns the optimal (and minimal) LIWORK. [in] liwork INTEGER The dimension of the array IWORK. LIWORK >= max(1,10*N). If LIWORK = -1, then a workspace query is assumed; the routine only calculates the optimal size of the IWORK array, returns this value as the first entry of the IWORK array, and no error message related to LIWORK is issued by XERBLA. [out] info INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value > 0: Internal error

## Further Details

Based on contributions by Inderjit Dhillon, IBM Almaden, USA Osni Marques, LBNL/NERSC, USA Ken Stanley, Computer Science Division, University of California at Berkeley, USA


text:
Advanced search

Browse ORBi by ORBi project The Open Access movement
ORBi is a project of

 References of "Millan-Gabet, R"      in Complete repository Arts & humanities   Archaeology   Art & art history   Classical & oriental studies   History   Languages & linguistics   Literature   Performing arts   Philosophy & ethics   Religion & theology   Multidisciplinary, general & others Business & economic sciences   Accounting & auditing   Production, distribution & supply chain management   Finance   General management & organizational theory   Human resources management   Management information systems   Marketing   Strategy & innovation   Quantitative methods in economics & management   General economics & history of economic thought   International economics   Macroeconomics & monetary economics   Microeconomics   Economic systems & public economics   Social economics   Special economic topics (health, labor, transportation…)   Multidisciplinary, general & others Engineering, computing & technology   Aerospace & aeronautics engineering   Architecture   Chemical engineering   Civil engineering   Computer science   Electrical & electronics engineering   Energy   Geological, petroleum & mining engineering   Materials science & engineering   Mechanical engineering   Multidisciplinary, general & others Human health sciences   Alternative medicine   Anesthesia & intensive care   Cardiovascular & respiratory systems   Dentistry & oral medicine   Dermatology   Endocrinology, metabolism & nutrition   Forensic medicine   Gastroenterology & hepatology   General & internal medicine   Geriatrics   Hematology   Immunology & infectious disease   Laboratory medicine & medical technology   Neurology   Oncology   Ophthalmology   Orthopedics, rehabilitation & sports medicine   Otolaryngology   Pediatrics   Pharmacy, pharmacology & toxicology   Psychiatry   Public health, health care sciences & services   Radiology, nuclear medicine & imaging   Reproductive medicine (gynecology, andrology, obstetrics)   Rheumatology   Surgery   Urology & nephrology   Multidisciplinary, general & others Law, criminology & political science   Civil law   Criminal law & procedure   Criminology   Economic & commercial law   European & international law   Judicial law   Metalaw, Roman law, history of law & comparative law   Political science, public administration & international relations   Public law   Social law   Tax law   Multidisciplinary, general & others Life sciences   Agriculture & agronomy   Anatomy (cytology, histology, embryology...) & physiology   Animal production & animal husbandry   Aquatic sciences & oceanology   Biochemistry, biophysics & molecular biology   Biotechnology   Entomology & pest control   Environmental sciences & ecology   Food science   Genetics & genetic processes   Microbiology   Phytobiology (plant sciences, forestry, mycology...)   Veterinary medicine & animal health   Zoology   Multidisciplinary, general & others Physical, chemical, mathematical & earth Sciences   Chemistry   Earth sciences & physical geography   Mathematics   Physics   Space science, astronomy & astrophysics   Multidisciplinary, general & others Social & behavioral sciences, psychology   Animal psychology, ethology & psychobiology   Anthropology   Communication & mass media   Education & instruction   Human geography & demography   Library & information sciences   Neurosciences & behavior   Regional & inter-regional studies   Social work & social policy   Sociology & social sciences   Social, industrial & organizational psychology   Theoretical & cognitive psychology   Treatment & clinical psychology   Multidisciplinary, general & others     Showing results 1 to 20 of 30 1 2     Structure of Herbig AeBe disks at the milliarcsecond scale. A statistical survey in the H band using PIONIER-VLTILazareff, B.; Berger, J.-P.; Kluska, J. et alin Astronomy and Astrophysics (in press)Context. It is now generally accepted that the near-infrared excess of Herbig AeBe stars originates in the dust of a circumstellar disk. Aims. The aims of this article are to infer the radial and vertical ... [more ▼]Context. It is now generally accepted that the near-infrared excess of Herbig AeBe stars originates in the dust of a circumstellar disk. Aims. The aims of this article are to infer the radial and vertical structure of these disks at scales of order one au, and the properties of the dust grains. Methods. The program objects (51 in total) were observed with the H-band (1.6micron) PIONIER/VLTI interferometer. The largest baselines allowed us to resolve (at least partially) structures of a few tenths of an au at typical distances of a few hundred parsecs. Dedicated UBVRIJHK photometric measurements were also obtained. Spectral and 2D geometrical parameters are extracted via fits of a few simple models: ellipsoids and broadened rings with azimuthal modulation. Model bias is mitigated by parallel fits of physical disk models. Sample statistics were evaluated against similar statistics for the physical disk models to infer properties of the sample objects as a group. Results. We find that dust at the inner rim of the disk has a sublimation temperature Tsub~1800K. A ring morphology is confirmed for approximately half the resolved objects; these rings are wide delta_r>=0.5. A wide ring favors a rim that, on the star-facing side, looks more like a knife edge than a doughnut. The data are also compatible with a the combination of a narrow ring and an inner disk of unspecified nature inside the dust sublimation radius. The disk inner part has a thickness z/r~0.2, flaring to z/r~0.5 in the outer part. We confirm the known luminosity-radius relation; a simple physical model is consistent with both the mean luminosity-radius relation and the ring relative width; however, a significant spread around the mean relation is present. In some of the objects we find a halo component, fully resolved at the shortest interferometer spacing, that is related to the HAeBe class. [less ▲]Detailed reference viewed: 18 (4 ULg) Overview of LBTI: a multipurpose facility for high spatial resolution observationsHinz, P. M.; Defrere, Denis ; Skemer, A. et alin Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series (2016, August 01)The Large Binocular Telescope Interferometer (LBTI) is a high spatial resolution instrument developed for coherent imaging and nulling interferometry using the 14.4 m baseline of the 2×8.4 m LBT. The ... [more ▼]The Large Binocular Telescope Interferometer (LBTI) is a high spatial resolution instrument developed for coherent imaging and nulling interferometry using the 14.4 m baseline of the 2×8.4 m LBT. The unique telescope design, comprising of the dual apertures on a common elevation-azimuth mount, enables a broad use of observing modes. The full system is comprised of dual adaptive optics systems, a near-infrared phasing camera, a 1-5 μm camera (called LMIRCam), and an 8-13 μm camera (called NOMIC). The key program for LBTI is the Hunt for Observable Signatures of Terrestrial planetary Systems (HOSTS), a survey using nulling interferometry to constrain the typical brightness from exozodiacal dust around nearby stars. Additional observations focus on the detection and characterization of giant planets in the thermal infrared, high spatial resolution imaging of complex scenes such as Jupiter's moon, Io, planets forming in transition disks, and the structure of active Galactic Nuclei (AGN). Several instrumental upgrades are currently underway to improve and expand the capabilities of LBTI. These include: Improving the performance and limiting magnitude of the parallel adaptive optics systems; quadrupling the field of view of LMIRcam (increasing to 20"x20"); adding an integral field spectrometry mode; and implementing a new algorithm for path length correction that accounts for dispersion due to atmospheric water vapor. We present the current architecture and performance of LBTI, as well as an overview of the upgrades. [less ▲]Detailed reference viewed: 8 (0 ULg) Enabling the direct detection of earth-sized exoplanets with the LBTI HOSTS project: a progress reportDanchi, W.; Bailey, V.; Bryden, G. et alin Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series (2016, August 01)NASA has funded a project called the Hunt for Observable Signatures of Terrestrial Systems (HOSTS) to survey nearby solar type stars to determine the amount of warm zodiacal dust in their habitable zones ... [more ▼]NASA has funded a project called the Hunt for Observable Signatures of Terrestrial Systems (HOSTS) to survey nearby solar type stars to determine the amount of warm zodiacal dust in their habitable zones. The goal is not only to determine the luminosity distribution function but also to know which individual stars have the least amount of zodiacal dust. It is important to have this information for future missions that directly image exoplanets as this dust is the main source of astrophysical noise for them. The HOSTS project utilizes the Large Binocular Telescope Interferometer (LBTI), which consists of two 8.4-m apertures separated by a 14.4-m baseline on Mt. Graham, Arizona. The LBTI operates in a nulling mode in the mid-infrared spectral window (8-13 μm), in which light from the two telescopes is coherently combined with a 180 degree phase shift between them, producing a dark fringe at the location of the target star. In doing so the starlight is greatly reduced, increasing the contrast, analogous to a coronagraph operating at shorter wavelengths. The LBTI is a unique instrument, having only three warm reflections before the starlight reaches cold mirrors, giving it the best photometric sensitivity of any interferometer operating in the mid-infrared. It also has a superb Adaptive Optics (AO) system giving it Strehl ratios greater than 98% at 10 μm. In 2014 into early 2015 LBTI was undergoing commissioning. The HOSTS project team passed its Operational Readiness Review (ORR) in April 2015. The team recently published papers on the target sample, modeling of the nulled disk images, and initial results such as the detection of warm dust around η Corvi. Recently a paper was published on the data pipeline and on-sky performance. An additional paper is in preparation on β Leo. We will discuss the scientific and programmatic context for the LBTI project, and we will report recent progress, new results, and plans for the science verification phase that started in February 2016, and for the survey. [less ▲]Detailed reference viewed: 17 (0 ULg) Nulling Data Reduction and On-sky Performance of the Large Binocular Telescope InterferometerDefrere, Denis ; Hinz, P. M.; Mennesson, B. et alin Astrophysical Journal (2016), 824The Large Binocular Telescope Interferometer (LBTI) is a versatile instrument designed for high angular resolution and high-contrast infrared imaging (1.5-13 μm). In this paper, we focus on the mid ... [more ▼]The Large Binocular Telescope Interferometer (LBTI) is a versatile instrument designed for high angular resolution and high-contrast infrared imaging (1.5-13 μm). In this paper, we focus on the mid-infrared (8-13 μm) nulling mode and present its theory of operation, data reduction, and on-sky performance as of the end of the commissioning phase in 2015 March. With an interferometric baseline of 14.4 m, the LBTI nuller is specifically tuned to resolve the habitable zone of nearby main-sequence stars, where warm exozodiacal dust emission peaks. Measuring the exozodi luminosity function of nearby main-sequence stars is a key milestone to prepare for future exo-Earth direct imaging instruments. Thanks to recent progress in wavefront control and phase stabilization, as well as in data reduction techniques, the LBTI demonstrated in 2015 February a calibrated null accuracy of 0.05% over a 3 hr long observing sequence on the bright nearby A3V star β Leo. This is equivalent to an exozodiacal disk density of 15-30 zodi for a Sun-like star located at 10 pc, depending on the adopted disk model. This result sets a new record for high-contrast mid-infrared interferometric imaging and opens a new window on the study of planetary systems. [less ▲]Detailed reference viewed: 29 (12 ULg) Models of the η Corvi Debris Disk from the Keck Interferometer, Spitzer, and HerschelLebreton, J.; Beichman, C.; Bryden, G. et alin Astrophysical Journal (2016), 817Debris disks are signposts of analogs to small-body populations of the solar system, often, however, with much higher masses and dust production rates. The disk associated with the nearby star η Crv is ... [more ▼]Debris disks are signposts of analogs to small-body populations of the solar system, often, however, with much higher masses and dust production rates. The disk associated with the nearby star η Crv is especially striking, as it shows strong mid- and far-infrared excesses despite an age of ∼1.4 Gyr. We undertake constructing a consistent model of the system that can explain a diverse collection of spatial and spectral data. We analyze Keck Interferometer Nuller measurements and revisit Spitzer and additional spectrophotometric data, as well as resolved Herschel images, to determine the dust spatial distribution in the inner exozodi and in the outer belt. We model in detail the two-component disk and the dust properties from the sub-AU scale to the outermost regions by fitting simultaneously all measurements against a large parameter space. The properties of the cold belt are consistent with a collisional cascade in a reservoir of ice-free planetesimals at 133 AU. It shows marginal evidence for asymmetries along the major axis. KIN enables us to establish that the warm dust consists of a ring that peaks between 0.2 and 0.8 AU. To reconcile this location with the ∼400 K dust temperature, very high albedo dust must be invoked, and a distribution of forsterite grains starting from micron sizes satisfies this criterion, while providing an excellent fit to the spectrum. We discuss additional constraints from the LBTI and near-infrared spectra, and we present predictions of what James Webb Space Telescope can unveil about this unusual object and whether it can detect unseen planets. [less ▲]Detailed reference viewed: 10 (3 ULg) Simultaneous Water Vapor and Dry Air Optical Path Length Measurements and Compensation with the Large Binocular Telescope InterferometerDefrere, Denis ; Hinz, P.; Downey, E. et alin Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series (2016)The Large Binocular Telescope Interferometer uses a near-infrared camera to measure the optical path length variations between the two AO-corrected apertures and provide high-angular resolution ... [more ▼]The Large Binocular Telescope Interferometer uses a near-infrared camera to measure the optical path length variations between the two AO-corrected apertures and provide high-angular resolution observations for all its science channels (1.5-13 microns). There is however a wavelength dependent component to the atmospheric turbulence, which can introduce optical path length errors when observing at a wavelength different from that of the fringe sensing camera. Water vapor in particular is highly dispersive and its effect must be taken into account for high-precision infrared interferometric observations as described previously for VLTI/MIDI or the Keck Interferometer Nuller. In this paper, we describe the new sensing approach that has been developed at the LBT to measure and monitor the optical path length fluctuations due to dry air and water vapor separately. After reviewing the current performance of the system for dry air seeing compensation, we present simultaneous H-, K-, and N-band observations that illustrate the feasibility of our feedforward approach to stabilize the path length fluctuations seen by the LBTI nuller. [less ▲]Detailed reference viewed: 18 (6 ULg) Exoplanet science with the LBTI: instrument status and plansDefrere, Denis ; Hinz, P.; Skemer, A. et alin Shaklan, Stuart (Ed.) Techniques and Instrumentation for Detection of Exoplanets VII (2015, September 16)The Large Binocular Telescope Interferometer (LBTI) is a strategic instrument of the LBT designed for high-sensitivity, high-contrast, and high-resolution infrared (1.5-13 $\mu$m) imaging of nearby ... [more ▼]The Large Binocular Telescope Interferometer (LBTI) is a strategic instrument of the LBT designed for high-sensitivity, high-contrast, and high-resolution infrared (1.5-13 $\mu$m) imaging of nearby planetary systems. To carry out a wide range of high-spatial resolution observations, it can combine the two AO-corrected 8.4-m apertures of the LBT in various ways including direct (non-interferometric) imaging, coronagraphy (APP and AGPM), Fizeau imaging, non-redundant aperture masking, and nulling interferometry. It also has broadband, narrowband, and spectrally dispersed capabilities. In this paper, we review the performance of these modes in terms of exoplanet science capabilities and describe recent instrumental milestones such as first-light Fizeau images (with the angular resolution of an equivalent 22.8-m telescope) and deep interferometric nulling observations. [less ▲]Detailed reference viewed: 35 (10 ULg) The VLTI/PIONIER near-infrared interferometric survey of southern T Tauri stars. I. First resultsAnthonioz, F.; Menard, F.; Pinte, C. et alin Astronomy and Astrophysics (2015), 574Context. The properties of the inner disks of bright Herbig AeBe stars have been studied with near-infrared (NIR) interferometry and high resolution spectroscopy. The continuum (dust) and a few molecular ... [more ▼]Context. The properties of the inner disks of bright Herbig AeBe stars have been studied with near-infrared (NIR) interferometry and high resolution spectroscopy. The continuum (dust) and a few molecular gas species have been studied close to the central star; however, sensitivity problems limit direct information about the inner disks of the fainter T Tauri stars.
Aims: Our aim is to measure some of the properties (inner radius, brightness profile, shape) of the inner regions of circumstellar disk surrounding southern T Tauri stars.
Methods: We performed a survey with the VLTI/PIONIER recombiner instrument at H-band of 21 T Tauri stars. The baselines used ranged from 11 m to 129 m, corresponding to a maximum resolution of ~3 mas (~0.45 au at 150 pc).
Results: Thirteen disks are resolved well and the visibility curves are fully sampled as a function of baseline in the range 45-130 m for these 13 objects. A simple qualitative examination of visibility profiles allows us to identify a rapid drop-off in the visibilities at short baselines(<10 Mlambda) in 8 resolved disks. This is indicative of a significant contribution from an extended (R> 3 au, at 150 pc) contribution of light from the disk. We demonstrate that this component is compatible with scattered light, providing strong support to an earlier prediction. The amplitude of the drop-off and the amount of dust thermal emission changes from source to source suggesting that each disk is different. A by-product of the survey is the identification of a new milli-arcsec separation binary: WW Cha. Spectroscopic and interferometric data of AK Sco have also been fitted with a binary + disk model.
Conclusions: The visibility data are reproduced well when thermal emission and scattering from dust are fully considered. The inner radii measured are consistent with the expected dust sublimation radii. The modelling of AK Sco suggests a likely coplanarity between the disk and the binary's orbital plane. [less ▲]Detailed reference viewed: 26 (3 ULg) First-light LBT Nulling Interferometric Observations: Warm Exozodiacal Dust Resolved within a Few AU of eta CrvDefrere, Denis ; Hinz, P. M.; Skemer, A. J. et alin Astrophysical Journal (2015), 799We report on the first nulling interferometric observations with the Large Binocular Telescope Interferometer (LBTI), resolving the N' band (9.81-12.41 μm) emission around the nearby main-sequence star η ... [more ▼]We report on the first nulling interferometric observations with the Large Binocular Telescope Interferometer (LBTI), resolving the N' band (9.81-12.41 μm) emission around the nearby main-sequence star η Crv (F2V, 1-2 Gyr). The measured source null depth amounts to 4.40% ± 0.35% over a field-of-view of 140 mas in radius (~2.6 AU for the distance of η Crv) and shows no significant variation over 35° of sky rotation. This relatively low null is unexpected given the total disk to star flux ratio measured by the Spitzer Infrared Spectrograph (IRS; ~23% across the N' band), suggesting that a significant fraction of the dust lies within the central nulled response of the LBTI (79 mas or 1.4 AU). Modeling of the warm disk shows that it cannot resemble a scaled version of the solar zodiacal cloud unless it is almost perpendicular to the outer disk imaged by Herschel. It is more likely that the inner and outer disks are coplanar and the warm dust is located at a distance of 0.5-1.0 AU, significantly closer than previously predicted by models of the IRS spectrum (~3 AU). The predicted disk sizes can be reconciled if the warm disk is not centrosymmetric, or if the dust particles are dominated by very small grains. Both possibilities hint that a recent collision has produced much of the dust. Finally, we discuss the implications for the presence of dust for the distance where the insolation is the same as Earth's (2.3 AU). [less ▲]Detailed reference viewed: 29 (9 ULg) Constraining the Exozodiacal Luminosity Function of Main-sequence Stars: Complete Results from the Keck Nuller Mid-infrared SurveysMennesson, B.; Millan-Gabet, R.; Serabyn, E. et alin Astrophysical Journal (2014), 797Forty-seven nearby main-sequence stars were surveyed with the Keck Interferometer mid-infrared Nulling instrument (KIN) between 2008 and 2011, searching for faint resolved emission from exozodiacal dust ... [more ▼]Forty-seven nearby main-sequence stars were surveyed with the Keck Interferometer mid-infrared Nulling instrument (KIN) between 2008 and 2011, searching for faint resolved emission from exozodiacal dust. Observations of a subset of the sample have already been reported, focusing essentially on stars with no previously known dust. Here we extend this previous analysis to the whole KIN sample, including 22 more stars with known near- and/or far-infrared excesses. In addition to an analysis similar to that of the first paper of this series, which was restricted to the 8-9 mum spectral region, we present measurements obtained in all 10 spectral channels covering the 8-13 mum instrumental bandwidth. Based on the 8-9 mum data alone, which provide the highest signal-to-noise measurements, only one star shows a large excess imputable to dust emission (eta Crv), while four more show a significant (>3sigma) excess: beta Leo, beta UMa, zeta Lep, and gamma Oph. Overall, excesses detected by KIN are more frequent around A-type stars than later spectral types. A statistical analysis of the measurements further indicates that stars with known far-infrared (lambda >= 70 mum) excesses have higher exozodiacal emission levels than stars with no previous indication of a cold outer disk. This statistical trend is observed regardless of spectral type and points to a dynamical connection between the inner (zodi-like) and outer (Kuiper-Belt-like) dust populations. The measured levels for such stars are clustering close to the KIN detection limit of a few hundred zodis and are indeed consistent with those expected from a populat [less ▲]Detailed reference viewed: 23 (8 ULg) Co-phasing the Large Binocular Telescope: status and performance of LBTI/PHASECamDefrere, Denis ; Hinz, P.; Downey, E. et alin Optical and Infrared Interferometry IV (2014, July 01)The Large Binocular Telescope Interferometer is a NASA-funded nulling and imaging instrument designed to coherently combine the two 8.4-m primary mirrors of the LBT for high-sensitivity, high-contrast ... [more ▼]The Large Binocular Telescope Interferometer is a NASA-funded nulling and imaging instrument designed to coherently combine the two 8.4-m primary mirrors of the LBT for high-sensitivity, high-contrast, and highresolution infrared imaging (1.5-13 μm). PHASECam is LBTI's near-infrared camera used to measure tip-tilt and phase variations between the two AO-corrected apertures and provide high-angular resolution observations. We report on the status of the system and describe its on-sky performance measured during the first semester of 2014. With a spatial resolution equivalent to that of a 22.8-meter telescope and the light-gathering power of single 11.8-meter mirror, the co-phased LBT can be considered to be a forerunner of the next-generation extremely large telescopes (ELT). [less ▲]Detailed reference viewed: 34 (11 ULg) The LBTI hunt for observable signatures of terrestrial systems (HOSTS) survey: a key NASA science program on the road to exoplanet imaging missionsDanchi, W.; Bailey, V.; Bryden, G. et alin Optical and Infrared Interferometry IV (2014, July 01)The Hunt for Observable Signatures of Terrestrial planetary Systems (HOSTS) program on the Large Binocular Telescope Interferometer (LBTI) will survey nearby stars for faint exozodiacal dust (exozodi ... [more ▼]The Hunt for Observable Signatures of Terrestrial planetary Systems (HOSTS) program on the Large Binocular Telescope Interferometer (LBTI) will survey nearby stars for faint exozodiacal dust (exozodi). This warm circumstellar dust, analogous to the interplanetary dust found in the vicinity of the Earth in our own system, is produced in comet breakups and asteroid collisions. Emission and/or scattered light from the exozodi will be the major source of astrophysical noise for a future space telescope aimed at direct imaging and spectroscopy of terrestrial planets (exo- Earths) around nearby stars. About 20% of nearby field stars have cold dust coming from planetesimals at large distances from the stars (Eiroa et al. 2013, A&A, 555, A11; Siercho et al. 2014, ApJ, 785, 33). Much less is known about exozodi; current detection limits for individual stars are at best ~ 500 times our solar system's level (aka. 500 zodi). LBTI-HOSTS will be the first survey capable of measuring exozodi at the 10 zodi level (3σ). Detections of warm dust will also reveal new information about planetary system architectures and evolution. We will describe the motivation for the survey and progress on target selection, not only the actual stars likely to be observed by such a mission but also those whose observation will enable sensible extrapolations for stars that will not be observed with LBTI. We briefly describe the detection of the debris disk around η Crv, which is the first scientific result from the LBTI coming from the commissioning of the instrument in December 2013, shortly after the first time the fringes were stabilized. [less ▲]Detailed reference viewed: 11 (1 ULg) Update on the LBTI: a versatile high-contrast and high-resolution infrared imager for a 23-m telescopeDefrere, Denis ; Hinz, P.; Skemer, A. et alin Improving the Performances of Current Optical Interferometers & Future Designs (2014, April 01)The Large Binocular Telescope Interferometer (LBTI) is a strategic instrument of the LBT designed for high-sensitivity, high-contrast, and high-resolution infrared imaging of nearby nearby planetary ... [more ▼]The Large Binocular Telescope Interferometer (LBTI) is a strategic instrument of the LBT designed for high-sensitivity, high-contrast, and high-resolution infrared imaging of nearby nearby planetary systems. It combines the two 8.4-m apertures of the LBT in various ways including direct (non-interferometric) or Fizeau imaging, non-redundant aperture masking, and nulling interferometry. First fringe-tracked observations were obtained in December 2013. We report in this paper on the status and recent progress of the LBTI with a particular focus on interferometry. [less ▲]Detailed reference viewed: 11 (1 ULg) PIONIER : A Four Telescope VLTI InstrumentLazareff, B.; Le Bouquin, J.-L.; Berger, J.-P. et alin Surdej, Jean; Le Coroller, Hervé; Arnold, Luc (Eds.) Improving the Performances of Current Optical Interferometers & Future Designs (2014, April 01)Pionier is a guest instrument, the first four-telescope recombiner at ESO's VLTI. We discuss salient design features and illustrate selected scientific results from the fiDetailed reference viewed: 14 (1 ULg) Modeling the η Corvi debris disk from the sub-AU scale to its outermost regionsLebreton, J.; Beichman, C. A.; Bryden, G. et alPoster (2014, March 01)Dusty debris disks surrounding main sequence stars are thought to be analogues to thepopulations of small bodies of the Solar System (asteroids, comets/icy bodies and dust grains), however with often much ... [more ▼]Dusty debris disks surrounding main sequence stars are thought to be analogues to thepopulations of small bodies of the Solar System (asteroids, comets/icy bodies and dust grains), however with often much higher masses and associated dust production rates. Mecanisms such as massive collisions or LHB-like events must therefore be invoked to justify their existence. This is especially striking for the nearby F2V star η Corvi that shows a very strong mid- and far-infrared excess despite an estimated age of ~1.4 Gyr (Lisse et al. 2012, Wyatt et al. 2005). We present new observations of the η Crv debris disk obtained in the far-infrared with Herschel/PACS and SPIRE and in the mid-infrared with the Keck Interferometer Nuller (Millan-Gabet et al. 2011). The Herschel/PACS images at 70, 100 and 160 μm reveal a well resolved belt of cold material at ~130 AU, as well as an unresolved component in the innermost parts of the system. This warmer counterpart is resolved in the mid-infrared as a strong null excess originating from within the ~2x4 AU field-of-view of the interferometer, which is reminiscent of the architecture of the Fomalhaut debris disk (Mennesson et al. 2012, Lebreton et al. 2013). The signature of warm silicate dust is also very clear in Spitzer/IRS high-resolution spectra (Chen et al. 2006) at intermediate wavelengths (10-35 μm). We undertake to establish a consistent model of the debris disk from the sub-AU scale to its outermost regions using the GRaTer radiative transfer code (Augereau et al. 1999a, Lebreton et al. 2013) by adjusting simultaneously the interferometric nulls, the resolved Herschel images and the spectro-photometric data against a large parameter space. Our analysis providesaccurate estimates of the fundamental parameters of the disk: its surface density profile, grain size distribution and mass, making it possible to unveil the origin of the dust and the relation between the cold (~50 K) Kuiper-like belt and the warm (~500 K) exo-zodiacal disk. We further discuss the possible existence of an additional dust population at intermediate temperatures and its nature. η Corvi will be of prime interest for future observations with the JWST. We finally make predictions of the ability of NIRCam and MIRI to image details in the disk at high contrast with both spatial and spectral resolution in order to obtain a better view of this complex planetary system. [less ▲]Detailed reference viewed: 9 (0 ULg) The Hunt for Observable Signatures of Terrestrial Planetary Systems (HOSTS)Defrere, Denis ; Hinz, P.; Bryden, G. et alConference (2014, March)The presence of large amounts of exozodiacal dust around nearby main sequence stars is considered as a potential threat for the direct imaging of Earth-like exoplanets and, hence, the search for ... [more ▼]The presence of large amounts of exozodiacal dust around nearby main sequence stars is considered as a potential threat for the direct imaging of Earth-like exoplanets and, hence, the search for biosignatures (Roberge et al. 2012). However, it is also considered as a signpost for the presence of terrestrial planets that might be hidden in the dust disk (Stark and Kuchner 2008). Characterizing exozodiacal dust around nearby sequence stars is therefore a crucial step toward one of the main goals of modern astronomy: finding extraterrestrial life. After briefly reviewing the latest results in this field, we present the exozodiacal dust survey on the Large Binocular Telescope Interferometer (LBTI). The survey is called HOSTS and is specifically designed to determine the prevalence and brightness of exozodiacal dust disks with the sensitivity required to prepare for future New Worlds Missions that will image Earth-like exoplanets. To achieve this objective, the LBTI science team has carefully established a balanced list of 50 nearby main-sequence stars that are likely candidates of these missions and/or can be observed with the best instrument performance (see companion abstract by Roberge et al.). Exozodiacal dust disk candidates detected by the Keck Interferometer Nuller will also be observed. The first results of the survey will be presented. To precisely detect exozodiacal dust, the LBTI combines the two 8-m primary mirrors of the LBT using N-band nulling interferometry. Interferometric combination provides the required angular resolution (70-90 mas) to resolve the habitable zone of nearby main sequence stars while nulling is used to subtract the stellar light and reach the required contrast of a few 10-4. A Kband fringe tracker ensures the stability of the null. The current performance of the instrument and the first nulling measurements will be presented. [less ▲]Detailed reference viewed: 16 (2 ULg) Searching for Faint Exozodiacal Disks: Keck Results and LBTI StatusDefrere, Denis ; Hinz, P.; Mennesson, B. et alin Exploring the Formation and Evolution of Planetary Systems (2014, January 01)The possible presence of dust in the habitable zone around nearby main-sequence stars is considered as a major hurdle toward the direct imaging of Earth-like extrasolar planets with future dedicated space ... [more ▼]The possible presence of dust in the habitable zone around nearby main-sequence stars is considered as a major hurdle toward the direct imaging of Earth-like extrasolar planets with future dedicated space-based telescopes (e.g., Roberge et al. 2012). In this context, NASA has funded two ground-based mid-infrared nulling interferometers to combine the large apertures available at the Keck Observatory and the Large Binocular Telescope (LBT). In this poster, we present the preliminary results of the extended survey carried out with the Keck Interferometer Nuller (KIN) between 2008 and 2011 and describe the forthcoming LBTI survey. [less ▲]Detailed reference viewed: 7 (1 ULg) Target Selection for the LBTI Hunt for Observable Signatures of Terrestrial Planetary SystemsWeinberger, Alycia J.; Roberge, A.; Kennedy, G. et alin American Astronomical Society Meeting Abstracts #223 (2014, January 01)The Hunt for Observable Signatures of Terrestrial planetary Systems (HOSTS) on the Large Binocular Telescope Interferometer (LBTI) will survey nearby stars for faint exozodiacal dust (exozodi). About 20 ... [more ▼]The Hunt for Observable Signatures of Terrestrial planetary Systems (HOSTS) on the Large Binocular Telescope Interferometer (LBTI) will survey nearby stars for faint exozodiacal dust (exozodi). About 20% of field stars have cold debris disks created by the collisions and evaporation of planetesimals. Much less is known about warm circumstellar dust, such as that found in the vicinity of the Earth in our own system. This dust is generated in asteroidal collisions and cometary breakups, and current detection limits are at best ~500 times our system's level, i.e. 500 zodi. LBTI-HOSTS will be the first survey capable of measuring exozodi at the 10 zodi level (3σ). Exozodi of this brightness would be the major source of astrophysical noise for a future space telescope aimed at direct imaging and spectroscopy of habitable zone terrestrial planets. Detections of warm dust will also reveal new information about planetary system architectures and evolution. We describe the target star selection by the LBTI Science Team to satisfy the goals of the HOSTS survey -- to fully inform target selection for a future exoEarth mission. We are interested in actual stars likely to be observed by a mission and stars whose observation will enable sensible extrapolations to those stars that cannot be observed. We integrated two approaches to generate the HOSTS target list. The mission-driven approach concentrates on F, G, and K-type stars that are the best targets for future direct observations of exoEarths, thereby providing model-independent “ground truth” dust observations. However, not every potential target of a future exoEarth mission can be observed with LBTI. The sensitivity-driven approach selects targets based only on what exozodi sensitivity could be achieved, without consideration of exoEarth mission constraints. This naturally selects more luminous stars (A and early F-type stars). In both cases, all stars are close enough to Earth such that their habitable zones are resolvable by LBTI and bright enough at N-band (10 μm) to provide excellent sensitivity. We also discuss observational and astrophysical motivations for excluding binaries of certain separations. [less ▲]Detailed reference viewed: 6 (1 ULg) Exozodi disk models for the HOSTS survey on the LBTIWyatt, Mark; Kennedy, G.; Skemer, A. et alin American Astronomical Society Meeting Abstracts #223 (2014, January 01)This poster describes a simple model for exozodiacal emission that was developed to interpret observations of the Hunt for Observable Signatures of Terrestrial planetary Systems (HOSTS) project on the ... [more ▼]This poster describes a simple model for exozodiacal emission that was developed to interpret observations of the Hunt for Observable Signatures of Terrestrial planetary Systems (HOSTS) project on the Large Binocular Telescope Interferometer (LBTI). HOSTS is a NASA-funded key science project using mid-infrared nulling interferometry at the LBTI to seach for faint exozodiacal dust (exozodi) in the habitable zones of nearby stars. The aim was to make a model that includes the fewest possible assumptions, so that it is easy to characterize how choices of model parameters affect what can be inferred from the observations. However the model is also sufficiently complex that it can be compared in a physically meaningful way with the level of dust in the Solar System, and can also be readily used to assess the impact of a detection (or of a non-detection) on the ability of a mission to detect Earth-like planets. Here we describe the model, and apply it to the sample of stars being searched by HOSTS to determine the zodi level (i.e., the number of Solar System zodiacal clouds) that would be needed for a detection for each star in the survey. Particular emphasis is given to our definition of a zodi, and what that means for stars of different luminosity, and a comparison is given between different zodi definitions justifying our final choice. The achievable exozodi levels range from 1-20 zodi for different stars in the prime sample for a 0.01% null depth, with a median level of 2.5 zodi. [less ▲]Detailed reference viewed: 12 (3 ULg) The VLTi/PIONIER survey of southern TTauri disksAnthonioz, F.; Ménard, F.; Pinte, C. et alin Booth, Mark; Matthews, Brenda; Graham, James (Eds.) Exploring the Formation and Evolution of Planetary Systems (2014, January 01)Studying the inner regions of protoplanetary disks (1-10 AU) is of importance to understand the formation of planets and the accretion process feeding the forming central star. Herbig AeBe stars are ... [more ▼]Studying the inner regions of protoplanetary disks (1-10 AU) is of importance to understand the formation of planets and the accretion process feeding the forming central star. Herbig AeBe stars are bright enough to be routinely observed by Near IR interferometers. The data for the fainter T Tauri stars is much more sparse. In this contribution we present the results of our ongoing survey at the VLTI. We used the PIONIER combiner that allows the simultaneous use of 4 telescopes, yielding 6 baselines and 3 independent closure phases at once. PIONIER's integrated optics technology makes it a sensitive instrument. We have observed 22 T Tauri stars so far, the largest survey for T Tauri stars to this date. Our results demonstrate the very significant contribution of an extended component to the interferometric signal. The extended component is different from source to source and the data, with several baselines, offer a way to improve our knowledge of the disk geometry and/or composition. These results validate an earlier study by Pinte et al. 2008 and show that the dust inner radii of T Tauri disks now appear to be in better agreement with the expected position of the dust sublimation radius, contrary to previous claims. [less ▲]Detailed reference viewed: 33 (0 ULg) 1 2

format: plain


text:
# Oxidation States of Nitrogen

In comparing the chemistry of the amines with alcohols and ethers, we discover many classes of related compounds in which nitrogen assumes higher oxidation states, in contrast to limited oxidation states of oxygen.

In this context, keep in mind that the oxidation state of elemental oxygen (O2) and nitrogen (N2) is defined as zero. The most prevalent state of covalently bonded oxygen is -2. This is the case for water, alcohols, ethers and carbonyl compounds. The only common higher oxidation state (-1) is found in the peroxides, R–O–O–R, where R=hydrogen, alkyl, aryl or acyl. Because of the low covalent bond energy of the peroxide bond (ca.35 kcal/mole), these compounds are widely used as free radical initiators, and are sometimes dangerously explosive in their reactivity (e.g. triacetone triperoxide used by terrorist bombers). Nitrogen compounds, on the other hand, encompass oxidation states of nitrogen ranging from -3, as in ammonia and amines, to +5, as in nitric acid. The following table lists some of the known organic compounds of nitrogen, having different oxidation states of that element. Some of these classes of compounds have been described; others will be discussed later.

Oxidation State

_3

_2

_1

0

+1

+3

Formulas
(names)

R3N

(amines)
R4N(+)

(ammonium)
C=N–R

(imines)
C≡N

(nitriles)

R2NNR2

(hydrazines)
C=NNR2

(hydrazones)

RN=NR

(azo cpd.)
R2NOH

(hydroxyl amine)
R3NO

(amine oxide)

N2

(nitrogen)
R–N2(+)

(diazonium)

R–N=O (nitroso)

R-NO2

(nitro)
RO–N=O

(nitrite ester)

### Amine Oxides and the Cope Elimination

Amine oxides are prepared by oxidizing 3º-amines or pyridines with hydrogen peroxide or peracids (e.g. ZOOH, where Z=H or acyl).

$R_3N: + ZOOH \rightarrow R_3N^{(+)}–O^{(–)} + ZOH$

Amine oxides are relatively weak bases, pKa ca. 4.5, compared with the parent amine. The coordinate covalent N–O function is polar, with the oxygen being a powerful hydrogen bond acceptor. If one of the alkyl substituents consists of a long chain, such as C12H25, the resulting amine oxide is an amphoteric surfactant and finds use in shampoos and other mild cleaning agents.

An elimination reaction, complementary to the Hofmann elimination, occurs when 3º-amine oxides are heated at temperatures of 150 to 200 ºC. This reaction is known as the Cope Elimination. It is commonly carried out by dropwise addition of an amine oxide solution to a heated tube packed with small glass beads. A stream of nitrogen gas flowing through the column carries the volatile alkene products to a chilled receiver. The nitrogen-containing product is a hydroxyl amine. Unlike the Hofmann elimination, this reaction takes place by a concerted cyclic reorganization, as shown in the following diagram. For such a mechanism, the beta-hydrogen and amine oxide moieties necessarily have a syn-relationship.

Cope elimination of diastereomeric amine oxides, such as those shown in examples #2 & 3 above, provide proof of the syn-relationship of the beta-hydrogen and amine oxide groups. These examples also demonstrate a strong regioselectivity favoring the more stable double bond.

#### Pyrolytic syn-Eliminations

Amine oxides are not the only functions that undergo a unimolecular syn-elimination on heating. To see examples of other cases Click Here