stegr#

sstegr

Functions

void sstegr(
    const char*          jobz,
    const char*          range,
    const INT            n,
          f32*  restrict D,
          f32*  restrict E,
    const f32            vl,
    const f32            vu,
    const INT            il,
    const INT            iu,
    const f32            abstol,
          INT*           m,
          f32*  restrict W,
          f32*  restrict Z,
    const INT            ldz,
          INT*  restrict isuppz,
          f32*  restrict work,
    const INT            lwork,
          INT*  restrict iwork,
    const INT            liwork,
          INT*           info
);

void sstegr(const char *jobz, const char *range, const INT n, f32 *restrict D, f32 *restrict E, const f32 vl, const f32 vu, const INT il, const INT iu, const f32 abstol, INT *m, f32 *restrict W, f32 *restrict Z, const INT ldz, INT *restrict isuppz, f32 *restrict work, const INT lwork, INT *restrict iwork, const INT liwork, INT *info)#

SSTEGR computes selected eigenvalues and, optionally, eigenvectors of a real symmetric tridiagonal matrix T.

Any such unreduced matrix has a well defined set of pairwise different real eigenvalues, the corresponding real eigenvectors are pairwise orthogonal.

The spectrum may be computed either completely or partially by specifying either an interval (vl,vu] or a range of indices il:iu for the desired eigenvalues.

SSTEGR is a compatibility wrapper around the improved SSTEMR routine. See SSTEMR for further details.

One important change is that the ABSTOL parameter no longer provides any benefit and hence is no longer used.

Note: SSTEGR and SSTEMR work only on machines which follow IEEE-754 floating-point standard in their handling of infinities and NaNs. Normal execution may create these exceptional values and hence may abort due to a floating point exception in environments which do not conform to the IEEE-754 standard.

Parameters

in

jobz

Specifies whether to compute eigenvalues only or eigenvectors as well. = ‘N’: Compute eigenvalues only = ‘V’: Compute eigenvalues and eigenvectors

in

range

Specifies the range of eigenvalues to compute. = ‘A’: All eigenvalues will be found = ‘V’: All eigenvalues in the half-open interval (vl,vu] will be found = ‘I’: The il-th through iu-th eigenvalues will be found

in

n

The order of the matrix. n >= 0.

inout

D

Double precision array, dimension (n). On entry, the n diagonal elements of the tridiagonal matrix T. On exit, D is overwritten.

inout

E

Double precision array, dimension (n). On entry, the (n-1) subdiagonal elements of the tridiagonal matrix T in elements 0 to n-2 of E. E[n-1] need not be set on input, but is used internally as workspace. On exit, E is overwritten.

in

vl

If range = ‘V’, the lower bound of the interval to be searched for eigenvalues. vl < vu. Not referenced if range = ‘A’ or ‘I’.

in

vu

If range = ‘V’, the upper bound of the interval to be searched for eigenvalues. vl < vu. Not referenced if range = ‘A’ or ‘I’.

in

il

If range = ‘I’, the index of the smallest eigenvalue to be returned (0-based). 0 <= il <= iu <= n-1, if n > 0. Not referenced if range = ‘A’ or ‘V’.

in

iu

If range = ‘I’, the index of the largest eigenvalue to be returned (0-based). 0 <= il <= iu <= n-1, if n > 0. Not referenced if range = ‘A’ or ‘V’.

in

abstol

Unused. Was the absolute error tolerance in previous versions.

out

m

The total number of eigenvalues found. 0 <= m <= n. If range = ‘A’, m = n, and if range = ‘I’, m = iu-il+1.

out

W

Double precision array, dimension (n). The first m elements contain the selected eigenvalues in ascending order.

out

Z

Double precision array, dimension (ldz, max(1,m)). If jobz = ‘V’, and if info = 0, then the first m columns of Z contain the orthonormal eigenvectors of the matrix T corresponding to the selected eigenvalues, with the i-th column of Z holding the eigenvector associated with W[i]. If jobz = ‘N’, 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 = ‘V’, the exact value of m is not known in advance and an upper bound must be used. Supplying n columns is always safe.

in

ldz

The leading dimension of the array Z. ldz >= 1, and if jobz = ‘V’, then 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 computed eigenvector is nonzero only in elements isuppz[2*i] through isuppz[2*i+1] (0-based). This is relevant in the case when the matrix is split. isuppz is only accessed when jobz is ‘V’ and n > 0.

out

work

Double precision array, dimension (lwork). On exit, if info = 0, work[0] returns the optimal (and minimal) lwork.

in

lwork

The dimension of the array work. lwork >= max(1,18*n) if jobz = ‘V’, and lwork >= max(1,12*n) if jobz = ‘N’. 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

iwork

Integer array, dimension (liwork). On exit, if info = 0, iwork[0] returns the optimal liwork.

in

liwork

The dimension of the array iwork. liwork >= max(1,10*n) if the eigenvectors are desired, and liwork >= max(1,8*n) if only the eigenvalues are to be computed. 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

Exit status:

= 0: successful exit
< 0: if info = -i, the i-th argument had an illegal value
> 0: if info = 1X, internal error in SLARRE, if info = 2X, internal error in SLARRV. Here, the digit X = ABS(iinfo) < 10, where iinfo is the nonzero error code returned by SLARRE or SLARRV.

dstegr

Functions

void dstegr(
    const char*          jobz,
    const char*          range,
    const INT            n,
          f64*  restrict D,
          f64*  restrict E,
    const f64            vl,
    const f64            vu,
    const INT            il,
    const INT            iu,
    const f64            abstol,
          INT*           m,
          f64*  restrict W,
          f64*  restrict Z,
    const INT            ldz,
          INT*  restrict isuppz,
          f64*  restrict work,
    const INT            lwork,
          INT*  restrict iwork,
    const INT            liwork,
          INT*           info
);

void dstegr(const char *jobz, const char *range, const INT n, f64 *restrict D, f64 *restrict E, const f64 vl, const f64 vu, const INT il, const INT iu, const f64 abstol, INT *m, f64 *restrict W, f64 *restrict Z, const INT ldz, INT *restrict isuppz, f64 *restrict work, const INT lwork, INT *restrict iwork, const INT liwork, INT *info)#

DSTEGR computes selected eigenvalues and, optionally, eigenvectors of a real symmetric tridiagonal matrix T.

Any such unreduced matrix has a well defined set of pairwise different real eigenvalues, the corresponding real eigenvectors are pairwise orthogonal.

The spectrum may be computed either completely or partially by specifying either an interval (vl,vu] or a range of indices il:iu for the desired eigenvalues.

DSTEGR is a compatibility wrapper around the improved DSTEMR routine. See DSTEMR for further details.

One important change is that the ABSTOL parameter no longer provides any benefit and hence is no longer used.

Note: DSTEGR and DSTEMR work only on machines which follow IEEE-754 floating-point standard in their handling of infinities and NaNs. Normal execution may create these exceptional values and hence may abort due to a floating point exception in environments which do not conform to the IEEE-754 standard.

Parameters

in

jobz

Specifies whether to compute eigenvalues only or eigenvectors as well. = ‘N’: Compute eigenvalues only = ‘V’: Compute eigenvalues and eigenvectors

in

range

Specifies the range of eigenvalues to compute. = ‘A’: All eigenvalues will be found = ‘V’: All eigenvalues in the half-open interval (vl,vu] will be found = ‘I’: The il-th through iu-th eigenvalues will be found

in

n

The order of the matrix. n >= 0.

inout

D

Double precision array, dimension (n). On entry, the n diagonal elements of the tridiagonal matrix T. On exit, D is overwritten.

inout

E

Double precision array, dimension (n). On entry, the (n-1) subdiagonal elements of the tridiagonal matrix T in elements 0 to n-2 of E. E[n-1] need not be set on input, but is used internally as workspace. On exit, E is overwritten.

in

vl

If range = ‘V’, the lower bound of the interval to be searched for eigenvalues. vl < vu. Not referenced if range = ‘A’ or ‘I’.

in

vu

If range = ‘V’, the upper bound of the interval to be searched for eigenvalues. vl < vu. Not referenced if range = ‘A’ or ‘I’.

in

il

If range = ‘I’, the index of the smallest eigenvalue to be returned (0-based). 0 <= il <= iu <= n-1, if n > 0. Not referenced if range = ‘A’ or ‘V’.

in

iu

If range = ‘I’, the index of the largest eigenvalue to be returned (0-based). 0 <= il <= iu <= n-1, if n > 0. Not referenced if range = ‘A’ or ‘V’.

in

abstol

Unused. Was the absolute error tolerance in previous versions.

out

m

The total number of eigenvalues found. 0 <= m <= n. If range = ‘A’, m = n, and if range = ‘I’, m = iu-il+1.

out

W

Double precision array, dimension (n). The first m elements contain the selected eigenvalues in ascending order.

out

Z

Double precision array, dimension (ldz, max(1,m)). If jobz = ‘V’, and if info = 0, then the first m columns of Z contain the orthonormal eigenvectors of the matrix T corresponding to the selected eigenvalues, with the i-th column of Z holding the eigenvector associated with W[i]. If jobz = ‘N’, 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 = ‘V’, the exact value of m is not known in advance and an upper bound must be used. Supplying n columns is always safe.

in

ldz

The leading dimension of the array Z. ldz >= 1, and if jobz = ‘V’, then 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 computed eigenvector is nonzero only in elements isuppz[2*i] through isuppz[2*i+1] (0-based). This is relevant in the case when the matrix is split. isuppz is only accessed when jobz is ‘V’ and n > 0.

out

work

Double precision array, dimension (lwork). On exit, if info = 0, work[0] returns the optimal (and minimal) lwork.

in

lwork

The dimension of the array work. lwork >= max(1,18*n) if jobz = ‘V’, and lwork >= max(1,12*n) if jobz = ‘N’. 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

iwork

Integer array, dimension (liwork). On exit, if info = 0, iwork[0] returns the optimal liwork.

in

liwork

The dimension of the array iwork. liwork >= max(1,10*n) if the eigenvectors are desired, and liwork >= max(1,8*n) if only the eigenvalues are to be computed. 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

Exit status:

= 0: successful exit
< 0: if info = -i, the i-th argument had an illegal value
> 0: if info = 1X, internal error in DLARRE, if info = 2X, internal error in DLARRV. Here, the digit X = ABS(iinfo) < 10, where iinfo is the nonzero error code returned by DLARRE or DLARRV.

cstegr

Functions

void cstegr(
    const char*          jobz,
    const char*          range,
    const INT            n,
          f32*  restrict D,
          f32*  restrict E,
    const f32            vl,
    const f32            vu,
    const INT            il,
    const INT            iu,
    const f32            abstol,
          INT*           m,
          f32*  restrict W,
          c64*  restrict Z,
    const INT            ldz,
          INT*  restrict isuppz,
          f32*  restrict work,
    const INT            lwork,
          INT*  restrict iwork,
    const INT            liwork,
          INT*           info
);

void cstegr(const char *jobz, const char *range, const INT n, f32 *restrict D, f32 *restrict E, const f32 vl, const f32 vu, const INT il, const INT iu, const f32 abstol, INT *m, f32 *restrict W, c64 *restrict Z, const INT ldz, INT *restrict isuppz, f32 *restrict work, const INT lwork, INT *restrict iwork, const INT liwork, INT *info)#

CSTEGR computes selected eigenvalues and, optionally, eigenvectors of a real symmetric tridiagonal matrix T.

Any such unreduced matrix has a well defined set of pairwise different real eigenvalues, the corresponding real eigenvectors are pairwise orthogonal.

The spectrum may be computed either completely or partially by specifying either an interval (vl,vu] or a range of indices il:iu for the desired eigenvalues.

CSTEGR is a compatibility wrapper around the improved CSTEMR routine. See CSTEMR for further details.

One important change is that the ABSTOL parameter no longer provides any benefit and hence is no longer used.

Note: CSTEGR and CSTEMR work only on machines which follow IEEE-754 floating-point standard in their handling of infinities and NaNs. Normal execution may create these exceptional values and hence may abort due to a floating point exception in environments which do not conform to the IEEE-754 standard.

Parameters

in

jobz

Specifies whether to compute eigenvalues only or eigenvectors as well. = ‘N’: Compute eigenvalues only = ‘V’: Compute eigenvalues and eigenvectors

in

range

Specifies the range of eigenvalues to compute. = ‘A’: All eigenvalues will be found = ‘V’: All eigenvalues in the half-open interval (vl,vu] will be found = ‘I’: The il-th through iu-th eigenvalues will be found

in

n

The order of the matrix. n >= 0.

inout

D

Single precision array, dimension (n). On entry, the n diagonal elements of the tridiagonal matrix T. On exit, D is overwritten.

inout

E

Single precision array, dimension (n). On entry, the (n-1) subdiagonal elements of the tridiagonal matrix T in elements 0 to n-2 of E. E[n-1] need not be set on input, but is used internally as workspace. On exit, E is overwritten.

in

vl

If range = ‘V’, the lower bound of the interval to be searched for eigenvalues. vl < vu. Not referenced if range = ‘A’ or ‘I’.

in

vu

If range = ‘V’, the upper bound of the interval to be searched for eigenvalues. vl < vu. Not referenced if range = ‘A’ or ‘I’.

in

il

If range = ‘I’, the index of the smallest eigenvalue to be returned (0-based). 0 <= il <= iu <= n-1, if n > 0. Not referenced if range = ‘A’ or ‘V’.

in

iu

If range = ‘I’, the index of the largest eigenvalue to be returned (0-based). 0 <= il <= iu <= n-1, if n > 0. Not referenced if range = ‘A’ or ‘V’.

in

abstol

Unused. Was the absolute error tolerance in previous versions.

out

m

The total number of eigenvalues found. 0 <= m <= n. If range = ‘A’, m = n, and if range = ‘I’, m = iu-il+1.

out

W

Single precision array, dimension (n). The first m elements contain the selected eigenvalues in ascending order.

out

Z

Single complex array, dimension (ldz, max(1,m)). If jobz = ‘V’, and if info = 0, then the first m columns of Z contain the orthonormal eigenvectors of the matrix T corresponding to the selected eigenvalues, with the i-th column of Z holding the eigenvector associated with W[i]. If jobz = ‘N’, 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 = ‘V’, the exact value of m is not known in advance and an upper bound must be used. Supplying n columns is always safe.

in

ldz

The leading dimension of the array Z. ldz >= 1, and if jobz = ‘V’, then 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 computed eigenvector is nonzero only in elements isuppz[2*i] through isuppz[2*i+1] (0-based). This is relevant in the case when the matrix is split. isuppz is only accessed when jobz is ‘V’ and n > 0.

out

work

Single precision array, dimension (lwork). On exit, if info = 0, work[0] returns the optimal (and minimal) lwork.

in

lwork

The dimension of the array work. lwork >= max(1,18*n) if jobz = ‘V’, and lwork >= max(1,12*n) if jobz = ‘N’. 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

iwork

Integer array, dimension (liwork). On exit, if info = 0, iwork[0] returns the optimal liwork.

in

liwork

The dimension of the array iwork. liwork >= max(1,10*n) if the eigenvectors are desired, and liwork >= max(1,8*n) if only the eigenvalues are to be computed. 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

Exit status:

= 0: successful exit
< 0: if info = -i, the i-th argument had an illegal value
> 0: if info = 1X, internal error in SLARRE, if info = 2X, internal error in CLARRV. Here, the digit X = ABS(iinfo) < 10, where iinfo is the nonzero error code returned by SLARRE or CLARRV.

zstegr

Functions

void zstegr(
    const char*          jobz,
    const char*          range,
    const INT            n,
          f64*  restrict D,
          f64*  restrict E,
    const f64            vl,
    const f64            vu,
    const INT            il,
    const INT            iu,
    const f64            abstol,
          INT*           m,
          f64*  restrict W,
          c128* restrict Z,
    const INT            ldz,
          INT*  restrict isuppz,
          f64*  restrict work,
    const INT            lwork,
          INT*  restrict iwork,
    const INT            liwork,
          INT*           info
);

void zstegr(const char *jobz, const char *range, const INT n, f64 *restrict D, f64 *restrict E, const f64 vl, const f64 vu, const INT il, const INT iu, const f64 abstol, INT *m, f64 *restrict W, c128 *restrict Z, const INT ldz, INT *restrict isuppz, f64 *restrict work, const INT lwork, INT *restrict iwork, const INT liwork, INT *info)#

ZSTEGR computes selected eigenvalues and, optionally, eigenvectors of a real symmetric tridiagonal matrix T.

Any such unreduced matrix has a well defined set of pairwise different real eigenvalues, the corresponding real eigenvectors are pairwise orthogonal.

The spectrum may be computed either completely or partially by specifying either an interval (vl,vu] or a range of indices il:iu for the desired eigenvalues.

ZSTEGR is a compatibility wrapper around the improved ZSTEMR routine. See ZSTEMR for further details.

One important change is that the ABSTOL parameter no longer provides any benefit and hence is no longer used.

Note: ZSTEGR and ZSTEMR work only on machines which follow IEEE-754 floating-point standard in their handling of infinities and NaNs. Normal execution may create these exceptional values and hence may abort due to a floating point exception in environments which do not conform to the IEEE-754 standard.

Parameters

in

jobz

Specifies whether to compute eigenvalues only or eigenvectors as well. = ‘N’: Compute eigenvalues only = ‘V’: Compute eigenvalues and eigenvectors

in

range

Specifies the range of eigenvalues to compute. = ‘A’: All eigenvalues will be found = ‘V’: All eigenvalues in the half-open interval (vl,vu] will be found = ‘I’: The il-th through iu-th eigenvalues will be found

in

n

The order of the matrix. n >= 0.

inout

D

Double precision array, dimension (n). On entry, the n diagonal elements of the tridiagonal matrix T. On exit, D is overwritten.

inout

E

Double precision array, dimension (n). On entry, the (n-1) subdiagonal elements of the tridiagonal matrix T in elements 0 to n-2 of E. E[n-1] need not be set on input, but is used internally as workspace. On exit, E is overwritten.

in

vl

If range = ‘V’, the lower bound of the interval to be searched for eigenvalues. vl < vu. Not referenced if range = ‘A’ or ‘I’.

in

vu

If range = ‘V’, the upper bound of the interval to be searched for eigenvalues. vl < vu. Not referenced if range = ‘A’ or ‘I’.

in

il

If range = ‘I’, the index of the smallest eigenvalue to be returned (0-based). 0 <= il <= iu <= n-1, if n > 0. Not referenced if range = ‘A’ or ‘V’.

in

iu

If range = ‘I’, the index of the largest eigenvalue to be returned (0-based). 0 <= il <= iu <= n-1, if n > 0. Not referenced if range = ‘A’ or ‘V’.

in

abstol

Unused. Was the absolute error tolerance in previous versions.

out

m

The total number of eigenvalues found. 0 <= m <= n. If range = ‘A’, m = n, and if range = ‘I’, m = iu-il+1.

out

W

Double precision array, dimension (n). The first m elements contain the selected eigenvalues in ascending order.

out

Z

Double complex array, dimension (ldz, max(1,m)). If jobz = ‘V’, and if info = 0, then the first m columns of Z contain the orthonormal eigenvectors of the matrix T corresponding to the selected eigenvalues, with the i-th column of Z holding the eigenvector associated with W[i]. If jobz = ‘N’, 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 = ‘V’, the exact value of m is not known in advance and an upper bound must be used. Supplying n columns is always safe.

in

ldz

The leading dimension of the array Z. ldz >= 1, and if jobz = ‘V’, then 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 computed eigenvector is nonzero only in elements isuppz[2*i] through isuppz[2*i+1] (0-based). This is relevant in the case when the matrix is split. isuppz is only accessed when jobz is ‘V’ and n > 0.

out

work

Double precision array, dimension (lwork). On exit, if info = 0, work[0] returns the optimal (and minimal) lwork.

in

lwork

The dimension of the array work. lwork >= max(1,18*n) if jobz = ‘V’, and lwork >= max(1,12*n) if jobz = ‘N’. 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

iwork

Integer array, dimension (liwork). On exit, if info = 0, iwork[0] returns the optimal liwork.

in

liwork

The dimension of the array iwork. liwork >= max(1,10*n) if the eigenvectors are desired, and liwork >= max(1,8*n) if only the eigenvalues are to be computed. 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

Exit status:

= 0: successful exit
< 0: if info = -i, the i-th argument had an illegal value
> 0: if info = 1X, internal error in DLARRE, if info = 2X, internal error in ZLARRV. Here, the digit X = ABS(iinfo) < 10, where iinfo is the nonzero error code returned by DLARRE or ZLARRV.

stegr#

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