--- name: "stein" section: "3" platform: "linux" package: "lapack-doc" version: "3.12.1-2" release: "Arch Linux 2026-11-01" platforms: ["linux"] url: "https://new.carta.tech/stein" --- # stein(3) stein: eig, inverse iteration ## NAME stein - stein: eig, inverse iteration ## SYNOPSIS ### Functions subroutine **cstein** (n, d, e, m, w, iblock, isplit, z, ldz, work, iwork, ifail, info)\ **CSTEIN** subroutine **dstein** (n, d, e, m, w, iblock, isplit, z, ldz, work, iwork, ifail, info)\ **DSTEIN** subroutine **sstein** (n, d, e, m, w, iblock, isplit, z, ldz, work, iwork, ifail, info)\ **SSTEIN** subroutine **zstein** (n, d, e, m, w, iblock, isplit, z, ldz, work, iwork, ifail, info)\ **ZSTEIN** ## Detailed Description ## Function Documentation ### subroutine cstein (integer n, real, dimension( \* ) d, real, dimension( \* ) e, integer m, real, dimension( \* ) w, integer, dimension( \* ) iblock, integer, dimension( \* ) isplit, complex, dimension( ldz, \* ) z, integer ldz, real, dimension( \* ) work, integer, dimension( \* ) iwork, integer, dimension( \* ) ifail, integer info) **CSTEIN** **Purpose:** ``` !> !> CSTEIN computes the eigenvectors of a real symmetric tridiagonal !> matrix T corresponding to specified eigenvalues, using inverse !> iteration. !> !> The maximum number of iterations allowed for each eigenvector is !> specified by an internal parameter MAXITS (currently set to 5). !> !> Although the eigenvectors are real, they are stored in a complex !> array, which may be passed to CUNMTR or CUPMTR for back !> transformation to the eigenvectors of a complex Hermitian matrix !> which was reduced to tridiagonal form. !> !> ``` **Parameters** *N* ``` !> N is INTEGER !> The order of the matrix. N >= 0. !> ``` *D* ``` !> D is REAL array, dimension (N) !> The n diagonal elements of the tridiagonal matrix T. !> ``` *E* ``` !> E is REAL array, dimension (N-1) !> The (n-1) subdiagonal elements of the tridiagonal matrix !> T, stored in elements 1 to N-1. !> ``` *M* ``` !> M is INTEGER !> The number of eigenvectors to be found. 0 <= M <= N. !> ``` *W* ``` !> W is REAL array, dimension (N) !> The first M elements of W contain the eigenvalues for !> which eigenvectors are to be computed. The eigenvalues !> should be grouped by split-off block and ordered from !> smallest to largest within the block. ( The output array !> W from SSTEBZ with ORDER = 'B' is expected here. ) !> ``` *IBLOCK* ``` !> IBLOCK is INTEGER array, dimension (N) !> The submatrix indices associated with the corresponding !> eigenvalues in W; IBLOCK(i)=1 if eigenvalue W(i) belongs to !> the first submatrix from the top, =2 if W(i) belongs to !> the second submatrix, etc. ( The output array IBLOCK !> from SSTEBZ is expected here. ) !> ``` *ISPLIT* ``` !> ISPLIT is INTEGER array, dimension (N) !> The splitting points, at which T breaks up into submatrices. !> The first submatrix consists of rows/columns 1 to !> ISPLIT( 1 ), the second of rows/columns ISPLIT( 1 )+1 !> through ISPLIT( 2 ), etc. !> ( The output array ISPLIT from SSTEBZ is expected here. ) !> ``` *Z* ``` !> Z is COMPLEX array, dimension (LDZ, M) !> The computed eigenvectors. The eigenvector associated !> with the eigenvalue W(i) is stored in the i-th column of !> Z. Any vector which fails to converge is set to its current !> iterate after MAXITS iterations. !> The imaginary parts of the eigenvectors are set to zero. !> ``` *LDZ* ``` !> LDZ is INTEGER !> The leading dimension of the array Z. LDZ >= max(1,N). !> ``` *WORK* ``` !> WORK is REAL array, dimension (5*N) !> ``` *IWORK* ``` !> IWORK is INTEGER array, dimension (N) !> ``` *IFAIL* ``` !> IFAIL is INTEGER array, dimension (M) !> On normal exit, all elements of IFAIL are zero. !> If one or more eigenvectors fail to converge after !> MAXITS iterations, then their indices are stored in !> array IFAIL. !> ``` *INFO* ``` !> INFO is INTEGER !> = 0: successful exit !> < 0: if INFO = -i, the i-th argument had an illegal value !> > 0: if INFO = i, then i eigenvectors failed to converge !> in MAXITS iterations. Their indices are stored in !> array IFAIL. !> ``` **Internal Parameters:** ``` !> MAXITS INTEGER, default = 5 !> The maximum number of iterations performed. !> !> EXTRA INTEGER, default = 2 !> The number of iterations performed after norm growth !> criterion is satisfied, should be at least 1. !> ``` **Author** Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Definition at line **180** of file **cstein.f**. ### subroutine dstein (integer n, double precision, dimension( \* ) d, double precision, dimension( \* ) e, integer m, double precision, dimension( \* ) w, integer, dimension( \* ) iblock, integer, dimension( \* ) isplit, double precision, dimension( ldz, \* ) z, integer ldz, double precision, dimension( \* ) work, integer, dimension( \* ) iwork, integer, dimension( \* ) ifail, integer info) **DSTEIN** **Purpose:** ``` !> !> DSTEIN computes the eigenvectors of a real symmetric tridiagonal !> matrix T corresponding to specified eigenvalues, using inverse !> iteration. !> !> The maximum number of iterations allowed for each eigenvector is !> specified by an internal parameter MAXITS (currently set to 5). !> ``` **Parameters** *N* ``` !> N is INTEGER !> The order of the matrix. N >= 0. !> ``` *D* ``` !> D is DOUBLE PRECISION array, dimension (N) !> The n diagonal elements of the tridiagonal matrix T. !> ``` *E* ``` !> E is DOUBLE PRECISION array, dimension (N-1) !> The (n-1) subdiagonal elements of the tridiagonal matrix !> T, in elements 1 to N-1. !> ``` *M* ``` !> M is INTEGER !> The number of eigenvectors to be found. 0 <= M <= N. !> ``` *W* ``` !> W is DOUBLE PRECISION array, dimension (N) !> The first M elements of W contain the eigenvalues for !> which eigenvectors are to be computed. The eigenvalues !> should be grouped by split-off block and ordered from !> smallest to largest within the block. ( The output array !> W from DSTEBZ with ORDER = 'B' is expected here. ) !> ``` *IBLOCK* ``` !> IBLOCK is INTEGER array, dimension (N) !> The submatrix indices associated with the corresponding !> eigenvalues in W; IBLOCK(i)=1 if eigenvalue W(i) belongs to !> the first submatrix from the top, =2 if W(i) belongs to !> the second submatrix, etc. ( The output array IBLOCK !> from DSTEBZ is expected here. ) !> ``` *ISPLIT* ``` !> ISPLIT is INTEGER array, dimension (N) !> The splitting points, at which T breaks up into submatrices. !> The first submatrix consists of rows/columns 1 to !> ISPLIT( 1 ), the second of rows/columns ISPLIT( 1 )+1 !> through ISPLIT( 2 ), etc. !> ( The output array ISPLIT from DSTEBZ is expected here. ) !> ``` *Z* ``` !> Z is DOUBLE PRECISION array, dimension (LDZ, M) !> The computed eigenvectors. The eigenvector associated !> with the eigenvalue W(i) is stored in the i-th column of !> Z. Any vector which fails to converge is set to its current !> iterate after MAXITS iterations. !> ``` *LDZ* ``` !> LDZ is INTEGER !> The leading dimension of the array Z. LDZ >= max(1,N). !> ``` *WORK* ``` !> WORK is DOUBLE PRECISION array, dimension (5*N) !> ``` *IWORK* ``` !> IWORK is INTEGER array, dimension (N) !> ``` *IFAIL* ``` !> IFAIL is INTEGER array, dimension (M) !> On normal exit, all elements of IFAIL are zero. !> If one or more eigenvectors fail to converge after !> MAXITS iterations, then their indices are stored in !> array IFAIL. !> ``` *INFO* ``` !> INFO is INTEGER !> = 0: successful exit. !> < 0: if INFO = -i, the i-th argument had an illegal value !> > 0: if INFO = i, then i eigenvectors failed to converge !> in MAXITS iterations. Their indices are stored in !> array IFAIL. !> ``` **Internal Parameters:** ``` !> MAXITS INTEGER, default = 5 !> The maximum number of iterations performed. !> !> EXTRA INTEGER, default = 2 !> The number of iterations performed after norm growth !> criterion is satisfied, should be at least 1. !> ``` **Author** Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Definition at line **172** of file **dstein.f**. ### subroutine sstein (integer n, real, dimension( \* ) d, real, dimension( \* ) e, integer m, real, dimension( \* ) w, integer, dimension( \* ) iblock, integer, dimension( \* ) isplit, real, dimension( ldz, \* ) z, integer ldz, real, dimension( \* ) work, integer, dimension( \* ) iwork, integer, dimension( \* ) ifail, integer info) **SSTEIN** **Purpose:** ``` !> !> SSTEIN computes the eigenvectors of a real symmetric tridiagonal !> matrix T corresponding to specified eigenvalues, using inverse !> iteration. !> !> The maximum number of iterations allowed for each eigenvector is !> specified by an internal parameter MAXITS (currently set to 5). !> ``` **Parameters** *N* ``` !> N is INTEGER !> The order of the matrix. N >= 0. !> ``` *D* ``` !> D is REAL array, dimension (N) !> The n diagonal elements of the tridiagonal matrix T. !> ``` *E* ``` !> E is REAL array, dimension (N-1) !> The (n-1) subdiagonal elements of the tridiagonal matrix !> T, in elements 1 to N-1. !> ``` *M* ``` !> M is INTEGER !> The number of eigenvectors to be found. 0 <= M <= N. !> ``` *W* ``` !> W is REAL array, dimension (N) !> The first M elements of W contain the eigenvalues for !> which eigenvectors are to be computed. The eigenvalues !> should be grouped by split-off block and ordered from !> smallest to largest within the block. ( The output array !> W from SSTEBZ with ORDER = 'B' is expected here. ) !> ``` *IBLOCK* ``` !> IBLOCK is INTEGER array, dimension (N) !> The submatrix indices associated with the corresponding !> eigenvalues in W; IBLOCK(i)=1 if eigenvalue W(i) belongs to !> the first submatrix from the top, =2 if W(i) belongs to !> the second submatrix, etc. ( The output array IBLOCK !> from SSTEBZ is expected here. ) !> ``` *ISPLIT* ``` !> ISPLIT is INTEGER array, dimension (N) !> The splitting points, at which T breaks up into submatrices. !> The first submatrix consists of rows/columns 1 to !> ISPLIT( 1 ), the second of rows/columns ISPLIT( 1 )+1 !> through ISPLIT( 2 ), etc. !> ( The output array ISPLIT from SSTEBZ is expected here. ) !> ``` *Z* ``` !> Z is REAL array, dimension (LDZ, M) !> The computed eigenvectors. The eigenvector associated !> with the eigenvalue W(i) is stored in the i-th column of !> Z. Any vector which fails to converge is set to its current !> iterate after MAXITS iterations. !> ``` *LDZ* ``` !> LDZ is INTEGER !> The leading dimension of the array Z. LDZ >= max(1,N). !> ``` *WORK* ``` !> WORK is REAL array, dimension (5*N) !> ``` *IWORK* ``` !> IWORK is INTEGER array, dimension (N) !> ``` *IFAIL* ``` !> IFAIL is INTEGER array, dimension (M) !> On normal exit, all elements of IFAIL are zero. !> If one or more eigenvectors fail to converge after !> MAXITS iterations, then their indices are stored in !> array IFAIL. !> ``` *INFO* ``` !> INFO is INTEGER !> = 0: successful exit. !> < 0: if INFO = -i, the i-th argument had an illegal value !> > 0: if INFO = i, then i eigenvectors failed to converge !> in MAXITS iterations. Their indices are stored in !> array IFAIL. !> ``` **Internal Parameters:** ``` !> MAXITS INTEGER, default = 5 !> The maximum number of iterations performed. !> !> EXTRA INTEGER, default = 2 !> The number of iterations performed after norm growth !> criterion is satisfied, should be at least 1. !> ``` **Author** Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Definition at line **172** of file **sstein.f**. ### subroutine zstein (integer n, double precision, dimension( \* ) d, double precision, dimension( \* ) e, integer m, double precision, dimension( \* ) w, integer, dimension( \* ) iblock, integer, dimension( \* ) isplit, complex\*16, dimension( ldz, \* ) z, integer ldz, double precision, dimension( \* ) work, integer, dimension( \* ) iwork, integer, dimension( \* ) ifail, integer info) **ZSTEIN** **Purpose:** ``` !> !> ZSTEIN computes the eigenvectors of a real symmetric tridiagonal !> matrix T corresponding to specified eigenvalues, using inverse !> iteration. !> !> The maximum number of iterations allowed for each eigenvector is !> specified by an internal parameter MAXITS (currently set to 5). !> !> Although the eigenvectors are real, they are stored in a complex !> array, which may be passed to ZUNMTR or ZUPMTR for back !> transformation to the eigenvectors of a complex Hermitian matrix !> which was reduced to tridiagonal form. !> !> ``` **Parameters** *N* ``` !> N is INTEGER !> The order of the matrix. N >= 0. !> ``` *D* ``` !> D is DOUBLE PRECISION array, dimension (N) !> The n diagonal elements of the tridiagonal matrix T. !> ``` *E* ``` !> E is DOUBLE PRECISION array, dimension (N-1) !> The (n-1) subdiagonal elements of the tridiagonal matrix !> T, stored in elements 1 to N-1. !> ``` *M* ``` !> M is INTEGER !> The number of eigenvectors to be found. 0 <= M <= N. !> ``` *W* ``` !> W is DOUBLE PRECISION array, dimension (N) !> The first M elements of W contain the eigenvalues for !> which eigenvectors are to be computed. The eigenvalues !> should be grouped by split-off block and ordered from !> smallest to largest within the block. ( The output array !> W from DSTEBZ with ORDER = 'B' is expected here. ) !> ``` *IBLOCK* ``` !> IBLOCK is INTEGER array, dimension (N) !> The submatrix indices associated with the corresponding !> eigenvalues in W; IBLOCK(i)=1 if eigenvalue W(i) belongs to !> the first submatrix from the top, =2 if W(i) belongs to !> the second submatrix, etc. ( The output array IBLOCK !> from DSTEBZ is expected here. ) !> ``` *ISPLIT* ``` !> ISPLIT is INTEGER array, dimension (N) !> The splitting points, at which T breaks up into submatrices. !> The first submatrix consists of rows/columns 1 to !> ISPLIT( 1 ), the second of rows/columns ISPLIT( 1 )+1 !> through ISPLIT( 2 ), etc. !> ( The output array ISPLIT from DSTEBZ is expected here. ) !> ``` *Z* ``` !> Z is COMPLEX*16 array, dimension (LDZ, M) !> The computed eigenvectors. The eigenvector associated !> with the eigenvalue W(i) is stored in the i-th column of !> Z. Any vector which fails to converge is set to its current !> iterate after MAXITS iterations. !> The imaginary parts of the eigenvectors are set to zero. !> ``` *LDZ* ``` !> LDZ is INTEGER !> The leading dimension of the array Z. LDZ >= max(1,N). !> ``` *WORK* ``` !> WORK is DOUBLE PRECISION array, dimension (5*N) !> ``` *IWORK* ``` !> IWORK is INTEGER array, dimension (N) !> ``` *IFAIL* ``` !> IFAIL is INTEGER array, dimension (M) !> On normal exit, all elements of IFAIL are zero. !> If one or more eigenvectors fail to converge after !> MAXITS iterations, then their indices are stored in !> array IFAIL. !> ``` *INFO* ``` !> INFO is INTEGER !> = 0: successful exit !> < 0: if INFO = -i, the i-th argument had an illegal value !> > 0: if INFO = i, then i eigenvectors failed to converge !> in MAXITS iterations. Their indices are stored in !> array IFAIL. !> ``` **Internal Parameters:** ``` !> MAXITS INTEGER, default = 5 !> The maximum number of iterations performed. !> !> EXTRA INTEGER, default = 2 !> The number of iterations performed after norm growth !> criterion is satisfied, should be at least 1. !> ``` **Author** Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Definition at line **180** of file **zstein.f**. ## Author Generated automatically by Doxygen for LAPACK from the source code.