--- name: "trsna" 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/trsna" --- # trsna(3) trsna: eig condition numbers ## NAME trsna - trsna: eig condition numbers ## SYNOPSIS ### Functions subroutine **ctrsna** (job, howmny, select, n, t, ldt, vl, ldvl, vr, ldvr, s, sep, mm, m, work, ldwork, rwork, info)\ **CTRSNA** subroutine **dtrsna** (job, howmny, select, n, t, ldt, vl, ldvl, vr, ldvr, s, sep, mm, m, work, ldwork, iwork, info)\ **DTRSNA** subroutine **strsna** (job, howmny, select, n, t, ldt, vl, ldvl, vr, ldvr, s, sep, mm, m, work, ldwork, iwork, info)\ **STRSNA** subroutine **ztrsna** (job, howmny, select, n, t, ldt, vl, ldvl, vr, ldvr, s, sep, mm, m, work, ldwork, rwork, info)\ **ZTRSNA** ## Detailed Description ## Function Documentation ### subroutine ctrsna (character job, character howmny, logical, dimension( \* ) select, integer n, complex, dimension( ldt, \* ) t, integer ldt, complex, dimension( ldvl, \* ) vl, integer ldvl, complex, dimension( ldvr, \* ) vr, integer ldvr, real, dimension( \* ) s, real, dimension( \* ) sep, integer mm, integer m, complex, dimension( ldwork, \* ) work, integer ldwork, real, dimension( \* ) rwork, integer info) **CTRSNA** **Purpose:** ``` !> !> CTRSNA estimates reciprocal condition numbers for specified !> eigenvalues and/or right eigenvectors of a complex upper triangular !> matrix T (or of any matrix Q*T*Q**H with Q unitary). !> ``` **Parameters** *JOB* ``` !> JOB is CHARACTER*1 !> Specifies whether condition numbers are required for !> eigenvalues (S) or eigenvectors (SEP): !> = 'E': for eigenvalues only (S); !> = 'V': for eigenvectors only (SEP); !> = 'B': for both eigenvalues and eigenvectors (S and SEP). !> ``` *HOWMNY* ``` !> HOWMNY is CHARACTER*1 !> = 'A': compute condition numbers for all eigenpairs; !> = 'S': compute condition numbers for selected eigenpairs !> specified by the array SELECT. !> ``` *SELECT* ``` !> SELECT is LOGICAL array, dimension (N) !> If HOWMNY = 'S', SELECT specifies the eigenpairs for which !> condition numbers are required. To select condition numbers !> for the j-th eigenpair, SELECT(j) must be set to .TRUE.. !> If HOWMNY = 'A', SELECT is not referenced. !> ``` *N* ``` !> N is INTEGER !> The order of the matrix T. N >= 0. !> ``` *T* ``` !> T is COMPLEX array, dimension (LDT,N) !> The upper triangular matrix T. !> ``` *LDT* ``` !> LDT is INTEGER !> The leading dimension of the array T. LDT >= max(1,N). !> ``` *VL* ``` !> VL is COMPLEX array, dimension (LDVL,M) !> If JOB = 'E' or 'B', VL must contain left eigenvectors of T !> (or of any Q*T*Q**H with Q unitary), corresponding to the !> eigenpairs specified by HOWMNY and SELECT. The eigenvectors !> must be stored in consecutive columns of VL, as returned by !> CHSEIN or CTREVC. !> If JOB = 'V', VL is not referenced. !> ``` *LDVL* ``` !> LDVL is INTEGER !> The leading dimension of the array VL. !> LDVL >= 1; and if JOB = 'E' or 'B', LDVL >= N. !> ``` *VR* ``` !> VR is COMPLEX array, dimension (LDVR,M) !> If JOB = 'E' or 'B', VR must contain right eigenvectors of T !> (or of any Q*T*Q**H with Q unitary), corresponding to the !> eigenpairs specified by HOWMNY and SELECT. The eigenvectors !> must be stored in consecutive columns of VR, as returned by !> CHSEIN or CTREVC. !> If JOB = 'V', VR is not referenced. !> ``` *LDVR* ``` !> LDVR is INTEGER !> The leading dimension of the array VR. !> LDVR >= 1; and if JOB = 'E' or 'B', LDVR >= N. !> ``` *S* ``` !> S is REAL array, dimension (MM) !> If JOB = 'E' or 'B', the reciprocal condition numbers of the !> selected eigenvalues, stored in consecutive elements of the !> array. Thus S(j), SEP(j), and the j-th columns of VL and VR !> all correspond to the same eigenpair (but not in general the !> j-th eigenpair, unless all eigenpairs are selected). !> If JOB = 'V', S is not referenced. !> ``` *SEP* ``` !> SEP is REAL array, dimension (MM) !> If JOB = 'V' or 'B', the estimated reciprocal condition !> numbers of the selected eigenvectors, stored in consecutive !> elements of the array. !> If JOB = 'E', SEP is not referenced. !> ``` *MM* ``` !> MM is INTEGER !> The number of elements in the arrays S (if JOB = 'E' or 'B') !> and/or SEP (if JOB = 'V' or 'B'). MM >= M. !> ``` *M* ``` !> M is INTEGER !> The number of elements of the arrays S and/or SEP actually !> used to store the estimated condition numbers. !> If HOWMNY = 'A', M is set to N. !> ``` *WORK* ``` !> WORK is COMPLEX array, dimension (LDWORK,N+6) !> If JOB = 'E', WORK is not referenced. !> ``` *LDWORK* ``` !> LDWORK is INTEGER !> The leading dimension of the array WORK. !> LDWORK >= 1; and if JOB = 'V' or 'B', LDWORK >= N. !> ``` *RWORK* ``` !> RWORK is REAL array, dimension (N) !> If JOB = 'E', RWORK is not referenced. !> ``` *INFO* ``` !> INFO is INTEGER !> = 0: successful exit !> < 0: if INFO = -i, the i-th argument had an illegal value !> ``` **Author** Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. **Further Details:** ``` !> !> The reciprocal of the condition number of an eigenvalue lambda is !> defined as !> !> S(lambda) = |v**H*u| / (norm(u)*norm(v)) !> !> where u and v are the right and left eigenvectors of T corresponding !> to lambda; v**H denotes the conjugate transpose of v, and norm(u) !> denotes the Euclidean norm. These reciprocal condition numbers always !> lie between zero (very badly conditioned) and one (very well !> conditioned). If n = 1, S(lambda) is defined to be 1. !> !> An approximate error bound for a computed eigenvalue W(i) is given by !> !> EPS * norm(T) / S(i) !> !> where EPS is the machine precision. !> !> The reciprocal of the condition number of the right eigenvector u !> corresponding to lambda is defined as follows. Suppose !> !> T = ( lambda c ) !> ( 0 T22 ) !> !> Then the reciprocal condition number is !> !> SEP( lambda, T22 ) = sigma-min( T22 - lambda*I ) !> !> where sigma-min denotes the smallest singular value. We approximate !> the smallest singular value by the reciprocal of an estimate of the !> one-norm of the inverse of T22 - lambda*I. If n = 1, SEP(1) is !> defined to be abs(T(1,1)). !> !> An approximate error bound for a computed right eigenvector VR(i) !> is given by !> !> EPS * norm(T) / SEP(i) !> ``` Definition at line **246** of file **ctrsna.f**. ### subroutine dtrsna (character job, character howmny, logical, dimension( \* ) select, integer n, double precision, dimension( ldt, \* ) t, integer ldt, double precision, dimension( ldvl, \* ) vl, integer ldvl, double precision, dimension( ldvr, \* ) vr, integer ldvr, double precision, dimension( \* ) s, double precision, dimension( \* ) sep, integer mm, integer m, double precision, dimension( ldwork, \* ) work, integer ldwork, integer, dimension( \* ) iwork, integer info) **DTRSNA** **Purpose:** ``` !> !> DTRSNA estimates reciprocal condition numbers for specified !> eigenvalues and/or right eigenvectors of a real upper !> quasi-triangular matrix T (or of any matrix Q*T*Q**T with Q !> orthogonal). !> !> T must be in Schur canonical form (as returned by DHSEQR), that is, !> block upper triangular with 1-by-1 and 2-by-2 diagonal blocks; each !> 2-by-2 diagonal block has its diagonal elements equal and its !> off-diagonal elements of opposite sign. !> ``` **Parameters** *JOB* ``` !> JOB is CHARACTER*1 !> Specifies whether condition numbers are required for !> eigenvalues (S) or eigenvectors (SEP): !> = 'E': for eigenvalues only (S); !> = 'V': for eigenvectors only (SEP); !> = 'B': for both eigenvalues and eigenvectors (S and SEP). !> ``` *HOWMNY* ``` !> HOWMNY is CHARACTER*1 !> = 'A': compute condition numbers for all eigenpairs; !> = 'S': compute condition numbers for selected eigenpairs !> specified by the array SELECT. !> ``` *SELECT* ``` !> SELECT is LOGICAL array, dimension (N) !> If HOWMNY = 'S', SELECT specifies the eigenpairs for which !> condition numbers are required. To select condition numbers !> for the eigenpair corresponding to a real eigenvalue w(j), !> SELECT(j) must be set to .TRUE.. To select condition numbers !> corresponding to a complex conjugate pair of eigenvalues w(j) !> and w(j+1), either SELECT(j) or SELECT(j+1) or both, must be !> set to .TRUE.. !> If HOWMNY = 'A', SELECT is not referenced. !> ``` *N* ``` !> N is INTEGER !> The order of the matrix T. N >= 0. !> ``` *T* ``` !> T is DOUBLE PRECISION array, dimension (LDT,N) !> The upper quasi-triangular matrix T, in Schur canonical form. !> ``` *LDT* ``` !> LDT is INTEGER !> The leading dimension of the array T. LDT >= max(1,N). !> ``` *VL* ``` !> VL is DOUBLE PRECISION array, dimension (LDVL,M) !> If JOB = 'E' or 'B', VL must contain left eigenvectors of T !> (or of any Q*T*Q**T with Q orthogonal), corresponding to the !> eigenpairs specified by HOWMNY and SELECT. The eigenvectors !> must be stored in consecutive columns of VL, as returned by !> DHSEIN or DTREVC. !> If JOB = 'V', VL is not referenced. !> ``` *LDVL* ``` !> LDVL is INTEGER !> The leading dimension of the array VL. !> LDVL >= 1; and if JOB = 'E' or 'B', LDVL >= N. !> ``` *VR* ``` !> VR is DOUBLE PRECISION array, dimension (LDVR,M) !> If JOB = 'E' or 'B', VR must contain right eigenvectors of T !> (or of any Q*T*Q**T with Q orthogonal), corresponding to the !> eigenpairs specified by HOWMNY and SELECT. The eigenvectors !> must be stored in consecutive columns of VR, as returned by !> DHSEIN or DTREVC. !> If JOB = 'V', VR is not referenced. !> ``` *LDVR* ``` !> LDVR is INTEGER !> The leading dimension of the array VR. !> LDVR >= 1; and if JOB = 'E' or 'B', LDVR >= N. !> ``` *S* ``` !> S is DOUBLE PRECISION array, dimension (MM) !> If JOB = 'E' or 'B', the reciprocal condition numbers of the !> selected eigenvalues, stored in consecutive elements of the !> array. For a complex conjugate pair of eigenvalues two !> consecutive elements of S are set to the same value. Thus !> S(j), SEP(j), and the j-th columns of VL and VR all !> correspond to the same eigenpair (but not in general the !> j-th eigenpair, unless all eigenpairs are selected). !> If JOB = 'V', S is not referenced. !> ``` *SEP* ``` !> SEP is DOUBLE PRECISION array, dimension (MM) !> If JOB = 'V' or 'B', the estimated reciprocal condition !> numbers of the selected eigenvectors, stored in consecutive !> elements of the array. For a complex eigenvector two !> consecutive elements of SEP are set to the same value. If !> the eigenvalues cannot be reordered to compute SEP(j), SEP(j) !> is set to 0; this can only occur when the true value would be !> very small anyway. !> If JOB = 'E', SEP is not referenced. !> ``` *MM* ``` !> MM is INTEGER !> The number of elements in the arrays S (if JOB = 'E' or 'B') !> and/or SEP (if JOB = 'V' or 'B'). MM >= M. !> ``` *M* ``` !> M is INTEGER !> The number of elements of the arrays S and/or SEP actually !> used to store the estimated condition numbers. !> If HOWMNY = 'A', M is set to N. !> ``` *WORK* ``` !> WORK is DOUBLE PRECISION array, dimension (LDWORK,N+6) !> If JOB = 'E', WORK is not referenced. !> ``` *LDWORK* ``` !> LDWORK is INTEGER !> The leading dimension of the array WORK. !> LDWORK >= 1; and if JOB = 'V' or 'B', LDWORK >= N. !> ``` *IWORK* ``` !> IWORK is INTEGER array, dimension (2*(N-1)) !> If JOB = 'E', IWORK is not referenced. !> ``` *INFO* ``` !> INFO is INTEGER !> = 0: successful exit !> < 0: if INFO = -i, the i-th argument had an illegal value !> ``` **Author** Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. **Further Details:** ``` !> !> The reciprocal of the condition number of an eigenvalue lambda is !> defined as !> !> S(lambda) = |v**T*u| / (norm(u)*norm(v)) !> !> where u and v are the right and left eigenvectors of T corresponding !> to lambda; v**T denotes the transpose of v, and norm(u) !> denotes the Euclidean norm. These reciprocal condition numbers always !> lie between zero (very badly conditioned) and one (very well !> conditioned). If n = 1, S(lambda) is defined to be 1. !> !> An approximate error bound for a computed eigenvalue W(i) is given by !> !> EPS * norm(T) / S(i) !> !> where EPS is the machine precision. !> !> The reciprocal of the condition number of the right eigenvector u !> corresponding to lambda is defined as follows. Suppose !> !> T = ( lambda c ) !> ( 0 T22 ) !> !> Then the reciprocal condition number is !> !> SEP( lambda, T22 ) = sigma-min( T22 - lambda*I ) !> !> where sigma-min denotes the smallest singular value. We approximate !> the smallest singular value by the reciprocal of an estimate of the !> one-norm of the inverse of T22 - lambda*I. If n = 1, SEP(1) is !> defined to be abs(T(1,1)). !> !> An approximate error bound for a computed right eigenvector VR(i) !> is given by !> !> EPS * norm(T) / SEP(i) !> ``` Definition at line **262** of file **dtrsna.f**. ### subroutine strsna (character job, character howmny, logical, dimension( \* ) select, integer n, real, dimension( ldt, \* ) t, integer ldt, real, dimension( ldvl, \* ) vl, integer ldvl, real, dimension( ldvr, \* ) vr, integer ldvr, real, dimension( \* ) s, real, dimension( \* ) sep, integer mm, integer m, real, dimension( ldwork, \* ) work, integer ldwork, integer, dimension( \* ) iwork, integer info) **STRSNA** **Purpose:** ``` !> !> STRSNA estimates reciprocal condition numbers for specified !> eigenvalues and/or right eigenvectors of a real upper !> quasi-triangular matrix T (or of any matrix Q*T*Q**T with Q !> orthogonal). !> !> T must be in Schur canonical form (as returned by SHSEQR), that is, !> block upper triangular with 1-by-1 and 2-by-2 diagonal blocks; each !> 2-by-2 diagonal block has its diagonal elements equal and its !> off-diagonal elements of opposite sign. !> ``` **Parameters** *JOB* ``` !> JOB is CHARACTER*1 !> Specifies whether condition numbers are required for !> eigenvalues (S) or eigenvectors (SEP): !> = 'E': for eigenvalues only (S); !> = 'V': for eigenvectors only (SEP); !> = 'B': for both eigenvalues and eigenvectors (S and SEP). !> ``` *HOWMNY* ``` !> HOWMNY is CHARACTER*1 !> = 'A': compute condition numbers for all eigenpairs; !> = 'S': compute condition numbers for selected eigenpairs !> specified by the array SELECT. !> ``` *SELECT* ``` !> SELECT is LOGICAL array, dimension (N) !> If HOWMNY = 'S', SELECT specifies the eigenpairs for which !> condition numbers are required. To select condition numbers !> for the eigenpair corresponding to a real eigenvalue w(j), !> SELECT(j) must be set to .TRUE.. To select condition numbers !> corresponding to a complex conjugate pair of eigenvalues w(j) !> and w(j+1), either SELECT(j) or SELECT(j+1) or both, must be !> set to .TRUE.. !> If HOWMNY = 'A', SELECT is not referenced. !> ``` *N* ``` !> N is INTEGER !> The order of the matrix T. N >= 0. !> ``` *T* ``` !> T is REAL array, dimension (LDT,N) !> The upper quasi-triangular matrix T, in Schur canonical form. !> ``` *LDT* ``` !> LDT is INTEGER !> The leading dimension of the array T. LDT >= max(1,N). !> ``` *VL* ``` !> VL is REAL array, dimension (LDVL,M) !> If JOB = 'E' or 'B', VL must contain left eigenvectors of T !> (or of any Q*T*Q**T with Q orthogonal), corresponding to the !> eigenpairs specified by HOWMNY and SELECT. The eigenvectors !> must be stored in consecutive columns of VL, as returned by !> SHSEIN or STREVC. !> If JOB = 'V', VL is not referenced. !> ``` *LDVL* ``` !> LDVL is INTEGER !> The leading dimension of the array VL. !> LDVL >= 1; and if JOB = 'E' or 'B', LDVL >= N. !> ``` *VR* ``` !> VR is REAL array, dimension (LDVR,M) !> If JOB = 'E' or 'B', VR must contain right eigenvectors of T !> (or of any Q*T*Q**T with Q orthogonal), corresponding to the !> eigenpairs specified by HOWMNY and SELECT. The eigenvectors !> must be stored in consecutive columns of VR, as returned by !> SHSEIN or STREVC. !> If JOB = 'V', VR is not referenced. !> ``` *LDVR* ``` !> LDVR is INTEGER !> The leading dimension of the array VR. !> LDVR >= 1; and if JOB = 'E' or 'B', LDVR >= N. !> ``` *S* ``` !> S is REAL array, dimension (MM) !> If JOB = 'E' or 'B', the reciprocal condition numbers of the !> selected eigenvalues, stored in consecutive elements of the !> array. For a complex conjugate pair of eigenvalues two !> consecutive elements of S are set to the same value. Thus !> S(j), SEP(j), and the j-th columns of VL and VR all !> correspond to the same eigenpair (but not in general the !> j-th eigenpair, unless all eigenpairs are selected). !> If JOB = 'V', S is not referenced. !> ``` *SEP* ``` !> SEP is REAL array, dimension (MM) !> If JOB = 'V' or 'B', the estimated reciprocal condition !> numbers of the selected eigenvectors, stored in consecutive !> elements of the array. For a complex eigenvector two !> consecutive elements of SEP are set to the same value. If !> the eigenvalues cannot be reordered to compute SEP(j), SEP(j) !> is set to 0; this can only occur when the true value would be !> very small anyway. !> If JOB = 'E', SEP is not referenced. !> ``` *MM* ``` !> MM is INTEGER !> The number of elements in the arrays S (if JOB = 'E' or 'B') !> and/or SEP (if JOB = 'V' or 'B'). MM >= M. !> ``` *M* ``` !> M is INTEGER !> The number of elements of the arrays S and/or SEP actually !> used to store the estimated condition numbers. !> If HOWMNY = 'A', M is set to N. !> ``` *WORK* ``` !> WORK is REAL array, dimension (LDWORK,N+6) !> If JOB = 'E', WORK is not referenced. !> ``` *LDWORK* ``` !> LDWORK is INTEGER !> The leading dimension of the array WORK. !> LDWORK >= 1; and if JOB = 'V' or 'B', LDWORK >= N. !> ``` *IWORK* ``` !> IWORK is INTEGER array, dimension (2*(N-1)) !> If JOB = 'E', IWORK is not referenced. !> ``` *INFO* ``` !> INFO is INTEGER !> = 0: successful exit !> < 0: if INFO = -i, the i-th argument had an illegal value !> ``` **Author** Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. **Further Details:** ``` !> !> The reciprocal of the condition number of an eigenvalue lambda is !> defined as !> !> S(lambda) = |v**T*u| / (norm(u)*norm(v)) !> !> where u and v are the right and left eigenvectors of T corresponding !> to lambda; v**T denotes the transpose of v, and norm(u) !> denotes the Euclidean norm. These reciprocal condition numbers always !> lie between zero (very badly conditioned) and one (very well !> conditioned). If n = 1, S(lambda) is defined to be 1. !> !> An approximate error bound for a computed eigenvalue W(i) is given by !> !> EPS * norm(T) / S(i) !> !> where EPS is the machine precision. !> !> The reciprocal of the condition number of the right eigenvector u !> corresponding to lambda is defined as follows. Suppose !> !> T = ( lambda c ) !> ( 0 T22 ) !> !> Then the reciprocal condition number is !> !> SEP( lambda, T22 ) = sigma-min( T22 - lambda*I ) !> !> where sigma-min denotes the smallest singular value. We approximate !> the smallest singular value by the reciprocal of an estimate of the !> one-norm of the inverse of T22 - lambda*I. If n = 1, SEP(1) is !> defined to be abs(T(1,1)). !> !> An approximate error bound for a computed right eigenvector VR(i) !> is given by !> !> EPS * norm(T) / SEP(i) !> ``` Definition at line **262** of file **strsna.f**. ### subroutine ztrsna (character job, character howmny, logical, dimension( \* ) select, integer n, complex\*16, dimension( ldt, \* ) t, integer ldt, complex\*16, dimension( ldvl, \* ) vl, integer ldvl, complex\*16, dimension( ldvr, \* ) vr, integer ldvr, double precision, dimension( \* ) s, double precision, dimension( \* ) sep, integer mm, integer m, complex\*16, dimension( ldwork, \* ) work, integer ldwork, double precision, dimension( \* ) rwork, integer info) **ZTRSNA** **Purpose:** ``` !> !> ZTRSNA estimates reciprocal condition numbers for specified !> eigenvalues and/or right eigenvectors of a complex upper triangular !> matrix T (or of any matrix Q*T*Q**H with Q unitary). !> ``` **Parameters** *JOB* ``` !> JOB is CHARACTER*1 !> Specifies whether condition numbers are required for !> eigenvalues (S) or eigenvectors (SEP): !> = 'E': for eigenvalues only (S); !> = 'V': for eigenvectors only (SEP); !> = 'B': for both eigenvalues and eigenvectors (S and SEP). !> ``` *HOWMNY* ``` !> HOWMNY is CHARACTER*1 !> = 'A': compute condition numbers for all eigenpairs; !> = 'S': compute condition numbers for selected eigenpairs !> specified by the array SELECT. !> ``` *SELECT* ``` !> SELECT is LOGICAL array, dimension (N) !> If HOWMNY = 'S', SELECT specifies the eigenpairs for which !> condition numbers are required. To select condition numbers !> for the j-th eigenpair, SELECT(j) must be set to .TRUE.. !> If HOWMNY = 'A', SELECT is not referenced. !> ``` *N* ``` !> N is INTEGER !> The order of the matrix T. N >= 0. !> ``` *T* ``` !> T is COMPLEX*16 array, dimension (LDT,N) !> The upper triangular matrix T. !> ``` *LDT* ``` !> LDT is INTEGER !> The leading dimension of the array T. LDT >= max(1,N). !> ``` *VL* ``` !> VL is COMPLEX*16 array, dimension (LDVL,M) !> If JOB = 'E' or 'B', VL must contain left eigenvectors of T !> (or of any Q*T*Q**H with Q unitary), corresponding to the !> eigenpairs specified by HOWMNY and SELECT. The eigenvectors !> must be stored in consecutive columns of VL, as returned by !> ZHSEIN or ZTREVC. !> If JOB = 'V', VL is not referenced. !> ``` *LDVL* ``` !> LDVL is INTEGER !> The leading dimension of the array VL. !> LDVL >= 1; and if JOB = 'E' or 'B', LDVL >= N. !> ``` *VR* ``` !> VR is COMPLEX*16 array, dimension (LDVR,M) !> If JOB = 'E' or 'B', VR must contain right eigenvectors of T !> (or of any Q*T*Q**H with Q unitary), corresponding to the !> eigenpairs specified by HOWMNY and SELECT. The eigenvectors !> must be stored in consecutive columns of VR, as returned by !> ZHSEIN or ZTREVC. !> If JOB = 'V', VR is not referenced. !> ``` *LDVR* ``` !> LDVR is INTEGER !> The leading dimension of the array VR. !> LDVR >= 1; and if JOB = 'E' or 'B', LDVR >= N. !> ``` *S* ``` !> S is DOUBLE PRECISION array, dimension (MM) !> If JOB = 'E' or 'B', the reciprocal condition numbers of the !> selected eigenvalues, stored in consecutive elements of the !> array. Thus S(j), SEP(j), and the j-th columns of VL and VR !> all correspond to the same eigenpair (but not in general the !> j-th eigenpair, unless all eigenpairs are selected). !> If JOB = 'V', S is not referenced. !> ``` *SEP* ``` !> SEP is DOUBLE PRECISION array, dimension (MM) !> If JOB = 'V' or 'B', the estimated reciprocal condition !> numbers of the selected eigenvectors, stored in consecutive !> elements of the array. !> If JOB = 'E', SEP is not referenced. !> ``` *MM* ``` !> MM is INTEGER !> The number of elements in the arrays S (if JOB = 'E' or 'B') !> and/or SEP (if JOB = 'V' or 'B'). MM >= M. !> ``` *M* ``` !> M is INTEGER !> The number of elements of the arrays S and/or SEP actually !> used to store the estimated condition numbers. !> If HOWMNY = 'A', M is set to N. !> ``` *WORK* ``` !> WORK is COMPLEX*16 array, dimension (LDWORK,N+6) !> If JOB = 'E', WORK is not referenced. !> ``` *LDWORK* ``` !> LDWORK is INTEGER !> The leading dimension of the array WORK. !> LDWORK >= 1; and if JOB = 'V' or 'B', LDWORK >= N. !> ``` *RWORK* ``` !> RWORK is DOUBLE PRECISION array, dimension (N) !> If JOB = 'E', RWORK is not referenced. !> ``` *INFO* ``` !> INFO is INTEGER !> = 0: successful exit !> < 0: if INFO = -i, the i-th argument had an illegal value !> ``` **Author** Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. **Further Details:** ``` !> !> The reciprocal of the condition number of an eigenvalue lambda is !> defined as !> !> S(lambda) = |v**H*u| / (norm(u)*norm(v)) !> !> where u and v are the right and left eigenvectors of T corresponding !> to lambda; v**H denotes the conjugate transpose of v, and norm(u) !> denotes the Euclidean norm. These reciprocal condition numbers always !> lie between zero (very badly conditioned) and one (very well !> conditioned). If n = 1, S(lambda) is defined to be 1. !> !> An approximate error bound for a computed eigenvalue W(i) is given by !> !> EPS * norm(T) / S(i) !> !> where EPS is the machine precision. !> !> The reciprocal of the condition number of the right eigenvector u !> corresponding to lambda is defined as follows. Suppose !> !> T = ( lambda c ) !> ( 0 T22 ) !> !> Then the reciprocal condition number is !> !> SEP( lambda, T22 ) = sigma-min( T22 - lambda*I ) !> !> where sigma-min denotes the smallest singular value. We approximate !> the smallest singular value by the reciprocal of an estimate of the !> one-norm of the inverse of T22 - lambda*I. If n = 1, SEP(1) is !> defined to be abs(T(1,1)). !> !> An approximate error bound for a computed right eigenvector VR(i) !> is given by !> !> EPS * norm(T) / SEP(i) !> ``` Definition at line **246** of file **ztrsna.f**. ## Author Generated automatically by Doxygen for LAPACK from the source code.