.TH "trsna" 3 "Wed Feb 7 2024 11:30:40" "Version 3.12.0" "LAPACK" \" -*- nroff -*- .ad l .nh .SH NAME trsna \- trsna: eig condition numbers .SH SYNOPSIS .br .PP .SS "Functions" .in +1c .ti -1c .RI "subroutine \fBctrsna\fP (job, howmny, select, n, t, ldt, vl, ldvl, vr, ldvr, s, sep, mm, m, work, ldwork, rwork, info)" .br .RI "\fBCTRSNA\fP " .ti -1c .RI "subroutine \fBdtrsna\fP (job, howmny, select, n, t, ldt, vl, ldvl, vr, ldvr, s, sep, mm, m, work, ldwork, iwork, info)" .br .RI "\fBDTRSNA\fP " .ti -1c .RI "subroutine \fBstrsna\fP (job, howmny, select, n, t, ldt, vl, ldvl, vr, ldvr, s, sep, mm, m, work, ldwork, iwork, info)" .br .RI "\fBSTRSNA\fP " .ti -1c .RI "subroutine \fBztrsna\fP (job, howmny, select, n, t, ldt, vl, ldvl, vr, ldvr, s, sep, mm, m, work, ldwork, rwork, info)" .br .RI "\fBZTRSNA\fP " .in -1c .SH "Detailed Description" .PP .SH "Function Documentation" .PP .SS "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)" .PP \fBCTRSNA\fP .PP \fBPurpose:\fP .RS 4 .PP .nf 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)\&. .fi .PP .RE .PP \fBParameters\fP .RS 4 \fIJOB\fP .PP .nf 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)\&. .fi .PP .br \fIHOWMNY\fP .PP .nf HOWMNY is CHARACTER*1 = 'A': compute condition numbers for all eigenpairs; = 'S': compute condition numbers for selected eigenpairs specified by the array SELECT\&. .fi .PP .br \fISELECT\fP .PP .nf 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\&. .fi .PP .br \fIN\fP .PP .nf N is INTEGER The order of the matrix T\&. N >= 0\&. .fi .PP .br \fIT\fP .PP .nf T is COMPLEX array, dimension (LDT,N) The upper triangular matrix T\&. .fi .PP .br \fILDT\fP .PP .nf LDT is INTEGER The leading dimension of the array T\&. LDT >= max(1,N)\&. .fi .PP .br \fIVL\fP .PP .nf 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\&. .fi .PP .br \fILDVL\fP .PP .nf LDVL is INTEGER The leading dimension of the array VL\&. LDVL >= 1; and if JOB = 'E' or 'B', LDVL >= N\&. .fi .PP .br \fIVR\fP .PP .nf 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\&. .fi .PP .br \fILDVR\fP .PP .nf LDVR is INTEGER The leading dimension of the array VR\&. LDVR >= 1; and if JOB = 'E' or 'B', LDVR >= N\&. .fi .PP .br \fIS\fP .PP .nf 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\&. .fi .PP .br \fISEP\fP .PP .nf 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\&. .fi .PP .br \fIMM\fP .PP .nf 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\&. .fi .PP .br \fIM\fP .PP .nf 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\&. .fi .PP .br \fIWORK\fP .PP .nf WORK is COMPLEX array, dimension (LDWORK,N+6) If JOB = 'E', WORK is not referenced\&. .fi .PP .br \fILDWORK\fP .PP .nf LDWORK is INTEGER The leading dimension of the array WORK\&. LDWORK >= 1; and if JOB = 'V' or 'B', LDWORK >= N\&. .fi .PP .br \fIRWORK\fP .PP .nf RWORK is REAL array, dimension (N) If JOB = 'E', RWORK is not referenced\&. .fi .PP .br \fIINFO\fP .PP .nf INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value .fi .PP .RE .PP \fBAuthor\fP .RS 4 Univ\&. of Tennessee .PP Univ\&. of California Berkeley .PP Univ\&. of Colorado Denver .PP NAG Ltd\&. .RE .PP \fBFurther Details:\fP .RS 4 .PP .nf 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) .fi .PP .RE .PP .SS "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)" .PP \fBDTRSNA\fP .PP \fBPurpose:\fP .RS 4 .PP .nf 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\&. .fi .PP .RE .PP \fBParameters\fP .RS 4 \fIJOB\fP .PP .nf 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)\&. .fi .PP .br \fIHOWMNY\fP .PP .nf HOWMNY is CHARACTER*1 = 'A': compute condition numbers for all eigenpairs; = 'S': compute condition numbers for selected eigenpairs specified by the array SELECT\&. .fi .PP .br \fISELECT\fP .PP .nf 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\&. .fi .PP .br \fIN\fP .PP .nf N is INTEGER The order of the matrix T\&. N >= 0\&. .fi .PP .br \fIT\fP .PP .nf T is DOUBLE PRECISION array, dimension (LDT,N) The upper quasi-triangular matrix T, in Schur canonical form\&. .fi .PP .br \fILDT\fP .PP .nf LDT is INTEGER The leading dimension of the array T\&. LDT >= max(1,N)\&. .fi .PP .br \fIVL\fP .PP .nf 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\&. .fi .PP .br \fILDVL\fP .PP .nf LDVL is INTEGER The leading dimension of the array VL\&. LDVL >= 1; and if JOB = 'E' or 'B', LDVL >= N\&. .fi .PP .br \fIVR\fP .PP .nf 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\&. .fi .PP .br \fILDVR\fP .PP .nf LDVR is INTEGER The leading dimension of the array VR\&. LDVR >= 1; and if JOB = 'E' or 'B', LDVR >= N\&. .fi .PP .br \fIS\fP .PP .nf 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\&. .fi .PP .br \fISEP\fP .PP .nf 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\&. .fi .PP .br \fIMM\fP .PP .nf 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\&. .fi .PP .br \fIM\fP .PP .nf 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\&. .fi .PP .br \fIWORK\fP .PP .nf WORK is DOUBLE PRECISION array, dimension (LDWORK,N+6) If JOB = 'E', WORK is not referenced\&. .fi .PP .br \fILDWORK\fP .PP .nf LDWORK is INTEGER The leading dimension of the array WORK\&. LDWORK >= 1; and if JOB = 'V' or 'B', LDWORK >= N\&. .fi .PP .br \fIIWORK\fP .PP .nf IWORK is INTEGER array, dimension (2*(N-1)) If JOB = 'E', IWORK is not referenced\&. .fi .PP .br \fIINFO\fP .PP .nf INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value .fi .PP .RE .PP \fBAuthor\fP .RS 4 Univ\&. of Tennessee .PP Univ\&. of California Berkeley .PP Univ\&. of Colorado Denver .PP NAG Ltd\&. .RE .PP \fBFurther Details:\fP .RS 4 .PP .nf 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) .fi .PP .RE .PP .SS "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)" .PP \fBSTRSNA\fP .PP \fBPurpose:\fP .RS 4 .PP .nf 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\&. .fi .PP .RE .PP \fBParameters\fP .RS 4 \fIJOB\fP .PP .nf 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)\&. .fi .PP .br \fIHOWMNY\fP .PP .nf HOWMNY is CHARACTER*1 = 'A': compute condition numbers for all eigenpairs; = 'S': compute condition numbers for selected eigenpairs specified by the array SELECT\&. .fi .PP .br \fISELECT\fP .PP .nf 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\&. .fi .PP .br \fIN\fP .PP .nf N is INTEGER The order of the matrix T\&. N >= 0\&. .fi .PP .br \fIT\fP .PP .nf T is REAL array, dimension (LDT,N) The upper quasi-triangular matrix T, in Schur canonical form\&. .fi .PP .br \fILDT\fP .PP .nf LDT is INTEGER The leading dimension of the array T\&. LDT >= max(1,N)\&. .fi .PP .br \fIVL\fP .PP .nf 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\&. .fi .PP .br \fILDVL\fP .PP .nf LDVL is INTEGER The leading dimension of the array VL\&. LDVL >= 1; and if JOB = 'E' or 'B', LDVL >= N\&. .fi .PP .br \fIVR\fP .PP .nf 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\&. .fi .PP .br \fILDVR\fP .PP .nf LDVR is INTEGER The leading dimension of the array VR\&. LDVR >= 1; and if JOB = 'E' or 'B', LDVR >= N\&. .fi .PP .br \fIS\fP .PP .nf 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\&. .fi .PP .br \fISEP\fP .PP .nf 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\&. .fi .PP .br \fIMM\fP .PP .nf 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\&. .fi .PP .br \fIM\fP .PP .nf 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\&. .fi .PP .br \fIWORK\fP .PP .nf WORK is REAL array, dimension (LDWORK,N+6) If JOB = 'E', WORK is not referenced\&. .fi .PP .br \fILDWORK\fP .PP .nf LDWORK is INTEGER The leading dimension of the array WORK\&. LDWORK >= 1; and if JOB = 'V' or 'B', LDWORK >= N\&. .fi .PP .br \fIIWORK\fP .PP .nf IWORK is INTEGER array, dimension (2*(N-1)) If JOB = 'E', IWORK is not referenced\&. .fi .PP .br \fIINFO\fP .PP .nf INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value .fi .PP .RE .PP \fBAuthor\fP .RS 4 Univ\&. of Tennessee .PP Univ\&. of California Berkeley .PP Univ\&. of Colorado Denver .PP NAG Ltd\&. .RE .PP \fBFurther Details:\fP .RS 4 .PP .nf 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) .fi .PP .RE .PP .SS "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)" .PP \fBZTRSNA\fP .PP \fBPurpose:\fP .RS 4 .PP .nf 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)\&. .fi .PP .RE .PP \fBParameters\fP .RS 4 \fIJOB\fP .PP .nf 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)\&. .fi .PP .br \fIHOWMNY\fP .PP .nf HOWMNY is CHARACTER*1 = 'A': compute condition numbers for all eigenpairs; = 'S': compute condition numbers for selected eigenpairs specified by the array SELECT\&. .fi .PP .br \fISELECT\fP .PP .nf 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\&. .fi .PP .br \fIN\fP .PP .nf N is INTEGER The order of the matrix T\&. N >= 0\&. .fi .PP .br \fIT\fP .PP .nf T is COMPLEX*16 array, dimension (LDT,N) The upper triangular matrix T\&. .fi .PP .br \fILDT\fP .PP .nf LDT is INTEGER The leading dimension of the array T\&. LDT >= max(1,N)\&. .fi .PP .br \fIVL\fP .PP .nf 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\&. .fi .PP .br \fILDVL\fP .PP .nf LDVL is INTEGER The leading dimension of the array VL\&. LDVL >= 1; and if JOB = 'E' or 'B', LDVL >= N\&. .fi .PP .br \fIVR\fP .PP .nf 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\&. .fi .PP .br \fILDVR\fP .PP .nf LDVR is INTEGER The leading dimension of the array VR\&. LDVR >= 1; and if JOB = 'E' or 'B', LDVR >= N\&. .fi .PP .br \fIS\fP .PP .nf 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\&. .fi .PP .br \fISEP\fP .PP .nf 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\&. .fi .PP .br \fIMM\fP .PP .nf 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\&. .fi .PP .br \fIM\fP .PP .nf 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\&. .fi .PP .br \fIWORK\fP .PP .nf WORK is COMPLEX*16 array, dimension (LDWORK,N+6) If JOB = 'E', WORK is not referenced\&. .fi .PP .br \fILDWORK\fP .PP .nf LDWORK is INTEGER The leading dimension of the array WORK\&. LDWORK >= 1; and if JOB = 'V' or 'B', LDWORK >= N\&. .fi .PP .br \fIRWORK\fP .PP .nf RWORK is DOUBLE PRECISION array, dimension (N) If JOB = 'E', RWORK is not referenced\&. .fi .PP .br \fIINFO\fP .PP .nf INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value .fi .PP .RE .PP \fBAuthor\fP .RS 4 Univ\&. of Tennessee .PP Univ\&. of California Berkeley .PP Univ\&. of Colorado Denver .PP NAG Ltd\&. .RE .PP \fBFurther Details:\fP .RS 4 .PP .nf 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) .fi .PP .RE .PP .SH "Author" .PP Generated automatically by Doxygen for LAPACK from the source code\&.