tgsna(3)

📦
version 3.12.1 · 9 versions
, in arch rolling

✨ NAME ✨

okay bestie, this is tgsna(3) — the page for tgsna, short for tgsna: reciprocal cond est. That's the whole name vibe: reciprocal condition estimation for the generalized eigenvalue problem. Slay. 💅

🦋 SYNOPSIS 🦋

🧚 Functions 🧚

Here are the four subroutine besties, exactly as they show up:

subroutine ctgsna (job, howmny, select, n, a, lda, b, ldb, vl, ldvl, vr,
    ldvr, s, dif, mm, m, work, lwork, iwork, info)
    CTGSNA
subroutine dtgsna (job, howmny, select, n, a, lda, b, ldb, vl, ldvl, vr,
    ldvr, s, dif, mm, m, work, lwork, iwork, info)
    DTGSNA
subroutine stgsna (job, howmny, select, n, a, lda, b, ldb, vl, ldvl, vr,
    ldvr, s, dif, mm, m, work, lwork, iwork, info)
    STGSNA
subroutine ztgsna (job, howmny, select, n, a, lda, b, ldb, vl, ldvl, vr,
    ldvr, s, dif, mm, m, work, lwork, iwork, info)
    ZTGSNA

🌸 Detailed Description 🌸

This is the umbrella section, and under it is Function Documentation — where the actual ctgsna tea lives. ☕✨

🎀 Function Documentation 🎀

💖 subroutine ctgsna 💖

subroutine ctgsna (character job, character howmny, logical, dimension( * )
    select, integer n, complex, dimension( lda, * ) a, integer lda, complex,
    dimension( ldb, * ) b, integer ldb, complex, dimension( ldvl, * ) vl,
    integer ldvl, complex, dimension( ldvr, * ) vr, integer ldvr, real,
    dimension( * ) s, real, dimension( * ) dif, integer mm, integer m,
    complex, dimension( * ) work, integer lwork, integer, dimension( * )
    iwork, integer info)
    CTGSNA

🌈 Purpose 🌈

CTGSNA estimates reciprocal condition numbers for specified eigenvalues and/or eigenvectors of a matrix pair (A, B).

Bestie, the pair (A, B) must be in generalized Schur canonical form — that means both A and B are upper triangular. No cap. 🫶

🍓 Parameters 🍓

  • JOB — JOB is CHARACTER*1. It says whether you want condition numbers for eigenvalues (S) or eigenvectors (DIF):
    • = 'E': for eigenvalues only (S);
    • = 'V': for eigenvectors only (DIF);
    • = 'B': for both eigenvalues and eigenvectors (S and DIF).
  • 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 corresponding j-th eigenvalue and/or eigenvector, SELECT(j) must be set to .TRUE.. If HOWMNY = 'A', SELECT is not referenced.
  • N — N is INTEGER. The order of the square matrix pair (A, B). N >= 0.
  • A — A is COMPLEX array, dimension (LDA,N). The upper triangular matrix A in the pair (A,B).
  • LDA — LDA is INTEGER. The leading dimension of the array A. LDA >= max(1,N).
  • B — B is COMPLEX array, dimension (LDB,N). The upper triangular matrix B in the pair (A, B).
  • LDB — LDB is INTEGER. The leading dimension of the array B. LDB >= max(1,N).
  • VL — VL is COMPLEX array, dimension (LDVL,M). If JOB = 'E' or 'B', VL must contain left eigenvectors of (A, B), corresponding to the eigenpairs specified by HOWMNY and SELECT. The eigenvectors must be stored in consecutive columns of VL, as returned by CTGEVC. 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 (A, B), corresponding to the eigenpairs specified by HOWMNY and SELECT. The eigenvectors must be stored in consecutive columns of VR, as returned by CTGEVC. If JOB = 'V', VR is not referenced.
  • LDVR — LDVR is INTEGER. The leading dimension of the array VR. LDVR >= 1; 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. If JOB = 'V', S is not referenced.
  • DIF — DIF 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 the eigenvalues cannot be reordered to compute DIF(j), DIF(j) is set to 0; this can only occur when the true value would be very small anyway. For each eigenvalue/vector specified by SELECT, DIF stores a Frobenius norm-based estimate of Difl. If JOB = 'E', DIF is not referenced.
  • MM — MM is INTEGER. The number of elements in the arrays S and DIF. MM >= M.
  • M — M is INTEGER. The number of elements of the arrays S and DIF used to store the specified condition numbers; for each selected eigenvalue one element is used. If HOWMNY = 'A', M is set to N.
  • WORK — WORK is COMPLEX array, dimension (MAX(1,LWORK)). On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
  • LWORK — LWORK is INTEGER. The dimension of the array WORK. LWORK >= max(1,N). If JOB = 'V' or 'B', LWORK >= max(1,2*N*N).
  • IWORK — IWORK is INTEGER array, dimension (N+2). 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 the i-th generalized eigenvalue w = (a, b) is defined as

S(I) = (|v**HAu|**2 + |v**HBu|**2)**(1/2) / (norm(u)*norm(v))

where u and v are the right and left eigenvectors of (A, B) corresponding to w; |z| denotes the absolute value of the complex number, and norm(u) denotes the 2-norm of the vector u. The pair (a, b) corresponds to an eigenvalue w = a/b (= v**HAu/v**HBu) of the matrix pair (A, B). If both a and b equal zero, then (A,B) is singular and S(I) = -1 is returned. 💀

An approximate error bound on the chordal distance between the i-th computed generalized eigenvalue w and the corresponding exact eigenvalue lambda is

chord(w, lambda) <=   EPS * norm(A, B) / S(I),

where EPS is the machine precision.

The reciprocal of the condition number of the right eigenvector u and left eigenvector v corresponding to the generalized eigenvalue w is defined as follows. Suppose

         (A, B) = ( a   *  ) ( b  *  )  1
                  ( 0  A22 ),( 0 B22 )  n-1
                    1  n-1     1 n-1

Then the reciprocal condition number DIF(I) is

Difl[(a, b), (A22, B22)]  = sigma-min( Zl )

where sigma-min(Zl) denotes the smallest singular value of

Zl = [ kron(a, In-1) -kron(1, A22) ]
     [ kron(b, In-1) -kron(1, B22) ].

Here In-1 is the identity matrix of size n-1 and X**H is the conjugate transpose of X. kron(X, Y) is the Kronecker product between the matrices X and Y.

We approximate the smallest singular value of Zl with an upper bound. This is done by CLATDF.

An approximate error bound for a computed eigenvector VL(i) or VR(i) is given by

            EPS * norm(A, B) / DIF(i).

See ref. [2-3] for more details and further references. 🌟

🧚 Contributors 🧚

Bo Kagstrom and Peter Poromaa, Department of Computing Science, Umea University, S-901 87 Umea, Sweden.

💜 References 💜

  • [1] B. Kagstrom; A Direct Method for Reordering Eigenvalues in the Generalized Real Schur Form of a Regular Matrix Pair (A, B), in M.S. Moonen et al (eds), Linear Algebra for Large Scale and Real-Time Applications, Kluwer Academic Publ. 1993, pp 195-218.
  • [2] B. Kagstrom and P. Poromaa; Computing Eigenspaces with Specified Eigenvalues of a Regular Matrix Pair (A, B) and Condition Estimation: Theory, Algorithms and Software, Report UMINF - 94.04, Department of Computing Science, Umea University, S-901 87 Umea, Sweden, 1994. Also as LAPACK Working Note 87. To appear in Numerical Algorithms, 1996.
  • [3] B. Kagstrom and P. Poromaa, LAPACK-Style Algorithms and Software for Solving the Generalized Sylvester Equation and Estimating the Separation between Regular Matrix Pairs, Report UMINF - 93.23, Department of Computing Science, Umea University, S-901 87 Umea, Sweden, December 1993, Revised April 1994, Also as LAPACK Working Note 75. To appear in ACM Trans. on Math. Software, Vol 22, No 1, 1996.

Definition at line 308 of file ctgsna.f.

   subroutine dtgsna (character job, character howmny, logical, dimension( * )
       select, integer n, double precision, dimension( lda, * ) a, integer lda,
       double precision, dimension( ldb, * ) b, integer ldb, double precision,
       dimension( ldvl, * ) vl, integer ldvl, double precision, dimension(
       ldvr, * ) vr, integer ldvr, double precision, dimension( * ) s, double
       precision, dimension( * ) dif, integer mm, integer m, double precision,
       dimension( * ) work, integer lwork, integer, dimension( * ) iwork,
       integer info)

DTGSNA

💖 Purpose 💖

Okay bestie, gather round: DTGSNA estimates the reciprocal condition numbers for specified eigenvalues and/or eigenvectors of a matrix pair (A, B) that’s in generalized real Schur canonical form — or of any matrix pair (Q*A*Z**T, Q*B*Z**T) with orthogonal matrices Q and Z, where Z**T denotes the transpose of Z. It’s giving linear algebra main character energy. ✨

The pair (A, B) must be in generalized real Schur form (as returned by DGGES), which means A is block upper triangular with 1-by-1 and 2-by-2 diagonal blocks, and B is upper triangular. No cap. 🫶

🌸 Parameters 🌸

  • JOB — CHARACTER*1. Specifies whether condition numbers are required for eigenvalues (S) or eigenvectors (DIF):
    • = 'E': for eigenvalues only (S);
    • = 'V': for eigenvectors only (DIF);
    • = 'B': for both eigenvalues and eigenvectors (S and DIF).
  • HOWMNY — CHARACTER*1. This is the selection vibe:
    • = 'A': compute condition numbers for all eigenpairs;
    • = 'S': compute condition numbers for selected eigenpairs specified by the array SELECT.
  • SELECT — 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 — INTEGER. The order of the square matrix pair (A, B). N >= 0.
  • A — DOUBLE PRECISION array, dimension (LDA,N). The upper quasi-triangular matrix A in the pair (A,B).
  • LDA — INTEGER. The leading dimension of the array A. LDA >= max(1,N).
  • B — DOUBLE PRECISION array, dimension (LDB,N). The upper triangular matrix B in the pair (A,B).
  • LDB — INTEGER. The leading dimension of the array B. LDB >= max(1,N).
  • VL — DOUBLE PRECISION array, dimension (LDVL,M). If JOB = 'E' or 'B', VL must contain left eigenvectors of (A, B) corresponding to the eigenpairs specified by HOWMNY and SELECT. The eigenvectors must be stored in consecutive columns of VL, as returned by DTGEVC. If JOB = 'V', VL is not referenced.
  • LDVL — INTEGER. The leading dimension of the array VL. LDVL >= 1. If JOB = 'E' or 'B', LDVL >= N.
  • VR — DOUBLE PRECISION array, dimension (LDVR,M). If JOB = 'E' or 'B', VR must contain right eigenvectors of (A, B) corresponding to the eigenpairs specified by HOWMNY and SELECT. The eigenvectors must be stored in consecutive columns of VR, as returned by DTGEVC. If JOB = 'V', VR is not referenced.
  • LDVR — INTEGER. The leading dimension of the array VR. LDVR >= 1. If JOB = 'E' or 'B', LDVR >= N.
  • S — DOUBLE PRECISION array, dimension (MM). If JOB = 'E' or 'B', this holds 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. So S(j), DIF(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.
  • DIF — DOUBLE PRECISION array, dimension (MM). If JOB = 'V' or 'B', this holds the estimated reciprocal condition numbers of the selected eigenvectors, stored in consecutive elements of the array. For a complex eigenvector, two consecutive elements of DIF are set to the same value. If the eigenvalues cannot be reordered to compute DIF(j), then DIF(j) is set to 0; this can only occur when the true value would be very small anyway. If JOB = 'E', DIF is not referenced.
  • MM — INTEGER. The number of elements in the arrays S and DIF. MM >= M.
  • M — INTEGER. The number of elements of the arrays S and DIF used to store the specified condition numbers. For each selected real eigenvalue, one element is used; for each selected complex conjugate pair of eigenvalues, two elements are used. If HOWMNY = 'A', M is set to N.
  • WORK — DOUBLE PRECISION array, dimension (MAX(1,LWORK)). On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
  • LWORK — INTEGER. The dimension of the array WORK. LWORK >= max(1,N). If JOB = 'V' or 'B', LWORK >= 2*N*(N+2)+16. 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. So yes, you can lowkey ask it how much space it needs first. 🫶
  • IWORK — INTEGER array, dimension (N + 6). If JOB = 'E', IWORK is not referenced.
  • INFO — 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 🦋

This section is literally empty in the original, bestie — no extra deets here. 🍵

Okay so bestie, let’s talk about the reciprocal of the condition number of a generalized eigenvalue w = (a, b) — it’s defined as:

S(w) = (|u**TAv|**2 + |u**TBv|**2)**(1/2) / (norm(u)*norm(v))

Here u and v are the left and right eigenvectors of (A, B) corresponding to w; |z| denotes the absolute value of the complex number, and norm(u) denotes the 2-norm of the vector u. The pair (a, b) corresponds to an eigenvalue w = a/b (= u**TAv/u**TBv) of the matrix pair (A, B). If both a and b equal zero, then (A B) is singular and S(I) = -1 is returned. 🫠

An approximate error bound on the chordal distance between the i-th computed generalized eigenvalue w and the corresponding exact eigenvalue lambda is:

chord(w, lambda) <= EPS * norm(A, B) / S(I)

where EPS is the machine precision. Literally the blueprint.

The reciprocal of the condition number DIF(i) of right eigenvector u and left eigenvector v corresponding to the generalized eigenvalue w is defined as follows:

a) If the i-th eigenvalue w = (a,b) is real

Suppose U and V are orthogonal transformations such that:

U**T*(A, B)*V  = (S, T) = ( a   *  ) ( b  *  )  1
                          ( 0  S22 ),( 0 T22 )  n-1
                            1  n-1     1 n-1

Then the reciprocal condition number DIF(i) is:

Difl((a, b), (S22, T22)) = sigma-min( Zl ),

where sigma-min(Zl) denotes the smallest singular value of the 2(n-1)-by-2(n-1) matrix:

Zl = [ kron(a, In-1)  -kron(1, S22) ]
     [ kron(b, In-1)  -kron(1, T22) ] .

Here In-1 is the identity matrix of size n-1. kron(X, Y) is the Kronecker product between the matrices X and Y.

Note that if the default method for computing DIF(i) is wanted (see DLATDF), then the parameter DIFDRI (see below) should be changed from 3 to 4 (routine DLATDF(IJOB = 2 will be used)). See DTGSYL` for more details.

b) If the i-th and (i+1)-th eigenvalues are complex conjugate pair,

Suppose U and V are orthogonal transformations such that:

U**T*(A, B)*V = (S, T) = ( S11  *   ) ( T11  *  )  2
                         ( 0    S22 ),( 0    T22) n-2
                           2    n-2     2    n-2

and (S11, T11) corresponds to the complex conjugate eigenvalue pair (w, conjg(w)). There exist unitary matrices U1 and V1 such that:

U1**T*S11*V1 = ( s11 s12 ) and U1**T*T11*V1 = ( t11 t12 )
               (  0  s22 )                    (  0  t22 )

where the generalized eigenvalues w = s11/t11 and conjg(w) = s22/t22.

Then the reciprocal condition number DIF(i) is bounded by:

min( d1, max( 1, |real(s11)/real(s22)| )*d2 )

where, d1 = Difl((s11, t11), (s22, t22)) = sigma-min(Z1), where Z1 is the complex 2-by-2 matrix:

Z1 =  [ s11  -s22 ]
      [ t11  -t22 ],

This is done by computing (using real arithmetic) the roots of the characteristical polynomial det(Z1**T * Z1 - lambda I), where Z1**T denotes the transpose of Z1 and det(X) denotes the determinant of X.

and d2 is an upper bound on Difl((S11, T11), (S22, T22)), i.e. an upper bound on sigma-min(Z2), where Z2 is (2n-2)-by-(2n-2):

Z2 = [ kron(S11**T, In-2)  -kron(I2, S22) ]
     [ kron(T11**T, In-2)  -kron(I2, T22) ]

Note that if the default method for computing DIF is wanted (see DLATDF), then the parameter DIFDRI (see below) should be changed from 3 to 4 (routine DLATDF(IJOB = 2 will be used)). See DTGSYL` for more details.

For each eigenvalue/vector specified by SELECT, DIF stores a Frobenius norm-based estimate of Difl.

An approximate error bound for the i-th computed eigenvector VL(i) or VR(i) is given by:

EPS * norm(A, B) / DIF(i).

See ref. [2-3] for more details and further references. 💅

Contributors

Shoutout to the legends: Bo Kagstrom and Peter Poromaa, Department of Computing Science, Umea University, S-901 87 Umea, Sweden. 👑

References

  1. [1] B. Kagstrom; A Direct Method for Reordering Eigenvalues in the Generalized Real Schur Form of a Regular Matrix Pair (A, B), in M.S. Moonen et al (eds), Linear Algebra for Large Scale and Real-Time Applications, Kluwer Academic Publ. 1993, pp 195-218.
  2. [2] B. Kagstrom and P. Poromaa; Computing Eigenspaces with Specified Eigenvalues of a Regular Matrix Pair (A, B) and Condition Estimation: Theory, Algorithms and Software, Report UMINF - 94.04, Department of Computing Science, Umea University, S-901 87 Umea, Sweden, 1994. Also as LAPACK Working Note 87. To appear in Numerical Algorithms, 1996.
  3. [3] B. Kagstrom and P. Poromaa, LAPACK-Style Algorithms and Software for Solving the Generalized Sylvester Equation and Estimating the Separation between Regular Matrix Pairs, Report UMINF - 93.23, Department of Computing Science, Umea University, S-901 87 Umea, Sweden, December 1993, Revised April 1994, Also as LAPACK Working Note 75. To appear in ACM Trans. on Math. Software, Vol 22, No 1, 1996.

Definition at line 378 of file dtgsna.f.

subroutine stgsna

subroutine stgsna (character job, character howmny, logical, dimension( * )
       select, integer n, real, dimension( lda, * ) a, integer lda, real,
       dimension( ldb, * ) b, integer ldb, real, dimension( ldvl, * ) vl,
       integer ldvl, real, dimension( ldvr, * ) vr, integer ldvr, real,
       dimension( * ) s, real, dimension( * ) dif, integer mm, integer m, real,
       dimension( * ) work, integer lwork, integer, dimension( * ) iwork,
       integer info)

STGSNA

Purpose:

STGSNA estimates reciprocal condition numbers for specified eigenvalues and/or eigenvectors of a matrix pair (A, B) in generalized real Schur canonical form (or of any matrix pair (Q*A*Z**T, Q*B*Z**T) with orthogonal matrices Q and Z, where Z**T denotes the transpose of Z). ✨

(A, B) must be in generalized real Schur form (as returned by SGGES), i.e. A is block upper triangular with 1-by-1 and 2-by-2 diagonal blocks. B is upper triangular. No cap.

Parameters

JOB

JOB is CHARACTER*1. It specifies whether condition numbers are required for eigenvalues (S) or eigenvectors (DIF). Pick your fighter:

  • = 'E': for eigenvalues only (S);
  • = 'V': for eigenvectors only (DIF);
  • = 'B': for both eigenvalues and eigenvectors (S and DIF).

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. Zero notes.

N

N is INTEGER. The order of the square matrix pair (A, B). N >= 0.

A

A is REAL array, dimension (LDA,N). The upper quasi-triangular matrix A in the pair (A,B).

LDA

LDA is INTEGER. The leading dimension of the array A. LDA >= max(1,N).

B

B is REAL array, dimension (LDB,N). The upper triangular matrix B in the pair (A,B).

LDB

LDB is INTEGER. The leading dimension of the array B. LDB >= max(1,N).

VL

VL is REAL array, dimension (LDVL,M).

If JOB = 'E' or 'B', VL must contain left eigenvectors of (A, B), corresponding to the eigenpairs specified by HOWMNY and SELECT. The eigenvectors must be stored in consecutive columns of VL, as returned by STGEVC.

If JOB = 'V', VL is not referenced.

LDVL

LDVL is INTEGER. The leading dimension of the array VL. LDVL >= 1. 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 (A, B), corresponding to the eigenpairs specified by HOWMNY and SELECT. The eigenvectors must be stored in consecutive columns ov VR, as returned by STGEVC.

If JOB = 'V', VR is not referenced.

LDVR

LDVR is INTEGER. The leading dimension of the array VR. LDVR >= 1. If JOB = 'E' or 'B', LDVR >= N.

S

  • 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), DIF(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.
  • DIF — DIF 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 DIF are set to the same value. If the eigenvalues cannot be reordered to compute DIF(j), DIF(j) is set to 0; this can only occur when the true value would be very small anyway. If JOB = 'E', DIF is not referenced.
  • MM — MM is INTEGER. The number of elements in the arrays S and DIF. MM >= M.
  • M — M is INTEGER. The number of elements of the arrays S and DIF used to store the specified condition numbers; for each selected real eigenvalue one element is used, and for each selected complex conjugate pair of eigenvalues, two elements are used. If HOWMNY = 'A', M is set to N.
  • WORK — WORK is REAL array, dimension (MAX(1,LWORK)). On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
  • LWORK — LWORK is INTEGER. The dimension of the array WORK. LWORK >= max(1,N). If JOB = 'V' or 'B' LWORK >= 2*N*(N+2)+16.

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.

  • IWORK — IWORK is INTEGER array, dimension (N + 6). 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 🧮✨

Okay bestie, this is the math part — stay with me, it's giving iconic linear algebra 💅

The reciprocal of the condition number of a generalized eigenvalue w = (a, b) is defined as

       S(w) = (|u**TAv|**2 + |u**TBv|**2)**(1/2) / (norm(u)*norm(v))

where u and v are the left and right eigenvectors of (A, B) corresponding to w; |z| denotes the absolute value of the complex number, and norm(u) denotes the 2-norm of the vector u. The pair (a, b) corresponds to an eigenvalue w = a/b (= u**TAv/u**TBv) of the matrix pair (A, B). If both a and b equal zero, then (A B) is singular and S(I) = -1 is returned.

An approximate error bound on the chordal distance between the i-th computed generalized eigenvalue w and the corresponding exact eigenvalue lambda is

       chord(w, lambda) <= EPS * norm(A, B) / S(I)

where EPS is the machine precision.

The reciprocal of the condition number DIF(i) of right eigenvector u and left eigenvector v corresponding to the generalized eigenvalue w is defined as follows:

a) If the i-th eigenvalue w = (a,b) is real

Suppose U and V are orthogonal transformations such that

              U**T*(A, B)*V  = (S, T) = ( a   *  ) ( b  *  )  1
                                        ( 0  S22 ),( 0 T22 )  n-1
                                          1  n-1     1 n-1

Then the reciprocal condition number DIF(i) is

                Difl((a, b), (S22, T22)) = sigma-min( Zl ),

where sigma-min(Zl) denotes the smallest singular value of the 2(n-1)-by-2(n-1) matrix

         Zl = [ kron(a, In-1)  -kron(1, S22) ]
              [ kron(b, In-1)  -kron(1, T22) ] .

Here In-1 is the identity matrix of size n-1. kron(X, Y) is the Kronecker product between the matrices X and Y.

Note that if the default method for computing DIF(i) is wanted (see SLATDF), then the parameter DIFDRI (see below) should be changed from 3 to 4 (routine SLATDF(IJOB = 2 will be used)). See STGSYL for more details.

b) If the i-th and (i+1)-th eigenvalues are complex conjugate pair,

Suppose U and V are orthogonal transformations such that

              U**T*(A, B)*V = (S, T) = ( S11  *   ) ( T11  *  )  2
                                       ( 0    S22 ),( 0    T22) n-2
                                         2    n-2     2    n-2

and (S11, T11) corresponds to the complex conjugate eigenvalue pair (w, conjg(w)). There exist unitary matrices U1 and V1 such that

       U1**T*S11*V1 = ( s11 s12 ) and U1**T*T11*V1 = ( t11 t12 )
                      (  0  s22 )                    (  0  t22 )

where the generalized eigenvalues w = s11/t11 and conjg(w) = s22/t22.

Then the reciprocal condition number DIF(i) is bounded by

         min( d1, max( 1, |real(s11)/real(s22)| )*d2 )

where, d1 = Difl((s11, t11), (s22, t22)) = sigma-min(Z1), where Z1 is the complex 2-by-2 matrix

              Z1 =  [ s11  -s22 ]
                    [ t11  -t22 ],

This is done by computing (using real arithmetic) the roots of the characteristical polynomial det(Z1**T * Z1 - lambda I), where Z1**T denotes the transpose of Z1 and det(X) denotes the determinant of X.

and d2 is an upper bound on Difl((S11, T11), (S22, T22)), i.e. an upper bound on sigma-min(Z2), where Z2 is (2n-2)-by-(2n-2)

              Z2 = [ kron(S11**T, In-2)  -kron(I2, S22) ]
                   [ kron(T11**T, In-2)  -kron(I2, T22) ]
Note that if the default method for computing DIF is wanted (see SLATDF), then the parameter DIFDRI (see below) should be changed from 3 to 4 (routine SLATDF(IJOB = 2 will be used)). See STGSYL for more details.

For each eigenvalue/vector specified by SELECT, DIF stores a Frobenius norm-based estimate of Difl.

An approximate error bound for the i-th computed eigenvector VL(i) or VR(i) is given by

             EPS * norm(A, B) / DIF(i).

See ref. [2-3] for more details and further references.

Contributors 🌸

Bo Kagstrom and Peter Poromaa, Department of Computing Science, Umea University, S-901 87 Umea, Sweden.

References 📚

  • [1] B. Kagstrom; A Direct Method for Reordering Eigenvalues in the Generalized Real Schur Form of a Regular Matrix Pair (A, B), in M.S. Moonen et al (eds), Linear Algebra for Large Scale and Real-Time Applications, Kluwer Academic Publ. 1993, pp 195-218.
  • [2] B. Kagstrom and P. Poromaa; Computing Eigenspaces with Specified Eigenvalues of a Regular Matrix Pair (A, B) and Condition Estimation: Theory, Algorithms and Software, Report UMINF - 94.04, Department of Computing Science, Umea University, S-901 87 Umea, Sweden, 1994. Also as LAPACK Working Note 87. To appear in Numerical Algorithms, 1996.
  • [3] B. Kagstrom and P. Poromaa, LAPACK-Style Algorithms and Software for Solving the Generalized Sylvester Equation and Estimating the Separation between Regular Matrix Pairs, Report UMINF - 93.23, Department of Computing Science, Umea University, S-901 87 Umea, Sweden, December 1993, Revised April 1994, Also as LAPACK Working Note 75. To appear in ACM Trans. on Math. Software, Vol 22, No 1, 1996.

Definition at line 378 of file stgsna.f.

   subroutine ztgsna (character job, character howmny, logical, dimension( * )
       select, integer n, complex*16, dimension( lda, * ) a, integer lda,
       complex*16, dimension( ldb, * ) b, integer ldb, complex*16, dimension(
       ldvl, * ) vl, integer ldvl, complex*16, dimension( ldvr, * ) vr, integer
       ldvr, double precision, dimension( * ) s, double precision, dimension( *
       ) dif, integer mm, integer m, complex*16, dimension( * ) work, integer
       lwork, integer, dimension( * ) iwork, integer info)
       ZTGSNA

ZTGSNA 💜

Purpose

ZTGSNA estimates reciprocal condition numbers for specified eigenvalues and/or eigenvectors of a matrix pair (A, B).

(A, B) must be in generalized Schur canonical form, that is, A and B are both upper triangular.

Parameters

  • JOB — JOB is CHARACTER*1. Specifies whether condition numbers are required for eigenvalues (S) or eigenvectors (DIF):
    • = 'E': for eigenvalues only (S);
    • = 'V': for eigenvectors only (DIF);
    • = 'B': for both eigenvalues and eigenvectors (S and DIF).
  • 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

okay bestie, we're picking up mid-argument-list and I'm obsessed with how much detail this is giving ✨ Here's the rest of the parameters, no skips, no cap:

  • 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 corresponding j-th eigenvalue and/or eigenvector, SELECT(j) must be set to .TRUE.. If HOWMNY = 'A', SELECT is not referenced.
  • N — N is INTEGER. The order of the square matrix pair (A, B). N >= 0.
  • A — A is COMPLEX*16 array, dimension (LDA,N). The upper triangular matrix A in the pair (A,B).
  • LDA — LDA is INTEGER. The leading dimension of the array A. LDA >= max(1,N).
  • B — B is COMPLEX*16 array, dimension (LDB,N). The upper triangular matrix B in the pair (A, B).
  • LDB — LDB is INTEGER. The leading dimension of the array B. LDB >= max(1,N).
  • VL — VL is COMPLEX*16 array, dimension (LDVL,M). If JOB = 'E' or 'B', VL must contain left eigenvectors of (A, B), corresponding to the eigenpairs specified by HOWMNY and SELECT. The eigenvectors must be stored in consecutive columns of VL, as returned by ZTGEVC. 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 (A, B), corresponding to the eigenpairs specified by HOWMNY and SELECT. The eigenvectors must be stored in consecutive columns of VR, as returned by ZTGEVC. If JOB = 'V', VR is not referenced.
  • LDVR — LDVR is INTEGER. The leading dimension of the array VR. LDVR >= 1; 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. If JOB = 'V', S is not referenced.
  • DIF — DIF 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 the eigenvalues cannot be reordered to compute DIF(j), DIF(j) is set to 0; this can only occur when the true value would be very small anyway. For each eigenvalue/vector specified by SELECT, DIF stores a Frobenius norm-based estimate of Difl. If JOB = 'E', DIF is not referenced.
  • MM — MM is INTEGER. The number of elements in the arrays S and DIF. MM >= M.
  • M — M is INTEGER. The number of elements of the arrays S and DIF used to store the specified condition numbers; for each selected eigenvalue one element is used. If HOWMNY = 'A', M is set to N.
  • WORK — WORK is COMPLEX*16 array, dimension (MAX(1,LWORK)). On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
  • LWORK — LWORK is INTEGER. The dimension of the array WORK. LWORK >= max(1,N). If JOB = 'V' or 'B', LWORK >= max(1,2*N*N).
  • IWORK — IWORK is INTEGER array, dimension (N+2). 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

the academic icons behind this whole thing:

  • Univ. of Tennessee
  • Univ. of California Berkeley
  • Univ. of Colorado Denver
  • NAG Ltd.

🔥 Further Details

okay, theory time and it's giving "linear algebra final boss" 😭 but it's all right here, bestie:

The reciprocal of the condition number of the i-th generalized eigenvalue w = (a, b) is defined as

        S(I) = (|v**HAu|**2 + |v**HBu|**2)**(1/2) / (norm(u)*norm(v))

where u and v are the right and left eigenvectors of (A, B) corresponding to w; |z| denotes the absolute value of the complex number, and norm(u) denotes the 2-norm of the vector u. The pair (a, b) corresponds to an eigenvalue w = a/b (= v**HAu/v**HBu) of the matrix pair (A, B). If both a and b equal zero, then (A,B) is singular and S(I) = -1 is returned.

An approximate error bound on the chordal distance between the i-th computed generalized eigenvalue w and the corresponding exact eigenvalue lambda is

        chord(w, lambda) <=   EPS * norm(A, B) / S(I),

where EPS is the machine precision.

The reciprocal of the condition number of the right eigenvector u and left eigenvector v corresponding to the generalized eigenvalue w is defined as follows. Suppose

                   (A, B) = ( a   *  ) ( b  *  )  1
                            ( 0  A22 ),( 0 B22 )  n-1
                              1  n-1     1 n-1

Then the reciprocal condition number DIF(I) is

        Difl[(a, b), (A22, B22)]  = sigma-min( Zl )

where sigma-min(Zl) denotes the smallest singular value of

         Zl = [ kron(a, In-1) -kron(1, A22) ]
              [ kron(b, In-1) -kron(1, B22) ].

Here In-1 is the identity matrix of size n-1 and X**H is the conjugate transpose of X. kron(X, Y) is the Kronecker product between the matrices X and Y.

We approximate the smallest singular value of Zl with an upper bound. This is done by ZLATDF.

An approximate error bound for a computed eigenvector VL(i) or VR(i) is given by

                      EPS * norm(A, B) / DIF(i).

See ref. [2-3] for more details and further references.

🌸 Contributors

Bo Kagstrom and Peter Poromaa, Department of Computing Science, Umea University, S-901 87 Umea, Sweden.

🎀 References

  1. [1] B. Kagstrom; A Direct Method for Reordering Eigenvalues in the Generalized Real Schur Form of a Regular Matrix Pair (A, B), in M.S. Moonen et al (eds), Linear Algebra for Large Scale and Real-Time Applications, Kluwer Academic Publ. 1993, pp 195-218.
  2. [2] B. Kagstrom and P. Poromaa; Computing Eigenspaces with Specified Eigenvalues of a Regular Matrix Pair (A, B) and Condition Estimation: Theory, Algorithms and Software, Report UMINF - 94.04, Department of Computing Science, Umea University, S-901 87 Umea, Sweden, 1994. Also as LAPACK Working Note 87. To appear in Numerical Algorithms, 1996.
  3. [3] B. Kagstrom and P. Poromaa, LAPACK-Style Algorithms and Software for Solving the Generalized Sylvester Equation and Estimating the Separation between Regular Matrix Pairs, Report UMINF - 93.23, Department of Computing Science, Umea University, S-901 87 Umea, Sweden, December 1993, Revised April 1994, Also as LAPACK Working Note 75. To appear in ACM Trans. on Math. Software, Vol 22, No 1, 1996.

Also, the source flex: Definition at line 308 of file ztgsna.f.

🤖 Author

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