--- 
:name: stgsna
:md5sum: df29cdbd8fb6f9df1af4856bd8b8b717
:category: :subroutine
:arguments: 
- job: 
    :type: char
    :intent: input
- howmny: 
    :type: char
    :intent: input
- select: 
    :type: logical
    :intent: input
    :dims: 
    - n
- n: 
    :type: integer
    :intent: input
- a: 
    :type: real
    :intent: input
    :dims: 
    - lda
    - n
- lda: 
    :type: integer
    :intent: input
- b: 
    :type: real
    :intent: input
    :dims: 
    - ldb
    - n
- ldb: 
    :type: integer
    :intent: input
- vl: 
    :type: real
    :intent: input
    :dims: 
    - ldvl
    - m
- ldvl: 
    :type: integer
    :intent: input
- vr: 
    :type: real
    :intent: input
    :dims: 
    - ldvr
    - m
- ldvr: 
    :type: integer
    :intent: input
- s: 
    :type: real
    :intent: output
    :dims: 
    - mm
- dif: 
    :type: real
    :intent: output
    :dims: 
    - mm
- mm: 
    :type: integer
    :intent: input
- m: 
    :type: integer
    :intent: output
- work: 
    :type: real
    :intent: output
    :dims: 
    - MAX(1,lwork)
- lwork: 
    :type: integer
    :intent: input
    :option: true
    :default: "(lsame_(&job,\"V\")||lsame_(&job,\"B\")) ? 2*n*n : n"
- iwork: 
    :type: integer
    :intent: workspace
    :dims: 
    - "lsame_(&job,\"E\") ? 0 : n + 6"
- info: 
    :type: integer
    :intent: output
:substitutions: 
  mm: m
:fortran_help: "      SUBROUTINE STGSNA( JOB, HOWMNY, SELECT, N, A, LDA, B, LDB, VL, LDVL, VR, LDVR, S, DIF, MM, M, WORK, LWORK, IWORK, INFO )\n\n\
  *  Purpose\n\
  *  =======\n\
  *\n\
  *  STGSNA estimates reciprocal condition numbers for specified\n\
  *  eigenvalues and/or eigenvectors of a matrix pair (A, B) in\n\
  *  generalized real Schur canonical form (or of any matrix pair\n\
  *  (Q*A*Z', Q*B*Z') with orthogonal matrices Q and Z, where\n\
  *  Z' denotes the transpose of Z.\n\
  *\n\
  *  (A, B) must be in generalized real Schur form (as returned by SGGES),\n\
  *  i.e. A is block upper triangular with 1-by-1 and 2-by-2 diagonal\n\
  *  blocks. B is upper triangular.\n\
  *\n\
  *\n\n\
  *  Arguments\n\
  *  =========\n\
  *\n\
  *  JOB     (input) CHARACTER*1\n\
  *          Specifies whether condition numbers are required for\n\
  *          eigenvalues (S) or eigenvectors (DIF):\n\
  *          = 'E': for eigenvalues only (S);\n\
  *          = 'V': for eigenvectors only (DIF);\n\
  *          = 'B': for both eigenvalues and eigenvectors (S and DIF).\n\
  *\n\
  *  HOWMNY  (input) CHARACTER*1\n\
  *          = 'A': compute condition numbers for all eigenpairs;\n\
  *          = 'S': compute condition numbers for selected eigenpairs\n\
  *                 specified by the array SELECT.\n\
  *\n\
  *  SELECT  (input) LOGICAL array, dimension (N)\n\
  *          If HOWMNY = 'S', SELECT specifies the eigenpairs for which\n\
  *          condition numbers are required. To select condition numbers\n\
  *          for the eigenpair corresponding to a real eigenvalue w(j),\n\
  *          SELECT(j) must be set to .TRUE.. To select condition numbers\n\
  *          corresponding to a complex conjugate pair of eigenvalues w(j)\n\
  *          and w(j+1), either SELECT(j) or SELECT(j+1) or both, must be\n\
  *          set to .TRUE..\n\
  *          If HOWMNY = 'A', SELECT is not referenced.\n\
  *\n\
  *  N       (input) INTEGER\n\
  *          The order of the square matrix pair (A, B). N >= 0.\n\
  *\n\
  *  A       (input) REAL array, dimension (LDA,N)\n\
  *          The upper quasi-triangular matrix A in the pair (A,B).\n\
  *\n\
  *  LDA     (input) INTEGER\n\
  *          The leading dimension of the array A. LDA >= max(1,N).\n\
  *\n\
  *  B       (input) REAL array, dimension (LDB,N)\n\
  *          The upper triangular matrix B in the pair (A,B).\n\
  *\n\
  *  LDB     (input) INTEGER\n\
  *          The leading dimension of the array B. LDB >= max(1,N).\n\
  *\n\
  *  VL      (input) REAL array, dimension (LDVL,M)\n\
  *          If JOB = 'E' or 'B', VL must contain left eigenvectors of\n\
  *          (A, B), corresponding to the eigenpairs specified by HOWMNY\n\
  *          and SELECT. The eigenvectors must be stored in consecutive\n\
  *          columns of VL, as returned by STGEVC.\n\
  *          If JOB = 'V', VL is not referenced.\n\
  *\n\
  *  LDVL    (input) INTEGER\n\
  *          The leading dimension of the array VL. LDVL >= 1.\n\
  *          If JOB = 'E' or 'B', LDVL >= N.\n\
  *\n\
  *  VR      (input) REAL array, dimension (LDVR,M)\n\
  *          If JOB = 'E' or 'B', VR must contain right eigenvectors of\n\
  *          (A, B), corresponding to the eigenpairs specified by HOWMNY\n\
  *          and SELECT. The eigenvectors must be stored in consecutive\n\
  *          columns ov VR, as returned by STGEVC.\n\
  *          If JOB = 'V', VR is not referenced.\n\
  *\n\
  *  LDVR    (input) INTEGER\n\
  *          The leading dimension of the array VR. LDVR >= 1.\n\
  *          If JOB = 'E' or 'B', LDVR >= N.\n\
  *\n\
  *  S       (output) REAL array, dimension (MM)\n\
  *          If JOB = 'E' or 'B', the reciprocal condition numbers of the\n\
  *          selected eigenvalues, stored in consecutive elements of the\n\
  *          array. For a complex conjugate pair of eigenvalues two\n\
  *          consecutive elements of S are set to the same value. Thus\n\
  *          S(j), DIF(j), and the j-th columns of VL and VR all\n\
  *          correspond to the same eigenpair (but not in general the\n\
  *          j-th eigenpair, unless all eigenpairs are selected).\n\
  *          If JOB = 'V', S is not referenced.\n\
  *\n\
  *  DIF     (output) REAL array, dimension (MM)\n\
  *          If JOB = 'V' or 'B', the estimated reciprocal condition\n\
  *          numbers of the selected eigenvectors, stored in consecutive\n\
  *          elements of the array. For a complex eigenvector two\n\
  *          consecutive elements of DIF are set to the same value. If\n\
  *          the eigenvalues cannot be reordered to compute DIF(j), DIF(j)\n\
  *          is set to 0; this can only occur when the true value would be\n\
  *          very small anyway.\n\
  *          If JOB = 'E', DIF is not referenced.\n\
  *\n\
  *  MM      (input) INTEGER\n\
  *          The number of elements in the arrays S and DIF. MM >= M.\n\
  *\n\
  *  M       (output) INTEGER\n\
  *          The number of elements of the arrays S and DIF used to store\n\
  *          the specified condition numbers; for each selected real\n\
  *          eigenvalue one element is used, and for each selected complex\n\
  *          conjugate pair of eigenvalues, two elements are used.\n\
  *          If HOWMNY = 'A', M is set to N.\n\
  *\n\
  *  WORK    (workspace/output) REAL array, dimension (MAX(1,LWORK))\n\
  *          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.\n\
  *\n\
  *  LWORK   (input) INTEGER\n\
  *          The dimension of the array WORK. LWORK >= max(1,N).\n\
  *          If JOB = 'V' or 'B' LWORK >= 2*N*(N+2)+16.\n\
  *\n\
  *          If LWORK = -1, then a workspace query is assumed; the routine\n\
  *          only calculates the optimal size of the WORK array, returns\n\
  *          this value as the first entry of the WORK array, and no error\n\
  *          message related to LWORK is issued by XERBLA.\n\
  *\n\
  *  IWORK   (workspace) INTEGER array, dimension (N + 6)\n\
  *          If JOB = 'E', IWORK is not referenced.\n\
  *\n\
  *  INFO    (output) INTEGER\n\
  *          =0: Successful exit\n\
  *          <0: If INFO = -i, the i-th argument had an illegal value\n\
  *\n\
  *\n\n\
  *  Further Details\n\
  *  ===============\n\
  *\n\
  *  The reciprocal of the condition number of a generalized eigenvalue\n\
  *  w = (a, b) is defined as\n\
  *\n\
  *       S(w) = (|u'Av|**2 + |u'Bv|**2)**(1/2) / (norm(u)*norm(v))\n\
  *\n\
  *  where u and v are the left and right eigenvectors of (A, B)\n\
  *  corresponding to w; |z| denotes the absolute value of the complex\n\
  *  number, and norm(u) denotes the 2-norm of the vector u.\n\
  *  The pair (a, b) corresponds to an eigenvalue w = a/b (= u'Av/u'Bv)\n\
  *  of the matrix pair (A, B). If both a and b equal zero, then (A B) is\n\
  *  singular and S(I) = -1 is returned.\n\
  *\n\
  *  An approximate error bound on the chordal distance between the i-th\n\
  *  computed generalized eigenvalue w and the corresponding exact\n\
  *  eigenvalue lambda is\n\
  *\n\
  *       chord(w, lambda) <= EPS * norm(A, B) / S(I)\n\
  *\n\
  *  where EPS is the machine precision.\n\
  *\n\
  *  The reciprocal of the condition number DIF(i) of right eigenvector u\n\
  *  and left eigenvector v corresponding to the generalized eigenvalue w\n\
  *  is defined as follows:\n\
  *\n\
  *  a) If the i-th eigenvalue w = (a,b) is real\n\
  *\n\
  *     Suppose U and V are orthogonal transformations such that\n\
  *\n\
  *                U'*(A, B)*V  = (S, T) = ( a   *  ) ( b  *  )  1\n\
  *                                        ( 0  S22 ),( 0 T22 )  n-1\n\
  *                                          1  n-1     1 n-1\n\
  *\n\
  *     Then the reciprocal condition number DIF(i) is\n\
  *\n\
  *                Difl((a, b), (S22, T22)) = sigma-min( Zl ),\n\
  *\n\
  *     where sigma-min(Zl) denotes the smallest singular value of the\n\
  *     2(n-1)-by-2(n-1) matrix\n\
  *\n\
  *         Zl = [ kron(a, In-1)  -kron(1, S22) ]\n\
  *              [ kron(b, In-1)  -kron(1, T22) ] .\n\
  *\n\
  *     Here In-1 is the identity matrix of size n-1. kron(X, Y) is the\n\
  *     Kronecker product between the matrices X and Y.\n\
  *\n\
  *     Note that if the default method for computing DIF(i) is wanted\n\
  *     (see SLATDF), then the parameter DIFDRI (see below) should be\n\
  *     changed from 3 to 4 (routine SLATDF(IJOB = 2 will be used)).\n\
  *     See STGSYL for more details.\n\
  *\n\
  *  b) If the i-th and (i+1)-th eigenvalues are complex conjugate pair,\n\
  *\n\
  *     Suppose U and V are orthogonal transformations such that\n\
  *\n\
  *                U'*(A, B)*V = (S, T) = ( S11  *   ) ( T11  *  )  2\n\
  *                                       ( 0    S22 ),( 0    T22) n-2\n\
  *                                         2    n-2     2    n-2\n\
  *\n\
  *     and (S11, T11) corresponds to the complex conjugate eigenvalue\n\
  *     pair (w, conjg(w)). There exist unitary matrices U1 and V1 such\n\
  *     that\n\
  *\n\
  *         U1'*S11*V1 = ( s11 s12 )   and U1'*T11*V1 = ( t11 t12 )\n\
  *                      (  0  s22 )                    (  0  t22 )\n\
  *\n\
  *     where the generalized eigenvalues w = s11/t11 and\n\
  *     conjg(w) = s22/t22.\n\
  *\n\
  *     Then the reciprocal condition number DIF(i) is bounded by\n\
  *\n\
  *         min( d1, max( 1, |real(s11)/real(s22)| )*d2 )\n\
  *\n\
  *     where, d1 = Difl((s11, t11), (s22, t22)) = sigma-min(Z1), where\n\
  *     Z1 is the complex 2-by-2 matrix\n\
  *\n\
  *              Z1 =  [ s11  -s22 ]\n\
  *                    [ t11  -t22 ],\n\
  *\n\
  *     This is done by computing (using real arithmetic) the\n\
  *     roots of the characteristical polynomial det(Z1' * Z1 - lambda I),\n\
  *     where Z1' denotes the conjugate transpose of Z1 and det(X) denotes\n\
  *     the determinant of X.\n\
  *\n\
  *     and d2 is an upper bound on Difl((S11, T11), (S22, T22)), i.e. an\n\
  *     upper bound on sigma-min(Z2), where Z2 is (2n-2)-by-(2n-2)\n\
  *\n\
  *              Z2 = [ kron(S11', In-2)  -kron(I2, S22) ]\n\
  *                   [ kron(T11', In-2)  -kron(I2, T22) ]\n\
  *\n\
  *     Note that if the default method for computing DIF is wanted (see\n\
  *     SLATDF), then the parameter DIFDRI (see below) should be changed\n\
  *     from 3 to 4 (routine SLATDF(IJOB = 2 will be used)). See STGSYL\n\
  *     for more details.\n\
  *\n\
  *  For each eigenvalue/vector specified by SELECT, DIF stores a\n\
  *  Frobenius norm-based estimate of Difl.\n\
  *\n\
  *  An approximate error bound for the i-th computed eigenvector VL(i) or\n\
  *  VR(i) is given by\n\
  *\n\
  *             EPS * norm(A, B) / DIF(i).\n\
  *\n\
  *  See ref. [2-3] for more details and further references.\n\
  *\n\
  *  Based on contributions by\n\
  *     Bo Kagstrom and Peter Poromaa, Department of Computing Science,\n\
  *     Umea University, S-901 87 Umea, Sweden.\n\
  *\n\
  *  References\n\
  *  ==========\n\
  *\n\
  *  [1] B. Kagstrom; A Direct Method for Reordering Eigenvalues in the\n\
  *      Generalized Real Schur Form of a Regular Matrix Pair (A, B), in\n\
  *      M.S. Moonen et al (eds), Linear Algebra for Large Scale and\n\
  *      Real-Time Applications, Kluwer Academic Publ. 1993, pp 195-218.\n\
  *\n\
  *  [2] B. Kagstrom and P. Poromaa; Computing Eigenspaces with Specified\n\
  *      Eigenvalues of a Regular Matrix Pair (A, B) and Condition\n\
  *      Estimation: Theory, Algorithms and Software,\n\
  *      Report UMINF - 94.04, Department of Computing Science, Umea\n\
  *      University, S-901 87 Umea, Sweden, 1994. Also as LAPACK Working\n\
  *      Note 87. To appear in Numerical Algorithms, 1996.\n\
  *\n\
  *  [3] B. Kagstrom and P. Poromaa, LAPACK-Style Algorithms and Software\n\
  *      for Solving the Generalized Sylvester Equation and Estimating the\n\
  *      Separation between Regular Matrix Pairs, Report UMINF - 93.23,\n\
  *      Department of Computing Science, Umea University, S-901 87 Umea,\n\
  *      Sweden, December 1993, Revised April 1994, Also as LAPACK Working\n\
  *      Note 75.  To appear in ACM Trans. on Math. Software, Vol 22,\n\
  *      No 1, 1996.\n\
  *\n\
  *  =====================================================================\n\
  *\n"