/* stgsna.f -- translated by f2c (version 20061008). You must link the resulting object file with libf2c: on Microsoft Windows system, link with libf2c.lib; on Linux or Unix systems, link with .../path/to/libf2c.a -lm or, if you install libf2c.a in a standard place, with -lf2c -lm -- in that order, at the end of the command line, as in cc *.o -lf2c -lm Source for libf2c is in /netlib/f2c/libf2c.zip, e.g., http://www.netlib.org/f2c/libf2c.zip */ #include "f2c.h" #include "blaswrap.h" /* Table of constant values */ static integer c__1 = 1; static real c_b19 = 1.f; static real c_b21 = 0.f; static integer c__2 = 2; static logical c_false = FALSE_; static integer c__3 = 3; /* Subroutine */ int stgsna_(char *job, char *howmny, logical *select, integer *n, real *a, integer *lda, real *b, integer *ldb, real *vl, integer *ldvl, real *vr, integer *ldvr, real *s, real *dif, integer * mm, integer *m, real *work, integer *lwork, integer *iwork, integer * info) { /* System generated locals */ integer a_dim1, a_offset, b_dim1, b_offset, vl_dim1, vl_offset, vr_dim1, vr_offset, i__1, i__2; real r__1, r__2; /* Builtin functions */ double sqrt(doublereal); /* Local variables */ integer i__, k; real c1, c2; integer n1, n2, ks, iz; real eps, beta, cond; logical pair; integer ierr; real uhav, uhbv; integer ifst; real lnrm; extern doublereal sdot_(integer *, real *, integer *, real *, integer *); integer ilst; real rnrm; extern /* Subroutine */ int slag2_(real *, integer *, real *, integer *, real *, real *, real *, real *, real *, real *); extern doublereal snrm2_(integer *, real *, integer *); real root1, root2, scale; extern logical lsame_(char *, char *); real uhavi, uhbvi; extern /* Subroutine */ int sgemv_(char *, integer *, integer *, real *, real *, integer *, real *, integer *, real *, real *, integer *); real tmpii; integer lwmin; logical wants; real tmpir, tmpri, dummy[1], tmprr; extern doublereal slapy2_(real *, real *); real dummy1[1], alphai, alphar; extern doublereal slamch_(char *); extern /* Subroutine */ int xerbla_(char *, integer *); logical wantbh, wantdf; extern /* Subroutine */ int slacpy_(char *, integer *, integer *, real *, integer *, real *, integer *), stgexc_(logical *, logical *, integer *, real *, integer *, real *, integer *, real *, integer *, real *, integer *, integer *, integer *, real *, integer *, integer *); logical somcon; real alprqt, smlnum; logical lquery; extern /* Subroutine */ int stgsyl_(char *, integer *, integer *, integer *, real *, integer *, real *, integer *, real *, integer *, real * , integer *, real *, integer *, real *, integer *, real *, real *, real *, integer *, integer *, integer *); /* -- LAPACK routine (version 3.2) -- */ /* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. */ /* November 2006 */ /* .. Scalar Arguments .. */ /* .. */ /* .. Array Arguments .. */ /* .. */ /* 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', Q*B*Z') with orthogonal matrices Q and Z, where */ /* Z' 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. */ /* Arguments */ /* ========= */ /* JOB (input) 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 (input) CHARACTER*1 */ /* = 'A': compute condition numbers for all eigenpairs; */ /* = 'S': compute condition numbers for selected eigenpairs */ /* specified by the array SELECT. */ /* SELECT (input) 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 (input) INTEGER */ /* The order of the square matrix pair (A, B). N >= 0. */ /* A (input) REAL array, dimension (LDA,N) */ /* The upper quasi-triangular matrix A in the pair (A,B). */ /* LDA (input) INTEGER */ /* The leading dimension of the array A. LDA >= max(1,N). */ /* B (input) REAL array, dimension (LDB,N) */ /* The upper triangular matrix B in the pair (A,B). */ /* LDB (input) INTEGER */ /* The leading dimension of the array B. LDB >= max(1,N). */ /* VL (input) 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 (input) INTEGER */ /* The leading dimension of the array VL. LDVL >= 1. */ /* If JOB = 'E' or 'B', LDVL >= N. */ /* VR (input) 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 (input) INTEGER */ /* The leading dimension of the array VR. LDVR >= 1. */ /* If JOB = 'E' or 'B', LDVR >= N. */ /* S (output) 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 (output) 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 (input) INTEGER */ /* The number of elements in the arrays S and DIF. MM >= M. */ /* M (output) 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 (workspace/output) REAL array, dimension (MAX(1,LWORK)) */ /* On exit, if INFO = 0, WORK(1) returns the optimal LWORK. */ /* LWORK (input) 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 (workspace) INTEGER array, dimension (N + 6) */ /* If JOB = 'E', IWORK is not referenced. */ /* INFO (output) INTEGER */ /* =0: Successful exit */ /* <0: If INFO = -i, the i-th argument had an illegal value */ /* Further Details */ /* =============== */ /* The reciprocal of the condition number of a generalized eigenvalue */ /* w = (a, b) is defined as */ /* S(w) = (|u'Av|**2 + |u'Bv|**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'Av/u'Bv) */ /* 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'*(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'*(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'*S11*V1 = ( s11 s12 ) and U1'*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' * Z1 - lambda I), */ /* where Z1' denotes the conjugate 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', In-2) -kron(I2, S22) ] */ /* [ kron(T11', 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. */ /* Based on contributions by */ /* 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. */ /* ===================================================================== */ /* .. Parameters .. */ /* .. */ /* .. Local Scalars .. */ /* .. */ /* .. Local Arrays .. */ /* .. */ /* .. External Functions .. */ /* .. */ /* .. External Subroutines .. */ /* .. */ /* .. Intrinsic Functions .. */ /* .. */ /* .. Executable Statements .. */ /* Decode and test the input parameters */ /* Parameter adjustments */ --select; a_dim1 = *lda; a_offset = 1 + a_dim1; a -= a_offset; b_dim1 = *ldb; b_offset = 1 + b_dim1; b -= b_offset; vl_dim1 = *ldvl; vl_offset = 1 + vl_dim1; vl -= vl_offset; vr_dim1 = *ldvr; vr_offset = 1 + vr_dim1; vr -= vr_offset; --s; --dif; --work; --iwork; /* Function Body */ wantbh = lsame_(job, "B"); wants = lsame_(job, "E") || wantbh; wantdf = lsame_(job, "V") || wantbh; somcon = lsame_(howmny, "S"); *info = 0; lquery = *lwork == -1; if (! wants && ! wantdf) { *info = -1; } else if (! lsame_(howmny, "A") && ! somcon) { *info = -2; } else if (*n < 0) { *info = -4; } else if (*lda < max(1,*n)) { *info = -6; } else if (*ldb < max(1,*n)) { *info = -8; } else if (wants && *ldvl < *n) { *info = -10; } else if (wants && *ldvr < *n) { *info = -12; } else { /* Set M to the number of eigenpairs for which condition numbers */ /* are required, and test MM. */ if (somcon) { *m = 0; pair = FALSE_; i__1 = *n; for (k = 1; k <= i__1; ++k) { if (pair) { pair = FALSE_; } else { if (k < *n) { if (a[k + 1 + k * a_dim1] == 0.f) { if (select[k]) { ++(*m); } } else { pair = TRUE_; if (select[k] || select[k + 1]) { *m += 2; } } } else { if (select[*n]) { ++(*m); } } } /* L10: */ } } else { *m = *n; } if (*n == 0) { lwmin = 1; } else if (lsame_(job, "V") || lsame_(job, "B")) { lwmin = (*n << 1) * (*n + 2) + 16; } else { lwmin = *n; } work[1] = (real) lwmin; if (*mm < *m) { *info = -15; } else if (*lwork < lwmin && ! lquery) { *info = -18; } } if (*info != 0) { i__1 = -(*info); xerbla_("STGSNA", &i__1); return 0; } else if (lquery) { return 0; } /* Quick return if possible */ if (*n == 0) { return 0; } /* Get machine constants */ eps = slamch_("P"); smlnum = slamch_("S") / eps; ks = 0; pair = FALSE_; i__1 = *n; for (k = 1; k <= i__1; ++k) { /* Determine whether A(k,k) begins a 1-by-1 or 2-by-2 block. */ if (pair) { pair = FALSE_; goto L20; } else { if (k < *n) { pair = a[k + 1 + k * a_dim1] != 0.f; } } /* Determine whether condition numbers are required for the k-th */ /* eigenpair. */ if (somcon) { if (pair) { if (! select[k] && ! select[k + 1]) { goto L20; } } else { if (! select[k]) { goto L20; } } } ++ks; if (wants) { /* Compute the reciprocal condition number of the k-th */ /* eigenvalue. */ if (pair) { /* Complex eigenvalue pair. */ r__1 = snrm2_(n, &vr[ks * vr_dim1 + 1], &c__1); r__2 = snrm2_(n, &vr[(ks + 1) * vr_dim1 + 1], &c__1); rnrm = slapy2_(&r__1, &r__2); r__1 = snrm2_(n, &vl[ks * vl_dim1 + 1], &c__1); r__2 = snrm2_(n, &vl[(ks + 1) * vl_dim1 + 1], &c__1); lnrm = slapy2_(&r__1, &r__2); sgemv_("N", n, n, &c_b19, &a[a_offset], lda, &vr[ks * vr_dim1 + 1], &c__1, &c_b21, &work[1], &c__1); tmprr = sdot_(n, &work[1], &c__1, &vl[ks * vl_dim1 + 1], & c__1); tmpri = sdot_(n, &work[1], &c__1, &vl[(ks + 1) * vl_dim1 + 1], &c__1); sgemv_("N", n, n, &c_b19, &a[a_offset], lda, &vr[(ks + 1) * vr_dim1 + 1], &c__1, &c_b21, &work[1], &c__1); tmpii = sdot_(n, &work[1], &c__1, &vl[(ks + 1) * vl_dim1 + 1], &c__1); tmpir = sdot_(n, &work[1], &c__1, &vl[ks * vl_dim1 + 1], & c__1); uhav = tmprr + tmpii; uhavi = tmpir - tmpri; sgemv_("N", n, n, &c_b19, &b[b_offset], ldb, &vr[ks * vr_dim1 + 1], &c__1, &c_b21, &work[1], &c__1); tmprr = sdot_(n, &work[1], &c__1, &vl[ks * vl_dim1 + 1], & c__1); tmpri = sdot_(n, &work[1], &c__1, &vl[(ks + 1) * vl_dim1 + 1], &c__1); sgemv_("N", n, n, &c_b19, &b[b_offset], ldb, &vr[(ks + 1) * vr_dim1 + 1], &c__1, &c_b21, &work[1], &c__1); tmpii = sdot_(n, &work[1], &c__1, &vl[(ks + 1) * vl_dim1 + 1], &c__1); tmpir = sdot_(n, &work[1], &c__1, &vl[ks * vl_dim1 + 1], & c__1); uhbv = tmprr + tmpii; uhbvi = tmpir - tmpri; uhav = slapy2_(&uhav, &uhavi); uhbv = slapy2_(&uhbv, &uhbvi); cond = slapy2_(&uhav, &uhbv); s[ks] = cond / (rnrm * lnrm); s[ks + 1] = s[ks]; } else { /* Real eigenvalue. */ rnrm = snrm2_(n, &vr[ks * vr_dim1 + 1], &c__1); lnrm = snrm2_(n, &vl[ks * vl_dim1 + 1], &c__1); sgemv_("N", n, n, &c_b19, &a[a_offset], lda, &vr[ks * vr_dim1 + 1], &c__1, &c_b21, &work[1], &c__1); uhav = sdot_(n, &work[1], &c__1, &vl[ks * vl_dim1 + 1], &c__1) ; sgemv_("N", n, n, &c_b19, &b[b_offset], ldb, &vr[ks * vr_dim1 + 1], &c__1, &c_b21, &work[1], &c__1); uhbv = sdot_(n, &work[1], &c__1, &vl[ks * vl_dim1 + 1], &c__1) ; cond = slapy2_(&uhav, &uhbv); if (cond == 0.f) { s[ks] = -1.f; } else { s[ks] = cond / (rnrm * lnrm); } } } if (wantdf) { if (*n == 1) { dif[ks] = slapy2_(&a[a_dim1 + 1], &b[b_dim1 + 1]); goto L20; } /* Estimate the reciprocal condition number of the k-th */ /* eigenvectors. */ if (pair) { /* Copy the 2-by 2 pencil beginning at (A(k,k), B(k, k)). */ /* Compute the eigenvalue(s) at position K. */ work[1] = a[k + k * a_dim1]; work[2] = a[k + 1 + k * a_dim1]; work[3] = a[k + (k + 1) * a_dim1]; work[4] = a[k + 1 + (k + 1) * a_dim1]; work[5] = b[k + k * b_dim1]; work[6] = b[k + 1 + k * b_dim1]; work[7] = b[k + (k + 1) * b_dim1]; work[8] = b[k + 1 + (k + 1) * b_dim1]; r__1 = smlnum * eps; slag2_(&work[1], &c__2, &work[5], &c__2, &r__1, &beta, dummy1, &alphar, dummy, &alphai); alprqt = 1.f; c1 = (alphar * alphar + alphai * alphai + beta * beta) * 2.f; c2 = beta * 4.f * beta * alphai * alphai; root1 = c1 + sqrt(c1 * c1 - c2 * 4.f); root2 = c2 / root1; root1 /= 2.f; /* Computing MIN */ r__1 = sqrt(root1), r__2 = sqrt(root2); cond = dmin(r__1,r__2); } /* Copy the matrix (A, B) to the array WORK and swap the */ /* diagonal block beginning at A(k,k) to the (1,1) position. */ slacpy_("Full", n, n, &a[a_offset], lda, &work[1], n); slacpy_("Full", n, n, &b[b_offset], ldb, &work[*n * *n + 1], n); ifst = k; ilst = 1; i__2 = *lwork - (*n << 1) * *n; stgexc_(&c_false, &c_false, n, &work[1], n, &work[*n * *n + 1], n, dummy, &c__1, dummy1, &c__1, &ifst, &ilst, &work[(*n * * n << 1) + 1], &i__2, &ierr); if (ierr > 0) { /* Ill-conditioned problem - swap rejected. */ dif[ks] = 0.f; } else { /* Reordering successful, solve generalized Sylvester */ /* equation for R and L, */ /* A22 * R - L * A11 = A12 */ /* B22 * R - L * B11 = B12, */ /* and compute estimate of Difl((A11,B11), (A22, B22)). */ n1 = 1; if (work[2] != 0.f) { n1 = 2; } n2 = *n - n1; if (n2 == 0) { dif[ks] = cond; } else { i__ = *n * *n + 1; iz = (*n << 1) * *n + 1; i__2 = *lwork - (*n << 1) * *n; stgsyl_("N", &c__3, &n2, &n1, &work[*n * n1 + n1 + 1], n, &work[1], n, &work[n1 + 1], n, &work[*n * n1 + n1 + i__], n, &work[i__], n, &work[n1 + i__], n, & scale, &dif[ks], &work[iz + 1], &i__2, &iwork[1], &ierr); if (pair) { /* Computing MIN */ r__1 = dmax(1.f,alprqt) * dif[ks]; dif[ks] = dmin(r__1,cond); } } } if (pair) { dif[ks + 1] = dif[ks]; } } if (pair) { ++ks; } L20: ; } work[1] = (real) lwmin; return 0; /* End of STGSNA */ } /* stgsna_ */