SGESC2 solves a system of linear equations


           A * X = scale* RHS


 with a general N-by-N matrix A using the LU factorization with


 complete pivoting computed by SGETC2.

N

          N is INTEGER


          The order of the matrix A.

A

          A is REAL array, dimension (LDA,N)


          On entry, the  LU part of the factorization of the n-by-n


          matrix A computed by SGETC2:  A = P * L * U * Q

LDA

          LDA is INTEGER


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

RHS

          RHS is REAL array, dimension (N).


          On entry, the right hand side vector b.


          On exit, the solution vector X.

IPIV

          IPIV is INTEGER array, dimension (N).


          The pivot indices; for 1 <= i <= N, row i of the


          matrix has been interchanged with row IPIV(i).

JPIV

          JPIV is INTEGER array, dimension (N).


          The pivot indices; for 1 <= j <= N, column j of the


          matrix has been interchanged with column JPIV(j).

SCALE

          SCALE is REAL


           On exit, SCALE contains the scale factor. SCALE is chosen


           0 <= SCALE <= 1 to prevent overflow in the solution.

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

 SGETC2 computes an LU factorization with complete pivoting of the


 n-by-n matrix A. The factorization has the form A = P * L * U * Q,


 where P and Q are permutation matrices, L is lower triangular with


 unit diagonal elements and U is upper triangular.


 This is the Level 2 BLAS algorithm.

N

          N is INTEGER


          The order of the matrix A. N >= 0.

A

          A is REAL array, dimension (LDA, N)


          On entry, the n-by-n matrix A to be factored.


          On exit, the factors L and U from the factorization


          A = P*L*U*Q; the unit diagonal elements of L are not stored.


          If U(k, k) appears to be less than SMIN, U(k, k) is given the


          value of SMIN, i.e., giving a nonsingular perturbed system.

LDA

          LDA is INTEGER


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

IPIV

          IPIV is INTEGER array, dimension(N).


          The pivot indices; for 1 <= i <= N, row i of the


          matrix has been interchanged with row IPIV(i).

JPIV

          JPIV is INTEGER array, dimension(N).


          The pivot indices; for 1 <= j <= N, column j of the


          matrix has been interchanged with column JPIV(j).

INFO

          INFO is INTEGER


           = 0: successful exit


           > 0: if INFO = k, U(k, k) is likely to produce overflow if


                we try to solve for x in Ax = b. So U is perturbed to


                avoid the overflow.

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

 SLANGE  returns the value of the one norm,  or the Frobenius norm, or


 the  infinity norm,  or the  element of  largest absolute value  of a


 real matrix A.

SLANGE

    SLANGE = ( max(abs(A(i,j))), NORM = 'M' or 'm'


             (


             ( norm1(A),         NORM = '1', 'O' or 'o'


             (


             ( normI(A),         NORM = 'I' or 'i'


             (


             ( normF(A),         NORM = 'F', 'f', 'E' or 'e'


 where  norm1  denotes the  one norm of a matrix (maximum column sum),


 normI  denotes the  infinity norm  of a matrix  (maximum row sum) and


 normF  denotes the  Frobenius norm of a matrix (square root of sum of


 squares).  Note that  max(abs(A(i,j)))  is not a consistent matrix norm.

NORM

          NORM is CHARACTER*1


          Specifies the value to be returned in SLANGE as described


          above.

M

          M is INTEGER


          The number of rows of the matrix A.  M >= 0.  When M = 0,


          SLANGE is set to zero.

N

          N is INTEGER


          The number of columns of the matrix A.  N >= 0.  When N = 0,


          SLANGE is set to zero.

A

          A is REAL array, dimension (LDA,N)


          The m by n matrix A.

LDA

          LDA is INTEGER


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

WORK

          WORK is REAL array, dimension (MAX(1,LWORK)),


          where LWORK >= M when NORM = 'I'; otherwise, WORK is not


          referenced.

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 SLAQGE equilibrates a general M by N matrix A using the row and


 column scaling factors in the vectors R and C.

M

          M is INTEGER


          The number of rows of the matrix A.  M >= 0.

N

          N is INTEGER


          The number of columns of the matrix A.  N >= 0.

A

          A is REAL array, dimension (LDA,N)


          On entry, the M by N matrix A.


          On exit, the equilibrated matrix.  See EQUED for the form of


          the equilibrated matrix.

LDA

          LDA is INTEGER


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

R

          R is REAL array, dimension (M)


          The row scale factors for A.

C

          C is REAL array, dimension (N)


          The column scale factors for A.

ROWCND

          ROWCND is REAL


          Ratio of the smallest R(i) to the largest R(i).

COLCND

          COLCND is REAL


          Ratio of the smallest C(i) to the largest C(i).

AMAX

          AMAX is REAL


          Absolute value of largest matrix entry.

EQUED

          EQUED is CHARACTER*1


          Specifies the form of equilibration that was done.


          = 'N':  No equilibration


          = 'R':  Row equilibration, i.e., A has been premultiplied by


                  diag(R).


          = 'C':  Column equilibration, i.e., A has been postmultiplied


                  by diag(C).


          = 'B':  Both row and column equilibration, i.e., A has been


                  replaced by diag(R) * A * diag(C).

  THRESH is a threshold value used to decide if row or column scaling


  should be done based on the ratio of the row or column scaling


  factors.  If ROWCND < THRESH, row scaling is done, and if


  COLCND < THRESH, column scaling is done.


  LARGE and SMALL are threshold values used to decide if row scaling


  should be done based on the absolute size of the largest matrix


  element.  If AMAX > LARGE or AMAX < SMALL, row scaling is done.

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 STGEX2 swaps adjacent diagonal blocks (A11, B11) and (A22, B22)


 of size 1-by-1 or 2-by-2 in an upper (quasi) triangular matrix pair


 (A, B) by an orthogonal equivalence transformation.


 (A, B) must be in generalized real Schur canonical 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.


 Optionally, the matrices Q and Z of generalized Schur vectors are


 updated.


        Q(in) * A(in) * Z(in)**T = Q(out) * A(out) * Z(out)**T


        Q(in) * B(in) * Z(in)**T = Q(out) * B(out) * Z(out)**T

WANTQ

          WANTQ is LOGICAL


          .TRUE. : update the left transformation matrix Q;


          .FALSE.: do not update Q.

WANTZ

          WANTZ is LOGICAL


          .TRUE. : update the right transformation matrix Z;


          .FALSE.: do not update Z.

N

          N is INTEGER


          The order of the matrices A and B. N >= 0.

A

          A is REAL array, dimension (LDA,N)


          On entry, the matrix A in the pair (A, B).


          On exit, the updated matrix A.

LDA

          LDA is INTEGER


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

B

          B is REAL array, dimension (LDB,N)


          On entry, the matrix B in the pair (A, B).


          On exit, the updated matrix B.

LDB

          LDB is INTEGER


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

Q

          Q is REAL array, dimension (LDQ,N)


          On entry, if WANTQ = .TRUE., the orthogonal matrix Q.


          On exit, the updated matrix Q.


          Not referenced if WANTQ = .FALSE..

LDQ

          LDQ is INTEGER


          The leading dimension of the array Q. LDQ >= 1.


          If WANTQ = .TRUE., LDQ >= N.

Z

          Z is REAL array, dimension (LDZ,N)


          On entry, if WANTZ =.TRUE., the orthogonal matrix Z.


          On exit, the updated matrix Z.


          Not referenced if WANTZ = .FALSE..

LDZ

          LDZ is INTEGER


          The leading dimension of the array Z. LDZ >= 1.


          If WANTZ = .TRUE., LDZ >= N.

J1

          J1 is INTEGER


          The index to the first block (A11, B11). 1 <= J1 <= N.

N1

          N1 is INTEGER


          The order of the first block (A11, B11). N1 = 0, 1 or 2.

N2

          N2 is INTEGER


          The order of the second block (A22, B22). N2 = 0, 1 or 2.

WORK

          WORK is REAL array, dimension (MAX(1,LWORK)).

LWORK

          LWORK is INTEGER


          The dimension of the array WORK.


          LWORK >=  MAX( N*(N2+N1), (N2+N1)*(N2+N1)*2 )

INFO

          INFO is INTEGER


            =0: Successful exit


            >0: If INFO = 1, the transformed matrix (A, B) would be


                too far from generalized Schur form; the blocks are


                not swapped and (A, B) and (Q, Z) are unchanged.


                The problem of swapping is too ill-conditioned.


            <0: If INFO = -16: LWORK is too small. Appropriate value


                for LWORK is returned in WORK(1).

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In the current code both weak and strong stability tests are performed. The user can omit the strong stability test by changing the internal logical parameter WANDS to .FALSE.. See ref. [2] for details.

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

  [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.