LAPACK
3.4.2
LAPACK: Linear Algebra PACKage

Functions/Subroutines  
subroutine  sgesc2 (N, A, LDA, RHS, IPIV, JPIV, SCALE) 
SGESC2 solves a system of linear equations using the LU factorization with complete pivoting computed by sgetc2.  
subroutine  sgetc2 (N, A, LDA, IPIV, JPIV, INFO) 
SGETC2 computes the LU factorization with complete pivoting of the general nbyn matrix.  
REAL function  slange (NORM, M, N, A, LDA, WORK) 
SLANGE returns the value of the 1norm, Frobenius norm, infinitynorm, or the largest absolute value of any element of a general rectangular matrix.  
subroutine  slaqge (M, N, A, LDA, R, C, ROWCND, COLCND, AMAX, EQUED) 
SLAQGE scales a general rectangular matrix, using row and column scaling factors computed by sgeequ.  
subroutine  stgex2 (WANTQ, WANTZ, N, A, LDA, B, LDB, Q, LDQ, Z, LDZ, J1, N1, N2, WORK, LWORK, INFO) 
STGEX2 swaps adjacent diagonal blocks in an upper (quasi) triangular matrix pair by an orthogonal equivalence transformation. 
This is the group of real auxiliary functions for GE matrices
subroutine sgesc2  (  integer  N, 
real, dimension( lda, * )  A,  
integer  LDA,  
real, dimension( * )  RHS,  
integer, dimension( * )  IPIV,  
integer, dimension( * )  JPIV,  
real  SCALE  
) 
SGESC2 solves a system of linear equations using the LU factorization with complete pivoting computed by sgetc2.
Download SGESC2 + dependencies [TGZ] [ZIP] [TXT]SGESC2 solves a system of linear equations A * X = scale* RHS with a general NbyN matrix A using the LU factorization with complete pivoting computed by SGETC2.
[in]  N  N is INTEGER The order of the matrix A. 
[in]  A  A is REAL array, dimension (LDA,N) On entry, the LU part of the factorization of the nbyn matrix A computed by SGETC2: A = P * L * U * Q 
[in]  LDA  LDA is INTEGER The leading dimension of the array A. LDA >= max(1, N). 
[in,out]  RHS  RHS is REAL array, dimension (N). On entry, the right hand side vector b. On exit, the solution vector X. 
[in]  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). 
[in]  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). 
[out]  SCALE  SCALE is REAL On exit, SCALE contains the scale factor. SCALE is chosen 0 <= SCALE <= 1 to prevent owerflow in the solution. 
Definition at line 115 of file sgesc2.f.
subroutine sgetc2  (  integer  N, 
real, dimension( lda, * )  A,  
integer  LDA,  
integer, dimension( * )  IPIV,  
integer, dimension( * )  JPIV,  
integer  INFO  
) 
SGETC2 computes the LU factorization with complete pivoting of the general nbyn matrix.
Download SGETC2 + dependencies [TGZ] [ZIP] [TXT]SGETC2 computes an LU factorization with complete pivoting of the nbyn 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.
[in]  N  N is INTEGER The order of the matrix A. N >= 0. 
[in,out]  A  A is REAL array, dimension (LDA, N) On entry, the nbyn 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. 
[in]  LDA  LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N). 
[out]  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). 
[out]  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). 
[out]  INFO  INFO is INTEGER = 0: successful exit > 0: if INFO = k, U(k, k) is likely to produce owerflow if we try to solve for x in Ax = b. So U is perturbed to avoid the overflow. 
Definition at line 112 of file sgetc2.f.
REAL function slange  (  character  NORM, 
integer  M,  
integer  N,  
real, dimension( lda, * )  A,  
integer  LDA,  
real, dimension( * )  WORK  
) 
SLANGE returns the value of the 1norm, Frobenius norm, infinitynorm, or the largest absolute value of any element of a general rectangular matrix.
Download SLANGE + dependencies [TGZ] [ZIP] [TXT]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 = ( 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.
[in]  NORM  NORM is CHARACTER*1 Specifies the value to be returned in SLANGE as described above. 
[in]  M  M is INTEGER The number of rows of the matrix A. M >= 0. When M = 0, SLANGE is set to zero. 
[in]  N  N is INTEGER The number of columns of the matrix A. N >= 0. When N = 0, SLANGE is set to zero. 
[in]  A  A is REAL array, dimension (LDA,N) The m by n matrix A. 
[in]  LDA  LDA is INTEGER The leading dimension of the array A. LDA >= max(M,1). 
[out]  WORK  WORK is REAL array, dimension (MAX(1,LWORK)), where LWORK >= M when NORM = 'I'; otherwise, WORK is not referenced. 
Definition at line 115 of file slange.f.
subroutine slaqge  (  integer  M, 
integer  N,  
real, dimension( lda, * )  A,  
integer  LDA,  
real, dimension( * )  R,  
real, dimension( * )  C,  
real  ROWCND,  
real  COLCND,  
real  AMAX,  
character  EQUED  
) 
SLAQGE scales a general rectangular matrix, using row and column scaling factors computed by sgeequ.
Download SLAQGE + dependencies [TGZ] [ZIP] [TXT]SLAQGE equilibrates a general M by N matrix A using the row and column scaling factors in the vectors R and C.
[in]  M  M is INTEGER The number of rows of the matrix A. M >= 0. 
[in]  N  N is INTEGER The number of columns of the matrix A. N >= 0. 
[in,out]  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. 
[in]  LDA  LDA is INTEGER The leading dimension of the array A. LDA >= max(M,1). 
[in]  R  R is REAL array, dimension (M) The row scale factors for A. 
[in]  C  C is REAL array, dimension (N) The column scale factors for A. 
[in]  ROWCND  ROWCND is REAL Ratio of the smallest R(i) to the largest R(i). 
[in]  COLCND  COLCND is REAL Ratio of the smallest C(i) to the largest C(i). 
[in]  AMAX  AMAX is REAL Absolute value of largest matrix entry. 
[out]  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.
Definition at line 142 of file slaqge.f.
subroutine stgex2  (  logical  WANTQ, 
logical  WANTZ,  
integer  N,  
real, dimension( lda, * )  A,  
integer  LDA,  
real, dimension( ldb, * )  B,  
integer  LDB,  
real, dimension( ldq, * )  Q,  
integer  LDQ,  
real, dimension( ldz, * )  Z,  
integer  LDZ,  
integer  J1,  
integer  N1,  
integer  N2,  
real, dimension( * )  WORK,  
integer  LWORK,  
integer  INFO  
) 
STGEX2 swaps adjacent diagonal blocks in an upper (quasi) triangular matrix pair by an orthogonal equivalence transformation.
Download STGEX2 + dependencies [TGZ] [ZIP] [TXT]STGEX2 swaps adjacent diagonal blocks (A11, B11) and (A22, B22) of size 1by1 or 2by2 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 1by1 and 2by2 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
[in]  WANTQ  WANTQ is LOGICAL .TRUE. : update the left transformation matrix Q; .FALSE.: do not update Q. 
[in]  WANTZ  WANTZ is LOGICAL .TRUE. : update the right transformation matrix Z; .FALSE.: do not update Z. 
[in]  N  N is INTEGER The order of the matrices A and B. N >= 0. 
[in,out]  A  A is REAL arrays, dimensions (LDA,N) On entry, the matrix A in the pair (A, B). On exit, the updated matrix A. 
[in]  LDA  LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N). 
[in,out]  B  B is REAL arrays, dimensions (LDB,N) On entry, the matrix B in the pair (A, B). On exit, the updated matrix B. 
[in]  LDB  LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N). 
[in,out]  Q  Q is REAL array, dimension (LDZ,N) On entry, if WANTQ = .TRUE., the orthogonal matrix Q. On exit, the updated matrix Q. Not referenced if WANTQ = .FALSE.. 
[in]  LDQ  LDQ is INTEGER The leading dimension of the array Q. LDQ >= 1. If WANTQ = .TRUE., LDQ >= N. 
[in,out]  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.. 
[in]  LDZ  LDZ is INTEGER The leading dimension of the array Z. LDZ >= 1. If WANTZ = .TRUE., LDZ >= N. 
[in]  J1  J1 is INTEGER The index to the first block (A11, B11). 1 <= J1 <= N. 
[in]  N1  N1 is INTEGER The order of the first block (A11, B11). N1 = 0, 1 or 2. 
[in]  N2  N2 is INTEGER The order of the second block (A22, B22). N2 = 0, 1 or 2. 
[out]  WORK  WORK is REAL array, dimension (MAX(1,LWORK)). 
[in]  LWORK  LWORK is INTEGER The dimension of the array WORK. LWORK >= MAX( N*(N2+N1), (N2+N1)*(N2+N1)*2 ) 
[out]  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 illconditioned. <0: If INFO = 16: LWORK is too small. Appropriate value for LWORK is returned in WORK(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 RealTime Applications, Kluwer Academic Publ. 1993, pp 195218. [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, S901 87 Umea, Sweden, 1994. Also as LAPACK Working Note 87. To appear in Numerical Algorithms, 1996.
Definition at line 221 of file stgex2.f.