Collaboration diagram for double:

Functions/Subroutines
subroutine	dgesc2 (N, A, LDA, RHS, IPIV, JPIV, SCALE)
	DGESC2 solves a system of linear equations using the LU factorization with complete pivoting computed by sgetc2.
subroutine	dgetc2 (N, A, LDA, IPIV, JPIV, INFO)
	DGETC2 computes the LU factorization with complete pivoting of the general n-by-n matrix.
DOUBLE PRECISION function	dlange (NORM, M, N, A, LDA, WORK)
	DLANGE returns the value of the 1-norm, Frobenius norm, infinity-norm, or the largest absolute value of any element of a general rectangular matrix.
subroutine	dlaqge (M, N, A, LDA, R, C, ROWCND, COLCND, AMAX, EQUED)
	DLAQGE scales a general rectangular matrix, using row and column scaling factors computed by sgeequ.
subroutine	dtgex2 (WANTQ, WANTZ, N, A, LDA, B, LDB, Q, LDQ, Z, LDZ, J1, N1, N2, WORK, LWORK, INFO)
	DTGEX2 swaps adjacent diagonal blocks in an upper (quasi) triangular matrix pair by an orthogonal equivalence transformation.

Detailed Description

This is the group of double auxiliary functions for GE matrices

Function/Subroutine Documentation

subroutine dgesc2	(	integer	N,
		double precision, dimension( lda, * )	A,
		integer	LDA,
		double precision, dimension( * )	RHS,
		integer, dimension( * )	IPIV,
		integer, dimension( * )	JPIV,
		double precision	SCALE
	)

DGESC2 solves a system of linear equations using the LU factorization with complete pivoting computed by sgetc2.

Download DGESC2 + dependencies [TGZ] [ZIP] [TXT]

Purpose:

 DGESC2 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 DGETC2.

Parameters:

[in]	N	N is INTEGER The order of the matrix A.
[in]	A	A is DOUBLE PRECISION array, dimension (LDA,N) On entry, the LU part of the factorization of the n-by-n matrix A computed by DGETC2: 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 DOUBLE PRECISION 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 DOUBLE PRECISION On exit, SCALE contains the scale factor. SCALE is chosen 0 <= SCALE <= 1 to prevent owerflow in the solution.

Author:: Univ. of Tennessee; Univ. of California Berkeley; Univ. of Colorado Denver; NAG Ltd.

Date:: September 2012

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

Definition at line 115 of file dgesc2.f.

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subroutine dgetc2	(	integer	N,
		double precision, dimension( lda, * )	A,
		integer	LDA,
		integer, dimension( * )	IPIV,
		integer, dimension( * )	JPIV,
		integer	INFO
	)

DGETC2 computes the LU factorization with complete pivoting of the general n-by-n matrix.

Download DGETC2 + dependencies [TGZ] [ZIP] [TXT]

Purpose:

 DGETC2 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.

Parameters:

[in]	N	N is INTEGER The order of the matrix A. N >= 0.
[in,out]	A	A is DOUBLE PRECISION 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 = PLU*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.

Author:: Univ. of Tennessee; Univ. of California Berkeley; Univ. of Colorado Denver; NAG Ltd.

Date:: September 2012

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

Definition at line 112 of file dgetc2.f.

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DOUBLE PRECISION function dlange	(	character	NORM,
		integer	M,
		integer	N,
		double precision, dimension( lda, * )	A,
		integer	LDA,
		double precision, dimension( * )	WORK
	)

DLANGE returns the value of the 1-norm, Frobenius norm, infinity-norm, or the largest absolute value of any element of a general rectangular matrix.

Download DLANGE + dependencies [TGZ] [ZIP] [TXT]

Purpose:

 DLANGE  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.

Returns:

DLANGE

    DLANGE = ( 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.

Parameters:

[in]	NORM	NORM is CHARACTER*1 Specifies the value to be returned in DLANGE as described above.
[in]	M	M is INTEGER The number of rows of the matrix A. M >= 0. When M = 0, DLANGE is set to zero.
[in]	N	N is INTEGER The number of columns of the matrix A. N >= 0. When N = 0, DLANGE is set to zero.
[in]	A	A is DOUBLE PRECISION 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 DOUBLE PRECISION array, dimension (MAX(1,LWORK)), where LWORK >= M when NORM = 'I'; otherwise, WORK is not referenced.

Author:: Univ. of Tennessee; Univ. of California Berkeley; Univ. of Colorado Denver; NAG Ltd.

Date:: September 2012

Definition at line 115 of file dlange.f.

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subroutine dlaqge	(	integer	M,
		integer	N,
		double precision, dimension( lda, * )	A,
		integer	LDA,
		double precision, dimension( * )	R,
		double precision, dimension( * )	C,
		double precision	ROWCND,
		double precision	COLCND,
		double precision	AMAX,
		character	EQUED
	)

DLAQGE scales a general rectangular matrix, using row and column scaling factors computed by sgeequ.

Download DLAQGE + dependencies [TGZ] [ZIP] [TXT]

Purpose:

 DLAQGE equilibrates a general M by N matrix A using the row and
 column scaling factors in the vectors R and C.

Parameters:

[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 DOUBLE PRECISION 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 DOUBLE PRECISION array, dimension (M) The row scale factors for A.
[in]	C	C is DOUBLE PRECISION array, dimension (N) The column scale factors for A.
[in]	ROWCND	ROWCND is DOUBLE PRECISION Ratio of the smallest R(i) to the largest R(i).
[in]	COLCND	COLCND is DOUBLE PRECISION Ratio of the smallest C(i) to the largest C(i).
[in]	AMAX	AMAX is DOUBLE PRECISION Absolute value of largest matrix entry.
[out]	EQUED	EQUED is CHARACTER1 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).

Internal Parameters:

  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.

Author:: Univ. of Tennessee; Univ. of California Berkeley; Univ. of Colorado Denver; NAG Ltd.

Date:: September 2012

Definition at line 142 of file dlaqge.f.

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subroutine dtgex2	(	logical	WANTQ,
		logical	WANTZ,
		integer	N,
		double precision, dimension( lda, * )	A,
		integer	LDA,
		double precision, dimension( ldb, * )	B,
		integer	LDB,
		double precision, dimension( ldq, * )	Q,
		integer	LDQ,
		double precision, dimension( ldz, * )	Z,
		integer	LDZ,
		integer	J1,
		integer	N1,
		integer	N2,
		double precision, dimension( * )	WORK,
		integer	LWORK,
		integer	INFO
	)

DTGEX2 swaps adjacent diagonal blocks in an upper (quasi) triangular matrix pair by an orthogonal equivalence transformation.

Download DTGEX2 + dependencies [TGZ] [ZIP] [TXT]

Purpose:

 DTGEX2 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 DGGES), 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

Parameters:

[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 DOUBLE PRECISION array, 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 DOUBLE PRECISION array, 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 DOUBLE PRECISION array, dimension (LDQ,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 DOUBLE PRECISION 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 DOUBLE PRECISION array, dimension (MAX(1,LWORK)).
[in]	LWORK	LWORK is INTEGER The dimension of the array WORK. LWORK >= MAX( 1, 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 ill-conditioned. <0: If INFO = -16: LWORK is too small. Appropriate value for LWORK is returned in WORK(1).

Author:: Univ. of Tennessee; Univ. of California Berkeley; Univ. of Colorado Denver; NAG Ltd.

Date:: September 2012

Further Details:: 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.

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.

Definition at line 221 of file dtgex2.f.

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Functions/Subroutines

Detailed Description

Function/Subroutine Documentation