REAL routines for general tridiagonal matrix
sgtcon
USAGE:
rcond, info = NumRu::Lapack.sgtcon( norm, dl, d, du, du2, ipiv, anorm, [:usage => usage, :help => help])
FORTRAN MANUAL
SUBROUTINE SGTCON( NORM, N, DL, D, DU, DU2, IPIV, ANORM, RCOND, WORK, IWORK, INFO )
* Purpose
* =======
*
* SGTCON estimates the reciprocal of the condition number of a real
* tridiagonal matrix A using the LU factorization as computed by
* SGTTRF.
*
* An estimate is obtained for norm(inv(A)), and the reciprocal of the
* condition number is computed as RCOND = 1 / (ANORM * norm(inv(A))).
*
* Arguments
* =========
*
* NORM (input) CHARACTER*1
* Specifies whether the 1-norm condition number or the
* infinity-norm condition number is required:
* = '1' or 'O': 1-norm;
* = 'I': Infinity-norm.
*
* N (input) INTEGER
* The order of the matrix A. N >= 0.
*
* DL (input) REAL array, dimension (N-1)
* The (n-1) multipliers that define the matrix L from the
* LU factorization of A as computed by SGTTRF.
*
* D (input) REAL array, dimension (N)
* The n diagonal elements of the upper triangular matrix U from
* the LU factorization of A.
*
* DU (input) REAL array, dimension (N-1)
* The (n-1) elements of the first superdiagonal of U.
*
* DU2 (input) REAL array, dimension (N-2)
* The (n-2) elements of the second superdiagonal of U.
*
* IPIV (input) INTEGER array, dimension (N)
* The pivot indices; for 1 <= i <= n, row i of the matrix was
* interchanged with row IPIV(i). IPIV(i) will always be either
* i or i+1; IPIV(i) = i indicates a row interchange was not
* required.
*
* ANORM (input) REAL
* If NORM = '1' or 'O', the 1-norm of the original matrix A.
* If NORM = 'I', the infinity-norm of the original matrix A.
*
* RCOND (output) REAL
* The reciprocal of the condition number of the matrix A,
* computed as RCOND = 1/(ANORM * AINVNM), where AINVNM is an
* estimate of the 1-norm of inv(A) computed in this routine.
*
* WORK (workspace) REAL array, dimension (2*N)
*
* IWORK (workspace) INTEGER array, dimension (N)
*
* INFO (output) INTEGER
* = 0: successful exit
* < 0: if INFO = -i, the i-th argument had an illegal value
*
* =====================================================================
*
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sgtrfs
USAGE:
ferr, berr, info, x = NumRu::Lapack.sgtrfs( trans, dl, d, du, dlf, df, duf, du2, ipiv, b, x, [:usage => usage, :help => help])
FORTRAN MANUAL
SUBROUTINE SGTRFS( TRANS, N, NRHS, DL, D, DU, DLF, DF, DUF, DU2, IPIV, B, LDB, X, LDX, FERR, BERR, WORK, IWORK, INFO )
* Purpose
* =======
*
* SGTRFS improves the computed solution to a system of linear
* equations when the coefficient matrix is tridiagonal, and provides
* error bounds and backward error estimates for the solution.
*
* Arguments
* =========
*
* TRANS (input) CHARACTER*1
* Specifies the form of the system of equations:
* = 'N': A * X = B (No transpose)
* = 'T': A**T * X = B (Transpose)
* = 'C': A**H * X = B (Conjugate transpose = Transpose)
*
* N (input) INTEGER
* The order of the matrix A. N >= 0.
*
* NRHS (input) INTEGER
* The number of right hand sides, i.e., the number of columns
* of the matrix B. NRHS >= 0.
*
* DL (input) REAL array, dimension (N-1)
* The (n-1) subdiagonal elements of A.
*
* D (input) REAL array, dimension (N)
* The diagonal elements of A.
*
* DU (input) REAL array, dimension (N-1)
* The (n-1) superdiagonal elements of A.
*
* DLF (input) REAL array, dimension (N-1)
* The (n-1) multipliers that define the matrix L from the
* LU factorization of A as computed by SGTTRF.
*
* DF (input) REAL array, dimension (N)
* The n diagonal elements of the upper triangular matrix U from
* the LU factorization of A.
*
* DUF (input) REAL array, dimension (N-1)
* The (n-1) elements of the first superdiagonal of U.
*
* DU2 (input) REAL array, dimension (N-2)
* The (n-2) elements of the second superdiagonal of U.
*
* IPIV (input) INTEGER array, dimension (N)
* The pivot indices; for 1 <= i <= n, row i of the matrix was
* interchanged with row IPIV(i). IPIV(i) will always be either
* i or i+1; IPIV(i) = i indicates a row interchange was not
* required.
*
* B (input) REAL array, dimension (LDB,NRHS)
* The right hand side matrix B.
*
* LDB (input) INTEGER
* The leading dimension of the array B. LDB >= max(1,N).
*
* X (input/output) REAL array, dimension (LDX,NRHS)
* On entry, the solution matrix X, as computed by SGTTRS.
* On exit, the improved solution matrix X.
*
* LDX (input) INTEGER
* The leading dimension of the array X. LDX >= max(1,N).
*
* FERR (output) REAL array, dimension (NRHS)
* The estimated forward error bound for each solution vector
* X(j) (the j-th column of the solution matrix X).
* If XTRUE is the true solution corresponding to X(j), FERR(j)
* is an estimated upper bound for the magnitude of the largest
* element in (X(j) - XTRUE) divided by the magnitude of the
* largest element in X(j). The estimate is as reliable as
* the estimate for RCOND, and is almost always a slight
* overestimate of the true error.
*
* BERR (output) REAL array, dimension (NRHS)
* The componentwise relative backward error of each solution
* vector X(j) (i.e., the smallest relative change in
* any element of A or B that makes X(j) an exact solution).
*
* WORK (workspace) REAL array, dimension (3*N)
*
* IWORK (workspace) INTEGER array, dimension (N)
*
* INFO (output) INTEGER
* = 0: successful exit
* < 0: if INFO = -i, the i-th argument had an illegal value
*
* Internal Parameters
* ===================
*
* ITMAX is the maximum number of steps of iterative refinement.
*
* =====================================================================
*
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sgtsv
USAGE:
info, dl, d, du, b = NumRu::Lapack.sgtsv( dl, d, du, b, [:usage => usage, :help => help])
FORTRAN MANUAL
SUBROUTINE SGTSV( N, NRHS, DL, D, DU, B, LDB, INFO )
* Purpose
* =======
*
* SGTSV solves the equation
*
* A*X = B,
*
* where A is an n by n tridiagonal matrix, by Gaussian elimination with
* partial pivoting.
*
* Note that the equation A'*X = B may be solved by interchanging the
* order of the arguments DU and DL.
*
* Arguments
* =========
*
* N (input) INTEGER
* The order of the matrix A. N >= 0.
*
* NRHS (input) INTEGER
* The number of right hand sides, i.e., the number of columns
* of the matrix B. NRHS >= 0.
*
* DL (input/output) REAL array, dimension (N-1)
* On entry, DL must contain the (n-1) sub-diagonal elements of
* A.
*
* On exit, DL is overwritten by the (n-2) elements of the
* second super-diagonal of the upper triangular matrix U from
* the LU factorization of A, in DL(1), ..., DL(n-2).
*
* D (input/output) REAL array, dimension (N)
* On entry, D must contain the diagonal elements of A.
*
* On exit, D is overwritten by the n diagonal elements of U.
*
* DU (input/output) REAL array, dimension (N-1)
* On entry, DU must contain the (n-1) super-diagonal elements
* of A.
*
* On exit, DU is overwritten by the (n-1) elements of the first
* super-diagonal of U.
*
* B (input/output) REAL array, dimension (LDB,NRHS)
* On entry, the N by NRHS matrix of right hand side matrix B.
* On exit, if INFO = 0, the N by NRHS solution matrix X.
*
* LDB (input) INTEGER
* The leading dimension of the array B. LDB >= max(1,N).
*
* INFO (output) INTEGER
* = 0: successful exit
* < 0: if INFO = -i, the i-th argument had an illegal value
* > 0: if INFO = i, U(i,i) is exactly zero, and the solution
* has not been computed. The factorization has not been
* completed unless i = N.
*
* =====================================================================
*
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sgtsvx
USAGE:
x, rcond, ferr, berr, info, dlf, df, duf, du2, ipiv = NumRu::Lapack.sgtsvx( fact, trans, dl, d, du, dlf, df, duf, du2, ipiv, b, [:usage => usage, :help => help])
FORTRAN MANUAL
SUBROUTINE SGTSVX( FACT, TRANS, N, NRHS, DL, D, DU, DLF, DF, DUF, DU2, IPIV, B, LDB, X, LDX, RCOND, FERR, BERR, WORK, IWORK, INFO )
* Purpose
* =======
*
* SGTSVX uses the LU factorization to compute the solution to a real
* system of linear equations A * X = B or A**T * X = B,
* where A is a tridiagonal matrix of order N and X and B are N-by-NRHS
* matrices.
*
* Error bounds on the solution and a condition estimate are also
* provided.
*
* Description
* ===========
*
* The following steps are performed:
*
* 1. If FACT = 'N', the LU decomposition is used to factor the matrix A
* as A = L * U, where L is a product of permutation and unit lower
* bidiagonal matrices and U is upper triangular with nonzeros in
* only the main diagonal and first two superdiagonals.
*
* 2. If some U(i,i)=0, so that U is exactly singular, then the routine
* returns with INFO = i. Otherwise, the factored form of A is used
* to estimate the condition number of the matrix A. If the
* reciprocal of the condition number is less than machine precision,
* INFO = N+1 is returned as a warning, but the routine still goes on
* to solve for X and compute error bounds as described below.
*
* 3. The system of equations is solved for X using the factored form
* of A.
*
* 4. Iterative refinement is applied to improve the computed solution
* matrix and calculate error bounds and backward error estimates
* for it.
*
* Arguments
* =========
*
* FACT (input) CHARACTER*1
* Specifies whether or not the factored form of A has been
* supplied on entry.
* = 'F': DLF, DF, DUF, DU2, and IPIV contain the factored
* form of A; DL, D, DU, DLF, DF, DUF, DU2 and IPIV
* will not be modified.
* = 'N': The matrix will be copied to DLF, DF, and DUF
* and factored.
*
* TRANS (input) CHARACTER*1
* Specifies the form of the system of equations:
* = 'N': A * X = B (No transpose)
* = 'T': A**T * X = B (Transpose)
* = 'C': A**H * X = B (Conjugate transpose = Transpose)
*
* N (input) INTEGER
* The order of the matrix A. N >= 0.
*
* NRHS (input) INTEGER
* The number of right hand sides, i.e., the number of columns
* of the matrix B. NRHS >= 0.
*
* DL (input) REAL array, dimension (N-1)
* The (n-1) subdiagonal elements of A.
*
* D (input) REAL array, dimension (N)
* The n diagonal elements of A.
*
* DU (input) REAL array, dimension (N-1)
* The (n-1) superdiagonal elements of A.
*
* DLF (input or output) REAL array, dimension (N-1)
* If FACT = 'F', then DLF is an input argument and on entry
* contains the (n-1) multipliers that define the matrix L from
* the LU factorization of A as computed by SGTTRF.
*
* If FACT = 'N', then DLF is an output argument and on exit
* contains the (n-1) multipliers that define the matrix L from
* the LU factorization of A.
*
* DF (input or output) REAL array, dimension (N)
* If FACT = 'F', then DF is an input argument and on entry
* contains the n diagonal elements of the upper triangular
* matrix U from the LU factorization of A.
*
* If FACT = 'N', then DF is an output argument and on exit
* contains the n diagonal elements of the upper triangular
* matrix U from the LU factorization of A.
*
* DUF (input or output) REAL array, dimension (N-1)
* If FACT = 'F', then DUF is an input argument and on entry
* contains the (n-1) elements of the first superdiagonal of U.
*
* If FACT = 'N', then DUF is an output argument and on exit
* contains the (n-1) elements of the first superdiagonal of U.
*
* DU2 (input or output) REAL array, dimension (N-2)
* If FACT = 'F', then DU2 is an input argument and on entry
* contains the (n-2) elements of the second superdiagonal of
* U.
*
* If FACT = 'N', then DU2 is an output argument and on exit
* contains the (n-2) elements of the second superdiagonal of
* U.
*
* IPIV (input or output) INTEGER array, dimension (N)
* If FACT = 'F', then IPIV is an input argument and on entry
* contains the pivot indices from the LU factorization of A as
* computed by SGTTRF.
*
* If FACT = 'N', then IPIV is an output argument and on exit
* contains the pivot indices from the LU factorization of A;
* row i of the matrix was interchanged with row IPIV(i).
* IPIV(i) will always be either i or i+1; IPIV(i) = i indicates
* a row interchange was not required.
*
* B (input) REAL array, dimension (LDB,NRHS)
* The N-by-NRHS right hand side matrix B.
*
* LDB (input) INTEGER
* The leading dimension of the array B. LDB >= max(1,N).
*
* X (output) REAL array, dimension (LDX,NRHS)
* If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X.
*
* LDX (input) INTEGER
* The leading dimension of the array X. LDX >= max(1,N).
*
* RCOND (output) REAL
* The estimate of the reciprocal condition number of the matrix
* A. If RCOND is less than the machine precision (in
* particular, if RCOND = 0), the matrix is singular to working
* precision. This condition is indicated by a return code of
* INFO > 0.
*
* FERR (output) REAL array, dimension (NRHS)
* The estimated forward error bound for each solution vector
* X(j) (the j-th column of the solution matrix X).
* If XTRUE is the true solution corresponding to X(j), FERR(j)
* is an estimated upper bound for the magnitude of the largest
* element in (X(j) - XTRUE) divided by the magnitude of the
* largest element in X(j). The estimate is as reliable as
* the estimate for RCOND, and is almost always a slight
* overestimate of the true error.
*
* BERR (output) REAL array, dimension (NRHS)
* The componentwise relative backward error of each solution
* vector X(j) (i.e., the smallest relative change in
* any element of A or B that makes X(j) an exact solution).
*
* WORK (workspace) REAL array, dimension (3*N)
*
* IWORK (workspace) INTEGER array, dimension (N)
*
* INFO (output) INTEGER
* = 0: successful exit
* < 0: if INFO = -i, the i-th argument had an illegal value
* > 0: if INFO = i, and i is
* <= N: U(i,i) is exactly zero. The factorization
* has not been completed unless i = N, but the
* factor U is exactly singular, so the solution
* and error bounds could not be computed.
* RCOND = 0 is returned.
* = N+1: U is nonsingular, but RCOND is less than machine
* precision, meaning that the matrix is singular
* to working precision. Nevertheless, the
* solution and error bounds are computed because
* there are a number of situations where the
* computed solution can be more accurate than the
* value of RCOND would suggest.
*
* =====================================================================
*
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sgttrf
USAGE:
du2, ipiv, info, dl, d, du = NumRu::Lapack.sgttrf( dl, d, du, [:usage => usage, :help => help])
FORTRAN MANUAL
SUBROUTINE SGTTRF( N, DL, D, DU, DU2, IPIV, INFO )
* Purpose
* =======
*
* SGTTRF computes an LU factorization of a real tridiagonal matrix A
* using elimination with partial pivoting and row interchanges.
*
* The factorization has the form
* A = L * U
* where L is a product of permutation and unit lower bidiagonal
* matrices and U is upper triangular with nonzeros in only the main
* diagonal and first two superdiagonals.
*
* Arguments
* =========
*
* N (input) INTEGER
* The order of the matrix A.
*
* DL (input/output) REAL array, dimension (N-1)
* On entry, DL must contain the (n-1) sub-diagonal elements of
* A.
*
* On exit, DL is overwritten by the (n-1) multipliers that
* define the matrix L from the LU factorization of A.
*
* D (input/output) REAL array, dimension (N)
* On entry, D must contain the diagonal elements of A.
*
* On exit, D is overwritten by the n diagonal elements of the
* upper triangular matrix U from the LU factorization of A.
*
* DU (input/output) REAL array, dimension (N-1)
* On entry, DU must contain the (n-1) super-diagonal elements
* of A.
*
* On exit, DU is overwritten by the (n-1) elements of the first
* super-diagonal of U.
*
* DU2 (output) REAL array, dimension (N-2)
* On exit, DU2 is overwritten by the (n-2) elements of the
* second super-diagonal of U.
*
* IPIV (output) INTEGER array, dimension (N)
* The pivot indices; for 1 <= i <= n, row i of the matrix was
* interchanged with row IPIV(i). IPIV(i) will always be either
* i or i+1; IPIV(i) = i indicates a row interchange was not
* required.
*
* INFO (output) INTEGER
* = 0: successful exit
* < 0: if INFO = -k, the k-th argument had an illegal value
* > 0: if INFO = k, U(k,k) is exactly zero. The factorization
* has been completed, but the factor U is exactly
* singular, and division by zero will occur if it is used
* to solve a system of equations.
*
* =====================================================================
*
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sgttrs
USAGE:
info, b = NumRu::Lapack.sgttrs( trans, dl, d, du, du2, ipiv, b, [:usage => usage, :help => help])
FORTRAN MANUAL
SUBROUTINE SGTTRS( TRANS, N, NRHS, DL, D, DU, DU2, IPIV, B, LDB, INFO )
* Purpose
* =======
*
* SGTTRS solves one of the systems of equations
* A*X = B or A'*X = B,
* with a tridiagonal matrix A using the LU factorization computed
* by SGTTRF.
*
* Arguments
* =========
*
* TRANS (input) CHARACTER*1
* Specifies the form of the system of equations.
* = 'N': A * X = B (No transpose)
* = 'T': A'* X = B (Transpose)
* = 'C': A'* X = B (Conjugate transpose = Transpose)
*
* N (input) INTEGER
* The order of the matrix A.
*
* NRHS (input) INTEGER
* The number of right hand sides, i.e., the number of columns
* of the matrix B. NRHS >= 0.
*
* DL (input) REAL array, dimension (N-1)
* The (n-1) multipliers that define the matrix L from the
* LU factorization of A.
*
* D (input) REAL array, dimension (N)
* The n diagonal elements of the upper triangular matrix U from
* the LU factorization of A.
*
* DU (input) REAL array, dimension (N-1)
* The (n-1) elements of the first super-diagonal of U.
*
* DU2 (input) REAL array, dimension (N-2)
* The (n-2) elements of the second super-diagonal of U.
*
* IPIV (input) INTEGER array, dimension (N)
* The pivot indices; for 1 <= i <= n, row i of the matrix was
* interchanged with row IPIV(i). IPIV(i) will always be either
* i or i+1; IPIV(i) = i indicates a row interchange was not
* required.
*
* B (input/output) REAL array, dimension (LDB,NRHS)
* On entry, the matrix of right hand side vectors B.
* On exit, B is overwritten by the solution vectors X.
*
* LDB (input) INTEGER
* The leading dimension of the array B. LDB >= max(1,N).
*
* INFO (output) INTEGER
* = 0: successful exit
* < 0: if INFO = -i, the i-th argument had an illegal value
*
* =====================================================================
*
* .. Local Scalars ..
LOGICAL NOTRAN
INTEGER ITRANS, J, JB, NB
* ..
* .. External Functions ..
INTEGER ILAENV
EXTERNAL ILAENV
* ..
* .. External Subroutines ..
EXTERNAL SGTTS2, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX, MIN
* ..
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sgtts2
USAGE:
b = NumRu::Lapack.sgtts2( itrans, dl, d, du, du2, ipiv, b, [:usage => usage, :help => help])
FORTRAN MANUAL
SUBROUTINE SGTTS2( ITRANS, N, NRHS, DL, D, DU, DU2, IPIV, B, LDB )
* Purpose
* =======
*
* SGTTS2 solves one of the systems of equations
* A*X = B or A'*X = B,
* with a tridiagonal matrix A using the LU factorization computed
* by SGTTRF.
*
* Arguments
* =========
*
* ITRANS (input) INTEGER
* Specifies the form of the system of equations.
* = 0: A * X = B (No transpose)
* = 1: A'* X = B (Transpose)
* = 2: A'* X = B (Conjugate transpose = Transpose)
*
* N (input) INTEGER
* The order of the matrix A.
*
* NRHS (input) INTEGER
* The number of right hand sides, i.e., the number of columns
* of the matrix B. NRHS >= 0.
*
* DL (input) REAL array, dimension (N-1)
* The (n-1) multipliers that define the matrix L from the
* LU factorization of A.
*
* D (input) REAL array, dimension (N)
* The n diagonal elements of the upper triangular matrix U from
* the LU factorization of A.
*
* DU (input) REAL array, dimension (N-1)
* The (n-1) elements of the first super-diagonal of U.
*
* DU2 (input) REAL array, dimension (N-2)
* The (n-2) elements of the second super-diagonal of U.
*
* IPIV (input) INTEGER array, dimension (N)
* The pivot indices; for 1 <= i <= n, row i of the matrix was
* interchanged with row IPIV(i). IPIV(i) will always be either
* i or i+1; IPIV(i) = i indicates a row interchange was not
* required.
*
* B (input/output) REAL array, dimension (LDB,NRHS)
* On entry, the matrix of right hand side vectors B.
* On exit, B is overwritten by the solution vectors X.
*
* LDB (input) INTEGER
* The leading dimension of the array B. LDB >= max(1,N).
*
* =====================================================================
*
* .. Local Scalars ..
INTEGER I, IP, J
REAL TEMP
* ..
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