COMPLEX*16 or DOUBLE COMPLEX routines for triangular, packed storage matrix
- ztpcon : Estimates the reciprocal of the condition number of a triangularmatrix in packed storage, in either the 1-norm or the infinity-norm.
- ztprfs : Provides forward and backward error bounds for the solutionof a triangular system of linear equations AX=B, A**T X=B orA**H X=B, where A is held in packed storage.
- ztptri : Computes the inverse of a triangular matrix in packed storage.
- ztptrs : Solves a triangular system of linear equations AX=B,A**T X=B or A**H X=B, where A is held in packed storage.
ztpcon
Estimates the reciprocal of the condition number of a triangularmatrix in packed storage, in either the 1-norm or the infinity-norm.
USAGE:
rcond, info = NumRu::Lapack.ztpcon( norm, uplo, diag, ap)
or
NumRu::Lapack.ztpcon # print help
FORTRAN MANUAL
SUBROUTINE ZTPCON( NORM, UPLO, DIAG, N, AP, RCOND, WORK, RWORK, INFO )
* Purpose
* =======
*
* ZTPCON estimates the reciprocal of the condition number of a packed
* triangular matrix A, in either the 1-norm or the infinity-norm.
*
* The norm of A is computed and an estimate is obtained for
* norm(inv(A)), then the reciprocal of the condition number is
* computed as
* RCOND = 1 / ( norm(A) * 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.
*
* UPLO (input) CHARACTER*1
* = 'U': A is upper triangular;
* = 'L': A is lower triangular.
*
* DIAG (input) CHARACTER*1
* = 'N': A is non-unit triangular;
* = 'U': A is unit triangular.
*
* N (input) INTEGER
* The order of the matrix A. N >= 0.
*
* AP (input) COMPLEX*16 array, dimension (N*(N+1)/2)
* The upper or lower triangular matrix A, packed columnwise in
* a linear array. The j-th column of A is stored in the array
* AP as follows:
* if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
* if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.
* If DIAG = 'U', the diagonal elements of A are not referenced
* and are assumed to be 1.
*
* RCOND (output) DOUBLE PRECISION
* The reciprocal of the condition number of the matrix A,
* computed as RCOND = 1/(norm(A) * norm(inv(A))).
*
* WORK (workspace) COMPLEX*16 array, dimension (2*N)
*
* RWORK (workspace) DOUBLE PRECISION 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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ztprfs
Provides forward and backward error bounds for the solutionof a triangular system of linear equations AX=B, A**T X=B orA**H X=B, where A is held in packed storage.
USAGE:
ferr, berr, info = NumRu::Lapack.ztprfs( uplo, trans, diag, ap, b, x)
or
NumRu::Lapack.ztprfs # print help
FORTRAN MANUAL
SUBROUTINE ZTPRFS( UPLO, TRANS, DIAG, N, NRHS, AP, B, LDB, X, LDX, FERR, BERR, WORK, RWORK, INFO )
* Purpose
* =======
*
* ZTPRFS provides error bounds and backward error estimates for the
* solution to a system of linear equations with a triangular packed
* coefficient matrix.
*
* The solution matrix X must be computed by ZTPTRS or some other
* means before entering this routine. ZTPRFS does not do iterative
* refinement because doing so cannot improve the backward error.
*
* Arguments
* =========
*
* UPLO (input) CHARACTER*1
* = 'U': A is upper triangular;
* = 'L': A is lower triangular.
*
* 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)
*
* DIAG (input) CHARACTER*1
* = 'N': A is non-unit triangular;
* = 'U': A is unit triangular.
*
* 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 matrices B and X. NRHS >= 0.
*
* AP (input) COMPLEX*16 array, dimension (N*(N+1)/2)
* The upper or lower triangular matrix A, packed columnwise in
* a linear array. The j-th column of A is stored in the array
* AP as follows:
* if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
* if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.
* If DIAG = 'U', the diagonal elements of A are not referenced
* and are assumed to be 1.
*
* B (input) COMPLEX*16 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) COMPLEX*16 array, dimension (LDX,NRHS)
* The solution matrix X.
*
* LDX (input) INTEGER
* The leading dimension of the array X. LDX >= max(1,N).
*
* FERR (output) DOUBLE PRECISION 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) DOUBLE PRECISION 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) COMPLEX*16 array, dimension (2*N)
*
* RWORK (workspace) DOUBLE PRECISION 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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ztptri
Computes the inverse of a triangular matrix in packed storage.
USAGE:
info, ap = NumRu::Lapack.ztptri( uplo, diag, n, ap)
or
NumRu::Lapack.ztptri # print help
FORTRAN MANUAL
SUBROUTINE ZTPTRI( UPLO, DIAG, N, AP, INFO )
* Purpose
* =======
*
* ZTPTRI computes the inverse of a complex upper or lower triangular
* matrix A stored in packed format.
*
* Arguments
* =========
*
* UPLO (input) CHARACTER*1
* = 'U': A is upper triangular;
* = 'L': A is lower triangular.
*
* DIAG (input) CHARACTER*1
* = 'N': A is non-unit triangular;
* = 'U': A is unit triangular.
*
* N (input) INTEGER
* The order of the matrix A. N >= 0.
*
* AP (input/output) COMPLEX*16 array, dimension (N*(N+1)/2)
* On entry, the upper or lower triangular matrix A, stored
* columnwise in a linear array. The j-th column of A is stored
* in the array AP as follows:
* if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
* if UPLO = 'L', AP(i + (j-1)*((2*n-j)/2) = A(i,j) for j<=i<=n.
* See below for further details.
* On exit, the (triangular) inverse of the original matrix, in
* the same packed storage format.
*
* INFO (output) INTEGER
* = 0: successful exit
* < 0: if INFO = -i, the i-th argument had an illegal value
* > 0: if INFO = i, A(i,i) is exactly zero. The triangular
* matrix is singular and its inverse can not be computed.
*
* Further Details
* ===============
*
* A triangular matrix A can be transferred to packed storage using one
* of the following program segments:
*
* UPLO = 'U': UPLO = 'L':
*
* JC = 1 JC = 1
* DO 2 J = 1, N DO 2 J = 1, N
* DO 1 I = 1, J DO 1 I = J, N
* AP(JC+I-1) = A(I,J) AP(JC+I-J) = A(I,J)
* 1 CONTINUE 1 CONTINUE
* JC = JC + J JC = JC + N - J + 1
* 2 CONTINUE 2 CONTINUE
*
* =====================================================================
*
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ztptrs
Solves a triangular system of linear equations AX=B,A**T X=B or A**H X=B, where A is held in packed storage.
USAGE:
info, b = NumRu::Lapack.ztptrs( uplo, trans, diag, n, ap, b)
or
NumRu::Lapack.ztptrs # print help
FORTRAN MANUAL
SUBROUTINE ZTPTRS( UPLO, TRANS, DIAG, N, NRHS, AP, B, LDB, INFO )
* Purpose
* =======
*
* ZTPTRS solves a triangular system of the form
*
* A * X = B, A**T * X = B, or A**H * X = B,
*
* where A is a triangular matrix of order N stored in packed format,
* and B is an N-by-NRHS matrix. A check is made to verify that A is
* nonsingular.
*
* Arguments
* =========
*
* UPLO (input) CHARACTER*1
* = 'U': A is upper triangular;
* = 'L': A is lower triangular.
*
* 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)
*
* DIAG (input) CHARACTER*1
* = 'N': A is non-unit triangular;
* = 'U': A is unit triangular.
*
* 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.
*
* AP (input) COMPLEX*16 array, dimension (N*(N+1)/2)
* The upper or lower triangular matrix A, packed columnwise in
* a linear array. The j-th column of A is stored in the array
* AP as follows:
* if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
* if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.
*
* B (input/output) COMPLEX*16 array, dimension (LDB,NRHS)
* On entry, the right hand side matrix B.
* On exit, if INFO = 0, the 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, the i-th diagonal element of A is zero,
* indicating that the matrix is singular and the
* solutions X have not been computed.
*
* =====================================================================
*
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