DPTSVX(1) LAPACK routine (version 3.2) DPTSVX(1)[top]NAMEDPTSVX - uses the factorization A = L*D*L**T to compute the solution to a real system of linear equations A*X = B, where A is an N-by-N symmet‐ ric positive definite tridiagonal matrix and X and B are N-by-NRHS matricesSYNOPSISSUBROUTINE DPTSVX( FACT, N, NRHS, D, E, DF, EF, B, LDB, X, LDX, RCOND, FERR, BERR, WORK, INFO ) CHARACTER FACT INTEGER INFO, LDB, LDX, N, NRHS DOUBLE PRECISION RCOND DOUBLE PRECISION B( LDB, * ), BERR( * ), D( * ), DF( * ), E( * ), EF( * ), FERR( * ), WORK( * ), X( LDX, * )PURPOSEDPTSVX uses the factorization A = L*D*L**T to compute the solution to a real system of linear equations A*X = B, where A is an N-by-N symmetric positive definite tridiagonal matrix and X and B are N-by-NRHS matri‐ ces. Error bounds on the solution and a condition estimate are also provided.DESCRIPTIONThe following steps are performed: 1. If FACT = 'N', the matrix A is factored as A = L*D*L**T, where L is a unit lower bidiagonal matrix and D is diagonal. The factorization can also be regarded as having the form A = U**T*D*U. 2. If the leading i-by-i principal minor is not positive definite, 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.ARGUMENTSFACT (input) CHARACTER*1 Specifies whether or not the factored form of A has been sup‐ plied on entry. = 'F': On entry, DF and EF contain the fac‐ tored form of A. D, E, DF, and EF will not be modified. = 'N': The matrix A will be copied to DF and EF and factored. 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. D (input) DOUBLE PRECISION array, dimension (N) The n diagonal elements of the tridiagonal matrix A. E (input) DOUBLE PRECISION array, dimension (N-1) The (n-1) subdiagonal elements of the tridiagonal matrix A. DF (input or output) DOUBLE PRECISION array, dimension (N) If FACT = 'F', then DF is an input argument and on entry con‐ tains the n diagonal elements of the diagonal matrix D from the L*D*L**T factorization of A. If FACT = 'N', then DF is an out‐ put argument and on exit contains the n diagonal elements of the diagonal matrix D from the L*D*L**T factorization of A. EF (input or output) DOUBLE PRECISION array, dimension (N-1) If FACT = 'F', then EF is an input argument and on entry con‐ tains the (n-1) subdiagonal elements of the unit bidiagonal factor L from the L*D*L**T factorization of A. If FACT = 'N', then EF is an output argument and on exit contains the (n-1) subdiagonal elements of the unit bidiagonal factor L from the L*D*L**T factorization of A. B (input) DOUBLE PRECISION 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) DOUBLE PRECISION array, dimension (LDX,NRHS) If INFO = 0 of 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) DOUBLE PRECISION 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) DOUBLE PRECISION array, dimension (NRHS) The forward error bound for each solution vector X(j) (the j-th column of the solution matrix X). If XTRUE is the true solu‐ tion 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). BERR (output) DOUBLE PRECISION array, dimension (NRHS) The componentwise relative backward error of each solution vec‐ tor X(j) (i.e., the smallest relative change in any element of A or B that makes X(j) an exact solution). WORK (workspace) DOUBLE PRECISION array, dimension (2*N) INFO (output) INTEGER = 0: successful exit < 0: if INFO =, the i-th argument had an illegal value > 0: if INFO = i, and i is <= N: the leading minor of order i of A is not positive defi‐ nite, so the factorization could not be completed, and the solution has not been 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. Nev‐ ertheless, 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. LAPACK routine (version 3.2) November 2008 DPTSVX(1)-i

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