org.netlib.lapack
Class Dlatps
java.lang.Object
org.netlib.lapack.Dlatps
public class Dlatps
- extends java.lang.Object
Following is the description from the original
Fortran source. For each array argument, the Java
version will include an integer offset parameter, so
the arguments may not match the description exactly.
Contact seymour@cs.utk.edu with any questions.
* ..
*
* Purpose
* =======
*
* DLATPS solves one of the triangular systems
*
* A *x = s*b or A'*x = s*b
*
* with scaling to prevent overflow, where A is an upper or lower
* triangular matrix stored in packed form. Here A' denotes the
* transpose of A, x and b are n-element vectors, and s is a scaling
* factor, usually less than or equal to 1, chosen so that the
* components of x will be less than the overflow threshold. If the
* unscaled problem will not cause overflow, the Level 2 BLAS routine
* DTPSV is called. If the matrix A is singular (A(j,j) = 0 for some j),
* then s is set to 0 and a non-trivial solution to A*x = 0 is returned.
*
* Arguments
* =========
*
* UPLO (input) CHARACTER*1
* Specifies whether the matrix A is upper or lower triangular.
* = 'U': Upper triangular
* = 'L': Lower triangular
*
* TRANS (input) CHARACTER*1
* Specifies the operation applied to A.
* = 'N': Solve A * x = s*b (No transpose)
* = 'T': Solve A'* x = s*b (Transpose)
* = 'C': Solve A'* x = s*b (Conjugate transpose = Transpose)
*
* DIAG (input) CHARACTER*1
* Specifies whether or not the matrix A is unit triangular.
* = 'N': Non-unit triangular
* = 'U': Unit triangular
*
* NORMIN (input) CHARACTER*1
* Specifies whether CNORM has been set or not.
* = 'Y': CNORM contains the column norms on entry
* = 'N': CNORM is not set on entry. On exit, the norms will
* be computed and stored in CNORM.
*
* N (input) INTEGER
* The order of the matrix A. N >= 0.
*
* AP (input) DOUBLE PRECISION 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.
*
* X (input/output) DOUBLE PRECISION array, dimension (N)
* On entry, the right hand side b of the triangular system.
* On exit, X is overwritten by the solution vector x.
*
* SCALE (output) DOUBLE PRECISION
* The scaling factor s for the triangular system
* A * x = s*b or A'* x = s*b.
* If SCALE = 0, the matrix A is singular or badly scaled, and
* the vector x is an exact or approximate solution to A*x = 0.
*
* CNORM (input or output) DOUBLE PRECISION array, dimension (N)
*
* If NORMIN = 'Y', CNORM is an input argument and CNORM(j)
* contains the norm of the off-diagonal part of the j-th column
* of A. If TRANS = 'N', CNORM(j) must be greater than or equal
* to the infinity-norm, and if TRANS = 'T' or 'C', CNORM(j)
* must be greater than or equal to the 1-norm.
*
* If NORMIN = 'N', CNORM is an output argument and CNORM(j)
* returns the 1-norm of the offdiagonal part of the j-th column
* of A.
*
* INFO (output) INTEGER
* = 0: successful exit
* < 0: if INFO = -k, the k-th argument had an illegal value
*
* Further Details
* ======= =======
*
* A rough bound on x is computed; if that is less than overflow, DTPSV
* is called, otherwise, specific code is used which checks for possible
* overflow or divide-by-zero at every operation.
*
* A columnwise scheme is used for solving A*x = b. The basic algorithm
* if A is lower triangular is
*
* x[1:n] := b[1:n]
* for j = 1, ..., n
* x(j) := x(j) / A(j,j)
* x[j+1:n] := x[j+1:n] - x(j) * A[j+1:n,j]
* end
*
* Define bounds on the components of x after j iterations of the loop:
* M(j) = bound on x[1:j]
* G(j) = bound on x[j+1:n]
* Initially, let M(0) = 0 and G(0) = max{x(i), i=1,...,n}.
*
* Then for iteration j+1 we have
* M(j+1) <= G(j) / | A(j+1,j+1) |
* G(j+1) <= G(j) + M(j+1) * | A[j+2:n,j+1] |
* <= G(j) ( 1 + CNORM(j+1) / | A(j+1,j+1) | )
*
* where CNORM(j+1) is greater than or equal to the infinity-norm of
* column j+1 of A, not counting the diagonal. Hence
*
* G(j) <= G(0) product ( 1 + CNORM(i) / | A(i,i) | )
* 1<=i<=j
* and
*
* |x(j)| <= ( G(0) / |A(j,j)| ) product ( 1 + CNORM(i) / |A(i,i)| )
* 1<=i< j
*
* Since |x(j)| <= M(j), we use the Level 2 BLAS routine DTPSV if the
* reciprocal of the largest M(j), j=1,..,n, is larger than
* max(underflow, 1/overflow).
*
* The bound on x(j) is also used to determine when a step in the
* columnwise method can be performed without fear of overflow. If
* the computed bound is greater than a large constant, x is scaled to
* prevent overflow, but if the bound overflows, x is set to 0, x(j) to
* 1, and scale to 0, and a non-trivial solution to A*x = 0 is found.
*
* Similarly, a row-wise scheme is used to solve A'*x = b. The basic
* algorithm for A upper triangular is
*
* for j = 1, ..., n
* x(j) := ( b(j) - A[1:j-1,j]' * x[1:j-1] ) / A(j,j)
* end
*
* We simultaneously compute two bounds
* G(j) = bound on ( b(i) - A[1:i-1,i]' * x[1:i-1] ), 1<=i<=j
* M(j) = bound on x(i), 1<=i<=j
*
* The initial values are G(0) = 0, M(0) = max{b(i), i=1,..,n}, and we
* add the constraint G(j) >= G(j-1) and M(j) >= M(j-1) for j >= 1.
* Then the bound on x(j) is
*
* M(j) <= M(j-1) * ( 1 + CNORM(j) ) / | A(j,j) |
*
* <= M(0) * product ( ( 1 + CNORM(i) ) / |A(i,i)| )
* 1<=i<=j
*
* and we can safely call DTPSV if 1/M(n) and 1/G(n) are both greater
* than max(underflow, 1/overflow).
*
* =====================================================================
*
* .. Parameters ..
|
Method Summary |
static void |
dlatps(java.lang.String uplo,
java.lang.String trans,
java.lang.String diag,
java.lang.String normin,
int n,
double[] ap,
int _ap_offset,
double[] x,
int _x_offset,
doubleW scale,
double[] cnorm,
int _cnorm_offset,
intW info)
|
| Methods inherited from class java.lang.Object |
clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait |
Dlatps
public Dlatps()
dlatps
public static void dlatps(java.lang.String uplo,
java.lang.String trans,
java.lang.String diag,
java.lang.String normin,
int n,
double[] ap,
int _ap_offset,
double[] x,
int _x_offset,
doubleW scale,
double[] cnorm,
int _cnorm_offset,
intW info)