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NAG[f04mcc] NAG[nag_real_cholesky_skyline_solve] - Approximate solution of real symmetric positive-definite variable-bandwidth simultaneous linear equations (coefficient matrix already factorized by f01mcc (nag_real_cholesky_skyline))
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Calling Sequence
f04mcc(selct, al, d, row, b, x, 'n'=n, 'nrhs'=nrhs, 'lal'=lal, 'tdb'=tdb, 'tdx'=tdx, 'fail'=fail)
nag_real_cholesky_skyline_solve(. . .)
Parameters
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al - Vector(1..lal, datatype=float[8]);
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On entry: the elements within the envelope of the lower triangular matrix  , taken in row by row order, as returned by f01mcc (nag_real_cholesky_skyline). The unit diagonal elements of  must be stored explicitly.
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d - Vector(1..n, datatype=float[8]);
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row - Vector(1..n, datatype=integer[kernelopts('wordsize')/8]);
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Constraint: ![1 <= row[i+1] and row[i+1] <= i+1](/support/helpjp/helpview.aspx?si=10087/file08071/math121.png) for  . .
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b - Matrix(1..n, 1..tdb, datatype=float[8], order=C_order);
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x - Matrix(1..n, 1..tdx, datatype=float[8], order=C_order);
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'n'=n - integer; (optional)
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Default value: the first dimension of the arrays d, row, b, x.
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On entry:  , the order of the matrix  .
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Constraint:  . .
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'nrhs'=nrhs - integer; (optional)
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Default value: the second dimension of the arrays b, x.
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On entry:  , the number of right-hand sides.
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Constraint:  . .
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'lal'=lal - integer; (optional)
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Default value: the dimension of the array al.
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On entry: the dimension of the array al as declared in the function from which nag_real_cholesky_skyline_solve (f04mcc) is called.
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Constraint: ![lal>=row[1]+row[2]+⋯+row[n]](/support/helpjp/helpview.aspx?si=10087/file08071/math230.png) . .
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'tdb'=tdb - integer; (optional)
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On entry: the second dimension of the array b as declared in the function from which nag_real_cholesky_skyline_solve (f04mcc) is called.
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Constraint:  . .
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'tdx'=tdx - integer; (optional)
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On entry: the second dimension of the array x as declared in the function from which nag_real_cholesky_skyline_solve (f04mcc) is called.
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Constraint:  . .
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'fail'=fail - table; (optional)
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The NAG error argument, see the documentation for NagError.
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Description
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Purpose
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nag_real_cholesky_skyline_solve (f04mcc) computes the approximate solution of a system of real linear equations with multiple right-hand sides,  , where  is a symmetric positive-definite variable-bandwidth matrix, which has previously been factorized by f01mcc (nag_real_cholesky_skyline). Related systems may also be solved.
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Description
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The normal use of nag_real_cholesky_skyline_solve (f04mcc) is the solution of the systems  , following a call of f01mcc (nag_real_cholesky_skyline) to determine the Cholesky factorization  of the symmetric positive-definite variable-bandwidth matrix  .
However, the function may be used to solve any one of the following systems of linear algebraic equations:
(1)
(2)
(3)
(4)
(5)
(6)
 denotes a unit lower triangular variable-bandwidth matrix of order  ,  a diagonal matrix of order  , and  a set of right-hand sides.
The matrix  is represented by the elements lying within its envelope, i.e., between the first non-zero of each row and the diagonal (see Section [Example] for an example). The width ![row[i+1]](/support/helpjp/helpview.aspx?si=10087/file08071/math344.png) of the  th row is the number of elements between the first non-zero element and the element on the diagonal inclusive.
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Error Indicators and Warnings
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"NE_2_INT_ARG_GT"
On entry, ![row[i+1]=< >](/support/helpjp/helpview.aspx?si=10087/file08071/math356.png) while  . These arguments must satisfy ![row[i+1] <= i+1](/support/helpjp/helpview.aspx?si=10087/file08071/math360.png) .
"NE_2_INT_ARG_LT"
On entry,  while ![row[1]+⋯+row[n]=< >](/support/helpjp/helpview.aspx?si=10087/file08071/math368.png) . These arguments must satisfy ![lal>=row[1]+⋯+row[n]](/support/helpjp/helpview.aspx?si=10087/file08071/math370.png) .
"NE_BAD_PARAM"
On entry, argument selct had an illegal value.
"NE_INT_ARG_LT"
On entry, n must not be less than 1:  .
"NE_NOT_UNIT_DIAG"
The lower triangular matrix  has at least one diagonal element which is not equal to unity. The first non-unit element has been located in the array ![al[< >+1]](/support/helpjp/helpview.aspx?si=10087/file08071/math392.png)
"NE_ZERO_DIAG"
The diagonal matrix  is singular as it has at least one zero element. The first zero element has been located in the array ![d[< >+1]](/support/helpjp/helpview.aspx?si=10087/file08071/math400.png)
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Accuracy
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The usual backward error analysis of the solution of triangular system applies: each computed solution vector is exact for slightly perturbed matrices  and  , as appropriate (see pages 25-27 and 54-55 of Wilkinson and Reinsch (1971)).
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Further Comments
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The time taken by nag_real_cholesky_skyline_solve (f04mcc) is approximately proportional to  , where ![p=row[1]+row[2]+⋯+row[n]](/support/helpjp/helpview.aspx?si=10087/file08071/math420.png) .
The function may be called with the same actual array supplied for the arguments b and x, in which case the solution matrix will overwrite the right-hand side matrix.
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Examples
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selct := "Nag_LDLTX":
n := 6:
nrhs := 2:
lal := 14:
tdb := 2:
tdx := 2:
al := Vector([1, 2, 1, 3, 1, 1, 5, 4, 1.5, 0.5, 1, 1.5, 5, 1], datatype=float[8]):
d := Vector([1, 1, 4, 16, 1, 16], datatype=float[8]):
arow := Vector([1, 2, 2, 1, 5, 3], datatype=integer[kernelopts('wordsize')/8]):
b := Matrix([[6, -10], [15, -21], [11, -3], [0, 24], [51, -39], [46, 67]], datatype=float[8], order='C_order'):
x := Matrix(6, 2, datatype=float[8], order='C_order'):
NAG:-f04mcc(selct, al, d, arow, b, x, 'n' = n, 'nrhs' = nrhs, 'lal' = lal, 'tdb' = tdb, 'tdx' = tdx):
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