bspline_defc_module Module

The point where the variance is desired

The number of discrete (X,Y) pairs for which dfc calculated a piece-wise polynomial curve.

The number of conditions that constrained the B-spline in dfc.

The order of the B-spline used in dfc. The value of NORD must satisfy 1 < NORD < 20 .

The number of knots in the array BKPT(). The value of NBKPT must satisfy NBKPT .GE. 2NORD.

The array of knots. Normally the problem data interval will be included between the limits BKPT(NORD) and BKPT(NBKPT-NORD+1). The additional end knots BKPT(I),I=1,...,NORD-1 and I=NBKPT-NORD+2,...,NBKPT, are required by dfc to compute the functions used to fit the data.

Real work array as used in dfc. See dfc for the required length of W(). The contents of W() must not be modified by the user if the variance function is desired.

number of points (size of xdata and ydata). Any non-negative value of NDATA is allowed. A negative value of NDATA is an error.

X data array. No sorting of XDATA(*) is required.

Y data array.

Y value standard deviation or uncertainty. A zero value for any entry of SDDATA(*) will weight that data point as 1. Otherwise the weight of that data point is the reciprocal of this entry.

B-spline order. (The order of the spline is one more than the degree of the piecewise polynomial defined on each interval. This is consistent with the B-spline package convention. For example, NORD=4 when we are using piecewise cubics.) NORD must be in the range 1 <= NORD <= 20.

The value of NBKPT must satisfy the condition NBKPT >= 2*NORD.

NBKPT knots of the B-spline. Normally the problem data interval will be included between the limits BKPT(NORD) and BKPT(NBKPT-NORD+1). The additional end knots BKPT(I),I=1,...,NORD-1 and I=NBKPT-NORD+2,...,NBKPT, are required to compute the functions used to fit the data. No sorting of BKPT(*) is required. Internal to DEFC the extreme end knots may be reduced and increased respectively to accommodate any data values that are exterior to the given knot values. The contents of BKPT(*) is not changed.

An integer flag, with one of two possible values (1 or 2), that directs the subprogram action with regard to new data points provided by the user:

An output flag that indicates the status of the curve fit:

If the output value of MDEOUT=1, this array contains the unknowns obtained from the least squares fitting process. These N=NBKPT-NORD parameters are the B-spline coefficients. For MDEOUT=2, not enough data was processed to uniquely determine the B-spline coefficients. In this case, and also when MDEOUT=-1, all values of COEFF(*) are set to zero.

The amount of working storage actually allocated for the working array W(*). This quantity is compared with the actual amount of storage needed in DEFC. Insufficient storage allocated for W(*) is an error. This feature was included in DEFC because misreading the storage formula for W(*) might very well lead to subtle and hard-to-find programming bugs.

Working Array. Its length is specified as an input parameter in LW as noted above. The contents of W(*) must not be modified by the user between calls to DEFC with values of MDEIN=1,2,2,... . The first (NBKPT-NORD+3)*(NORD+1) entries of W(*) are acceptable as direct input to DFC for an "old problem" only when MDEOUT=1 or 2.

G(MDG,NB+1)

The number of rows in the working array G(*,*). The value of MDG should be >= MU. The value of MU is defined in the abstract of these subprograms.

The bandwidth of the data matrix A.

Input Set by the user to the value 1 before the first call to DBNDAC. Its subsequent value is controlled by DBNDAC to set up for the next call to DBNDAC.

Input Index of the row of G(*,*) where the user is to place the new block of data (C F). Set by the user to the value 1 before the first call to DBNDAC. Its subsequent value is controlled by DBNDAC. A value of IR > MDG is considered an error.

Set by the user to indicate the number of new rows of data in the block

Set by the user to indicate the index of the first nonzero column in that set of rows (E F) = (0 C 0 F) being processed.

Set by the user to one of the values 1, 2, or 3. These values respectively indicate that the solution of AX = B, YR = H or RZ = W is required.

G(MDG,NB+1)

The number of rows in the working array G(*,*). The value of MDG should be >= MU. The value of MU is defined in the abstract of these subprograms.

This argument has the same meaning and contents as following the last call to DBNDAC.

This argument has the same meaning and contents as following the last call to DBNDAC.

This argument has the same meaning and contents as following the last call to DBNDAC.

X(N)

The number of variables in the solution vector. If any of the N diagonal terms are zero the subroutine DBNDSL prints an appropriate message. This condition is considered an error.

If MODE=1, RNORM is the Euclidean length of the residual vector AX-B. When MODE=2 or 3 RNORM` is set to zero.

JW : added

JW : added

JW : added

1 or 2 to select algorithm H1 or H2 .

the index of the pivot element.

If L1 <= M the transformation will be constructed to zero elements indexed from L1 through M. If L1 > M the subroutine does an identity transformation.

see l1

On entry to H1 U() contains the pivot vector. On exit from H1 U() and UP contain quantities defining the vector U of the Householder transformation. On entry to H2 U() and UP should contain quantities previously computed by H1. These will not be modified by H2.

the storage increment between elements of U.

see u

On entry to H1 or H2 C() contains a matrix which will be regarded as a set of vectors to which the Householder transformation is to be applied. On exit C() contains the set of transformed vectors.

Storage increment between elements of vectors in C().

Storage increment between vectors in C().

Number of vectors in C() to be transformed. If NCV <= 0 no operations will be done on C().

number of values in array DX to be sorted

control parameter: * Kflag < 0 : sort DX in decreasing order and optionally carry DY along. * Kflag > 0 : sort DX in increasing order and optionally carry DY along.

array of values to be sorted (usually abscissas)

array to be (optionally) carried along

array of values to be sorted

array to be (optionally) carried along

number of points (size of xdata and ydata). Any non-negative value of NDATA is allowed. A negative value of NDATA is an error.

X data array. No sorting of XDATA(*) is required.

Y data array.

Y value standard deviation or uncertainty. A zero value for any entry of SDDATA(*) will weight that data point as 1. Otherwise the weight of that data point is the reciprocal of this entry.

B-spline order. (The order of the spline is one more than the degree of the piecewise polynomial defined on each interval. This is consistent with the B-spline package convention. For example, NORD=4 when we are using piecewise cubics.) NORD must be in the range 1 <= NORD <= 20.

The value of NBKPT must satisfy the condition NBKPT >= 2*NORD.

NBKPT knots of the B-spline. Normally the problem data interval will be included between the limits BKPT(NORD) and BKPT(NBKPT-NORD+1). The additional end knots BKPT(I),I=1,...,NORD-1 and I=NBKPT-NORD+2,...,NBKPT, are required to compute the functions used to fit the data. No sorting of BKPT(*) is required. Internal to DEFC the extreme end knots may be reduced and increased respectively to accommodate any data values that are exterior to the given knot values. The contents of BKPT(*) is not changed.

The number of conditions that constrain the B-spline is NCONST. A constraint is specified by an (X,Y) pair in the arrays XCONST() and YCONST(), and by the type of constraint and derivative value encoded in the array NDERIV(*).

X value of constraint. No sorting of XCONST(*) is required.

Y value of constraint

The value of NDERIV(*) is determined as follows. Suppose the I-th constraint applies to the J-th derivative of the B-spline. (Any non-negative value of J < NORD is permitted. In particular the value J=0 refers to the B-spline itself.) For this I-th constraint, set

Input

If the output value of MODE=0 or 1, this array contains the unknowns obtained from the least squares fitting process. These N=NBKPT-NORD parameters are the B-spline coefficients. For MODE=1, the equality constraints are contradictory. To make the fitting process more robust, the equality constraints are satisfied in a least squares sense. In this case the array COEFF() contains B-spline coefficients for this extended concept of a solution. If MODE=-1,2 or 3 on output, the array COEFF() is undefined.

real work array of length IW(1). The contents of W(*) must not be modified by the user if the variance function is desired.

integer work array of length IW(2)

A(MDA,N). The array A(,) initially contains the M by N matrix A of the least squares problem AX = B. The first dimensioning parameter of the array A(,) is MDA, which must satisfy MDA>=M Either M>=N or M<N is permitted. There is no restriction on the rank of A. The condition MDA<M is considered an error.

actual leading dimension of a

(B(MDB,NB) or B(M)). If NB = 0 the subroutine will perform the orthogonal decomposition but will make no references to the array B(). If NB>0 the array B() must initially contain the M by NB matrix B of the least squares problem AX = B. If NB>=2 the array B() must be doubly subscripted with first dimensioning parameter MDB>=MAX(M,N). If NB = 1 the array B() may be either doubly or singly subscripted. In the latter case the value of MDB is arbitrary but it should be set to some valid integer value such as MDB = M.

actual leading dimension of b

Absolute tolerance parameter provided by user for pseudorank determination.

Set by the subroutine to indicate the pseudorank of A.

RNORM(NB). On return, RNORM(J) will contain the Euclidean norm of the residual vector for the problem defined by the J-th column vector of the array B(,) for J = 1,...,NB.

H(N). Array of working space used by DHFTI. On return, contains elements of the pre-multiplying Householder transformations used to compute the minimum Euclidean length solution. not generally required by the user.

G(N). Array of working space used by DHFTI. On return, contain elements of the post-multiplying Householder transformations used to compute the minimum Euclidean length solution. not generally required by the user.

IP(N). Array of working space used by DHFTI. Array in which the subroutine records indices describing the permutation of column vectors. not generally required by the user.

A(MDA,N+1), where N=N1+N2.

X(N), where N=N1+N2.

The value of MODE indicates the status of the computation after returning to the user.

WS((M+2)*(N+7)), where N=N1+N2. This is a slight overestimate for WS(*).

IS(M+N+1), where N=N1+N2.

W(*,*) contains:

contain (resp) var. dimension of W(*,*), and matrix dimensions.

contain (resp) var. dimension of W(*,*), and matrix dimensions.

contain (resp) var. dimension of W(*,*), and matrix dimensions.

contain (resp) var. dimension of W(*,*), and matrix dimensions.

Program option vector.

Solution vector(unless MODE=2)

length of AX-B.

Working storage of dimension K+N+(MG+2)*(N+7), where K=MAX(MA+MG,N).

IP(MG+2*N+1) Integer working storage