public static void r8po_sl(double[] a, int lda, int n, ref double[] b, int aIndex = 0, int bIndex = 0)
//****************************************************************************80
//
// Purpose:
//
// R8PO_SL solves a linear system factored by R8PO_FA.
//
// Discussion:
//
// A division by zero will occur if the input factor contains
// a zero on the diagonal. Technically this indicates
// singularity but it is usually caused by improper subroutine
// arguments. It will not occur if the subroutines are called
// correctly and INFO == 0.
//
// Licensing:
//
// This code is distributed under the GNU LGPL license.
//
// Modified:
//
// 23 May 2005
//
// Author:
//
// FORTRAN77 original version by Dongarra, Moler, Bunch, Stewart.
// C++ version by John Burkardt.
//
// Reference:
//
// Dongarra, Moler, Bunch and Stewart,
// LINPACK User's Guide,
// SIAM, (Society for Industrial and Applied Mathematics),
// 3600 University City Science Center,
// Philadelphia, PA, 19104-2688.
// ISBN 0-89871-172-X
//
// Parameters:
//
// Input, double A[LDA*N], the output from R8PO_FA.
//
// Input, int LDA, the leading dimension of the array A.
//
// Input, int N, the order of the matrix.
//
// Input/output, double B[N]. On input, the right hand side.
// On output, the solution.
//
{
int k;
double t;
//
// Solve R' * Y = B.
//
for (k = 1; k <= n; k++)
{
t = BLAS1D.ddot(k - 1, a, 1, b, 1, aIndex + 0 + (k - 1) * lda, bIndex);
b[bIndex + k - 1] = (b[bIndex + k - 1] - t) / a[aIndex + k - 1 + (k - 1) * lda];
}
//
// Solve R * X = Y.
//
for (k = n; 1 <= k; k--)
{
b[bIndex + k - 1] /= a[aIndex + k - 1 + (k - 1) * lda];
t = -b[bIndex + k - 1];
BLAS1D.daxpy(k - 1, t, a, 1, ref b, 1, aIndex + 0 + (k - 1) * lda, bIndex);
}
}
public static int r8po_fa(ref double[] a, int lda, int n)
//****************************************************************************80
//
// Purpose:
//
// R8PO_FA factors a real symmetric positive definite matrix.
//
// Licensing:
//
// This code is distributed under the GNU LGPL license.
//
// Modified:
//
// 23 May 2005
//
// Author:
//
// FORTRAN77 original version by Dongarra, Moler, Bunch, Stewart.
// C++ version by John Burkardt.
//
// Reference:
//
// Dongarra, Moler, Bunch and Stewart,
// LINPACK User's Guide,
// SIAM, (Society for Industrial and Applied Mathematics),
// 3600 University City Science Center,
// Philadelphia, PA, 19104-2688.
// ISBN 0-89871-172-X
//
// Parameters:
//
// Input/output, double A[LDA*N]. On input, the symmetric matrix
// to be factored. Only the diagonal and upper triangle are used.
// On output, an upper triangular matrix R so that A = R'*R
// where R' is the transpose. The strict lower triangle is unaltered.
// If INFO /= 0, the factorization is not complete.
//
// Input, int LDA, the leading dimension of the array A.
//
// Input, int N, the order of the matrix.
//
// Output, int R8PO_FA, error flag.
// 0, for normal return.
// K, signals an error condition. The leading minor of order K is not
// positive definite.
//
{
int info;
int j;
for (j = 1; j <= n; j++)
{
double s = 0.0;
int k;
for (k = 1; k <= j - 1; k++)
{
double t = a[k - 1 + (j - 1) * lda] -
BLAS1D.ddot(k - 1, a, 1, a, 1, +0 + (k - 1) * lda, +0 + (j - 1) * lda);
t /= a[k - 1 + (k - 1) * lda];
a[k - 1 + (j - 1) * lda] = t;
s += t * t;
}
s = a[j - 1 + (j - 1) * lda] - s;
switch (s)
{
case <= 0.0:
info = j;
return(info);
default:
a[j - 1 + (j - 1) * lda] = Math.Sqrt(s);
break;
}
}
info = 0;
return(info);
}
public static double dpoco(ref double[] a, int lda, int n, ref double[] z)
//****************************************************************************80
//
// Purpose:
//
// DPOCO factors a real symmetric positive definite matrix and estimates its condition.
//
// Discussion:
//
// If RCOND is not needed, DPOFA is slightly faster.
//
// To solve A*X = B, follow DPOCO by DPOSL.
//
// To compute inverse(A)*C, follow DPOCO by DPOSL.
//
// To compute determinant(A), follow DPOCO by DPODI.
//
// To compute inverse(A), follow DPOCO by DPODI.
//
// Licensing:
//
// This code is distributed under the GNU LGPL license.
//
// Modified:
//
// 06 June 2005
//
// Author:
//
// Original FORTRAN77 version by Jack Dongarra, Cleve Moler, Jim Bunch,
// Pete Stewart.
// C++ version by John Burkardt.
//
// Reference:
//
// Jack Dongarra, Cleve Moler, Jim Bunch and Pete Stewart,
// LINPACK User's Guide,
// SIAM, (Society for Industrial and Applied Mathematics),
// 3600 University City Science Center,
// Philadelphia, PA, 19104-2688.
// ISBN 0-89871-172-X
//
// Parameters:
//
// Input/output, double A[LDA*N]. On input, the symmetric
// matrix to be factored. Only the diagonal and upper triangle are used.
// On output, an upper triangular matrix R so that A = R'*R where R'
// is the transpose. The strict lower triangle is unaltered.
// If INFO /= 0, the factorization is not complete.
//
// Input, int LDA, the leading dimension of the array A.
//
// Input, int N, the order of the matrix.
//
// Output, double Z[N], a work vector whose contents are usually
// unimportant. If A is close to a singular matrix, then Z is an
// approximate null vector in the sense that
// norm(A*Z) = RCOND * norm(A) * norm(Z).
// If INFO /= 0, Z is unchanged.
//
// Output, double DPOCO, an estimate of the reciprocal
// condition of A. For the system A*X = B, relative perturbations in
// A and B of size EPSILON may cause relative perturbations in X of
// size EPSILON/RCOND. If RCOND is so small that the logical expression
// 1.0D+00 + RCOND == 1.0D+00
// is true, then A may be singular to working precision. In particular,
// RCOND is zero if exact singularity is detected or the estimate underflows.
//
{
int i;
int j;
int k;
double rcond;
double s;
double t;
//
// Find norm of A using only upper half.
//
for (j = 1; j <= n; j++)
{
z[j - 1] = BLAS1D.dasum(j, a, 1, index: +0 + (j - 1) * lda);
for (i = 1; i <= j - 1; i++)
{
z[i - 1] += Math.Abs(a[i - 1 + (j - 1) * lda]);
}
}
double anorm = 0.0;
for (i = 1; i <= n; i++)
{
anorm = Math.Max(anorm, z[i - 1]);
}
//
// Factor.
//
int info = DPOFA.dpofa(ref a, lda, n);
if (info != 0)
{
rcond = 0.0;
return(rcond);
}
//
// RCOND = 1/(norm(A)*(estimate of norm(inverse(A)))).
//
// Estimate = norm(Z)/norm(Y) where A*Z = Y and A*Y = E.
//
// The components of E are chosen to cause maximum local
// growth in the elements of W where R'*W = E.
//
// The vectors are frequently rescaled to avoid overflow.
//
// Solve R' * W = E.
//
double ek = 1.0;
for (i = 1; i <= n; i++)
{
z[i - 1] = 0.0;
}
for (k = 1; k <= n; k++)
{
if (z[k - 1] != 0.0)
{
ek *= typeMethods.r8_sign(-z[k - 1]);
}
if (a[k - 1 + (k - 1) * lda] < Math.Abs(ek - z[k - 1]))
{
s = a[k - 1 + (k - 1) * lda] / Math.Abs(ek - z[k - 1]);
for (i = 1; i <= n; i++)
{
z[i - 1] = s * z[i - 1];
}
ek = s * ek;
}
double wk = ek - z[k - 1];
double wkm = -ek - z[k - 1];
s = Math.Abs(wk);
double sm = Math.Abs(wkm);
wk /= a[k - 1 + (k - 1) * lda];
wkm /= a[k - 1 + (k - 1) * lda];
if (k + 1 <= n)
{
for (j = k + 1; j <= n; j++)
{
sm += Math.Abs(z[j - 1] + wkm * a[k - 1 + (j - 1) * lda]);
z[j - 1] += wk * a[k - 1 + (j - 1) * lda];
s += Math.Abs(z[j - 1]);
}
if (s < sm)
{
t = wkm - wk;
wk = wkm;
for (j = k + 1; j <= n; j++)
{
z[j - 1] += t * a[k - 1 + (j - 1) * lda];
}
}
}
z[k - 1] = wk;
}
s = BLAS1D.dasum(n, z, 1);
for (i = 1; i <= n; i++)
{
z[i - 1] /= s;
}
//
// Solve R * Y = W.
//
for (k = n; 1 <= k; k--)
{
if (a[k - 1 + (k - 1) * lda] < Math.Abs(z[k - 1]))
{
s = a[k - 1 + (k - 1) * lda] / Math.Abs(z[k - 1]);
for (i = 1; i <= n; i++)
{
z[i - 1] = s * z[i - 1];
}
}
z[k - 1] /= a[k - 1 + (k - 1) * lda];
t = -z[k - 1];
BLAS1D.daxpy(k - 1, t, a, 1, ref z, 1, xIndex: +0 + (k - 1) * lda);
}
s = BLAS1D.dasum(n, z, 1);
for (i = 1; i <= n; i++)
{
z[i - 1] /= s;
}
double ynorm = 1.0;
//
// Solve R' * V = Y.
//
for (k = 1; k <= n; k++)
{
z[k - 1] -= BLAS1D.ddot(k - 1, a, 1, z, 1, xIndex: +0 + (k - 1) * lda);
if (a[k - 1 + (k - 1) * lda] < Math.Abs(z[k - 1]))
{
s = a[k - 1 + (k - 1) * lda] / Math.Abs(z[k - 1]);
for (i = 1; i <= n; i++)
{
z[i - 1] = s * z[i - 1];
}
ynorm = s * ynorm;
}
z[k - 1] /= a[k - 1 + (k - 1) * lda];
}
s = 1.0 / BLAS1D.dasum(n, z, 1);
for (i = 1; i <= n; i++)
{
z[i - 1] = s * z[i - 1];
}
ynorm = s * ynorm;
//
// Solve R * Z = V.
//
for (k = n; 1 <= k; k--)
{
if (a[k - 1 + (k - 1) * lda] < Math.Abs(z[k - 1]))
{
s = a[k - 1 + (k - 1) * lda] / Math.Abs(z[k - 1]);
for (i = 1; i <= n; i++)
{
z[i - 1] = s * z[i - 1];
}
ynorm = s * ynorm;
}
z[k - 1] /= a[k - 1 + (k - 1) * lda];
t = -z[k - 1];
BLAS1D.daxpy(k - 1, t, a, 1, ref z, 1, xIndex: +0 + (k - 1) * lda);
}
//
// Make ZNORM = 1.0.
//
s = 1.0 / BLAS1D.dasum(n, z, 1);
for (i = 1; i <= n; i++)
{
z[i - 1] = s * z[i - 1];
}
ynorm = s * ynorm;
if (anorm != 0.0)
{
rcond = ynorm / anorm;
}
else
{
rcond = 0.0;
}
return(rcond);
}
public static double dgeco(ref double[] a, int lda, int n, ref int[] ipvt, ref double[] z)
//****************************************************************************80
//
// Purpose:
//
// DGECO factors a real matrix and estimates its condition number.
//
// Discussion:
//
// If RCOND is not needed, DGEFA is slightly faster.
//
// To solve A * X = B, follow DGECO by DGESL.
//
// To compute inverse ( A ) * C, follow DGECO by DGESL.
//
// To compute determinant ( A ), follow DGECO by DGEDI.
//
// To compute inverse ( A ), follow DGECO by DGEDI.
//
// For the system A * X = B, relative perturbations in A and B
// of size EPSILON may cause relative perturbations in X of size
// EPSILON/RCOND.
//
// If RCOND is so small that the logical expression
// 1.0D+00 + RCOND == 1.0D+00
// is true, then A may be singular to working precision. In particular,
// RCOND is zero if exact singularity is detected or the estimate
// underflows.
//
// Licensing:
//
// This code is distributed under the GNU LGPL license.
//
// Modified:
//
// 25 May 2005
//
// Author:
//
// Original FORTRAN77 version by Jack Dongarra, Cleve Moler, Jim Bunch,
// Pete Stewart.
// C++ version by John Burkardt.
//
// Reference:
//
// Jack Dongarra, Cleve Moler, Jim Bunch and Pete Stewart,
// LINPACK User's Guide,
// SIAM, (Society for Industrial and Applied Mathematics),
// 3600 University City Science Center,
// Philadelphia, PA, 19104-2688.
// ISBN 0-89871-172-X
//
// Parameters:
//
// Input/output, double A[LDA*N]. On input, a matrix to be
// factored. On output, the LU factorization of the matrix.
//
// Input, int LDA, the leading dimension of the array A.
//
// Input, int N, the order of the matrix A.
//
// Output, int IPVT[N], the pivot indices.
//
// Output, double Z[N], a work vector whose contents are usually
// unimportant. If A is close to a singular matrix, then Z is an
// approximate null vector in the sense that
// norm ( A * Z ) = RCOND * norm ( A ) * norm ( Z ).
//
// Output, double DGECO, the value of RCOND, an estimate
// of the reciprocal condition number of A.
//
{
int i;
int j;
int k;
int l;
double rcond;
double s;
double t;
//
// Compute the L1 norm of A.
//
double anorm = 0.0;
for (j = 1; j <= n; j++)
{
anorm = Math.Max(anorm, BLAS1D.dasum(n, a, 1, index: +0 + (j - 1) * lda));
}
//
// Compute the LU factorization.
//
DGEFA.dgefa(ref a, lda, n, ref ipvt);
//
// RCOND = 1 / ( norm(A) * (estimate of norm(inverse(A))) )
//
// estimate of norm(inverse(A)) = norm(Z) / norm(Y)
//
// where
// A * Z = Y
// and
// A' * Y = E
//
// The components of E are chosen to cause maximum local growth in the
// elements of W, where U'*W = E. The vectors are frequently rescaled
// to avoid overflow.
//
// Solve U' * W = E.
//
double ek = 1.0;
for (i = 1; i <= n; i++)
{
z[i - 1] = 0.0;
}
for (k = 1; k <= n; k++)
{
if (z[k - 1] != 0.0)
{
ek *= typeMethods.r8_sign(-z[k - 1]);
}
if (Math.Abs(a[k - 1 + (k - 1) * lda]) < Math.Abs(ek - z[k - 1]))
{
s = Math.Abs(a[k - 1 + (k - 1) * lda]) / Math.Abs(ek - z[k - 1]);
for (i = 1; i <= n; i++)
{
z[i - 1] = s * z[i - 1];
}
ek = s * ek;
}
double wk = ek - z[k - 1];
double wkm = -ek - z[k - 1];
s = Math.Abs(wk);
double sm = Math.Abs(wkm);
if (a[k - 1 + (k - 1) * lda] != 0.0)
{
wk /= a[k - 1 + (k - 1) * lda];
wkm /= a[k - 1 + (k - 1) * lda];
}
else
{
wk = 1.0;
wkm = 1.0;
}
if (k + 1 <= n)
{
for (j = k + 1; j <= n; j++)
{
sm += Math.Abs(z[j - 1] + wkm * a[k - 1 + (j - 1) * lda]);
z[j - 1] += wk * a[k - 1 + (j - 1) * lda];
s += Math.Abs(z[j - 1]);
}
if (s < sm)
{
t = wkm - wk;
wk = wkm;
for (i = k + 1; i <= n; i++)
{
z[i - 1] += t * a[k - 1 + (i - 1) * lda];
}
}
}
z[k - 1] = wk;
}
t = BLAS1D.dasum(n, z, 1);
for (i = 1; i <= n; i++)
{
z[i - 1] /= t;
}
//
// Solve L' * Y = W
//
for (k = n; 1 <= k; k--)
{
z[k - 1] += BLAS1D.ddot(n - k, a, 1, z, 1, xIndex: +k + (k - 1) * lda, yIndex: +k);
switch (Math.Abs(z[k - 1]))
{
case > 1.0:
{
t = Math.Abs(z[k - 1]);
for (i = 1; i <= n; i++)
{
z[i - 1] /= t;
}
break;
}
}
l = ipvt[k - 1];
t = z[l - 1];
z[l - 1] = z[k - 1];
z[k - 1] = t;
}
t = BLAS1D.dasum(n, z, 1);
for (i = 1; i <= n; i++)
{
z[i - 1] /= t;
}
double ynorm = 1.0;
//
// Solve L * V = Y.
//
for (k = 1; k <= n; k++)
{
l = ipvt[k - 1];
t = z[l - 1];
z[l - 1] = z[k - 1];
z[k - 1] = t;
for (i = k + 1; i <= n; i++)
{
z[i - 1] += t * a[i - 1 + (k - 1) * lda];
}
switch (Math.Abs(z[k - 1]))
{
case > 1.0:
{
ynorm /= Math.Abs(z[k - 1]);
t = Math.Abs(z[k - 1]);
for (i = 1; i <= n; i++)
{
z[i - 1] /= t;
}
break;
}
}
}
s = BLAS1D.dasum(n, z, 1);
for (i = 1; i <= n; i++)
{
z[i - 1] /= s;
}
ynorm /= s;
//
// Solve U * Z = V.
//
for (k = n; 1 <= k; k--)
{
if (Math.Abs(a[k - 1 + (k - 1) * lda]) < Math.Abs(z[k - 1]))
{
s = Math.Abs(a[k - 1 + (k - 1) * lda]) / Math.Abs(z[k - 1]);
for (i = 1; i <= n; i++)
{
z[i - 1] = s * z[i - 1];
}
ynorm = s * ynorm;
}
if (a[k - 1 + (k - 1) * lda] != 0.0)
{
z[k - 1] /= a[k - 1 + (k - 1) * lda];
}
else
{
z[k - 1] = 1.0;
}
for (i = 1; i <= k - 1; i++)
{
z[i - 1] -= z[k - 1] * a[i - 1 + (k - 1) * lda];
}
}
//
// Normalize Z in the L1 norm.
//
s = 1.0 / BLAS1D.dasum(n, z, 1);
for (i = 1; i <= n; i++)
{
z[i - 1] = s * z[i - 1];
}
ynorm = s * ynorm;
if (anorm != 0.0)
{
rcond = ynorm / anorm;
}
else
{
rcond = 0.0;
}
return(rcond);
}
public static int dchdd(ref double[] r, int ldr, int p, double[] x, ref double[] z, int ldz,
int nz, double[] y, ref double[] rho, ref double[] c, ref double[] s)
//****************************************************************************80
//
// Purpose:
//
// DCHDD downdates an augmented Cholesky decomposition.
//
// Discussion:
//
// DCHDD can also downdate the triangular factor of an augmented QR
// decomposition.
//
// Specifically, given an upper triangular matrix R of order P, a
// row vector X, a column vector Z, and a scalar Y, DCHDD
// determines an orthogonal matrix U and a scalar ZETA such that
//
// (R Z ) (RR ZZ)
// U * ( ) = ( ),
// (0 ZETA) ( X Y)
//
// where RR is upper triangular.
//
// If R and Z have been obtained from the factorization of a least squares
// problem, then RR and ZZ are the factors corresponding to the problem
// with the observation (X,Y) removed. In this case, if RHO
// is the norm of the residual vector, then the norm of
// the residual vector of the downdated problem is
// sqrt ( RHO * RHO - ZETA * ZETA ). DCHDD will simultaneously downdate
// several triplets (Z, Y, RHO) along with R.
//
// For a less terse description of what DCHDD does and how
// it may be applied, see the LINPACK guide.
//
// The matrix U is determined as the product U(1)*...*U(P)
// where U(I) is a rotation in the (P+1,I)-plane of the form
//
// ( C(I) -S(I) )
// ( ).
// ( S(I) C(I) )
//
// The rotations are chosen so that C(I) is real.
//
// The user is warned that a given downdating problem may be impossible
// to accomplish or may produce inaccurate results. For example, this
// can happen if X is near a vector whose removal will reduce the
// rank of R. Beware.
//
// Licensing:
//
// This code is distributed under the GNU LGPL license.
//
// Modified:
//
// 23 June 2009
//
// Author:
//
// Original FORTRAN77 version by Jack Dongarra, Cleve Moler, Jim Bunch,
// Pete Stewart.
// C++ version by John Burkardt.
//
// Reference:
//
// Jack Dongarra, Cleve Moler, Jim Bunch and Pete Stewart,
// LINPACK User's Guide,
// SIAM, (Society for Industrial and Applied Mathematics),
// 3600 University City Science Center,
// Philadelphia, PA, 19104-2688.
// ISBN 0-89871-172-X
//
// Parameters:
//
// Input/output, double R[LDR*P], the upper triangular matrix that
// is to be downdated. The part of R below the diagonal is not referenced.
//
// Input, int LDR, the leading dimension of the array R.
// LDR must be at least P.
//
// Input, int P, the order of the matrix R.
//
// Input, double X[P], the row vector that is to be removed from R.
//
// Input/output, double Z[LDZ*NZ], an array of NZ P-vectors
// which are to be downdated along with R.
//
// Input, int LDZ, the leading dimension of the array Z.
// LDZ must be at least P.
//
// Input, int NZ, the number of vectors to be downdated.
// NZ may be zero, in which case Z, Y, and RHO are not referenced.
//
// Input, double Y[NZ], the scalars for the downdating of
// the vectors Z.
//
// Input/output, double RHO[NZ], the norms of the residual vectors.
// On output these have been changed along with R and Z.
//
// Output, double C[P], S[P], the cosines and sines of the
// transforming rotations.
//
// Output, int DCHDD, return flag.
// 0, the entire downdating was successful.
// -1, if R could not be downdated. In this case, all quantities
// are left unaltered.
// 1, if some RHO could not be downdated. The offending RHO's are
// set to -1.
//
{
int i;
int ii;
int j;
//
// Solve R' * A = X, placing the result in the array S.
//
int info = 0;
s[0] = x[0] / r[0 + 0 * ldr];
for (j = 2; j <= p; j++)
{
s[j - 1] = x[j - 1] - BLAS1D.ddot(j - 1, r, 1, s, 1, xIndex: +0 + (j - 1) * ldr);
s[j - 1] /= r[j - 1 + (j - 1) * ldr];
}
double norm = BLAS1D.dnrm2(p, s, 1);
switch (norm)
{
case >= 1.0:
info = -1;
return(info);
}
double alpha = Math.Sqrt(1.0 - norm * norm);
//
// Determine the transformations.
//
for (ii = 1; ii <= p; ii++)
{
i = p - ii + 1;
double scale = alpha + Math.Abs(s[i - 1]);
double a = alpha / scale;
double b = s[i - 1] / scale;
norm = Math.Sqrt(a * a + b * b);
c[i - 1] = a / norm;
s[i - 1] = b / norm;
alpha = scale * norm;
}
//
// Apply the transformations to R.
//
for (j = 1; j <= p; j++)
{
double xx = 0.0;
for (ii = 1; ii <= j; ii++)
{
i = j - ii + 1;
double t = c[i - 1] * xx + s[i - 1] * r[i - 1 + (j - 1) * ldr];
r[i - 1 + (j - 1) * ldr] = c[i - 1] * r[i - 1 + (j - 1) * ldr] - s[i - 1] * xx;
xx = t;
}
}
//
// If required, downdate Z and RHO.
//
for (j = 1; j <= nz; j++)
{
double zeta = y[j - 1];
for (i = 1; i <= p; i++)
{
z[i - 1 + (j - 1) * ldz] = (z[i - 1 + (j - 1) * ldz] - s[i - 1] * zeta) / c[i - 1];
zeta = c[i - 1] * zeta - s[i - 1] * z[i - 1 + (j - 1) * ldz];
}
double azeta = Math.Abs(zeta);
if (rho[j - 1] < azeta)
{
info = 1;
rho[j - 1] = -1.0;
}
else
{
rho[j - 1] *= Math.Sqrt(1.0 - Math.Pow(azeta / rho[j - 1], 2));
}
}
return(info);
}
private static void ddot_test()
//****************************************************************************80
//
// Purpose:
//
// DDOT_TEST demonstrates DDOT.
//
// Licensing:
//
// This code is distributed under the GNU LGPL license.
//
// Modified:
//
// 15 May 2006
//
// Author:
//
// John Burkardt
//
{
const int N = 5;
const int LDA = 10;
const int LDB = 7;
const int LDC = 6;
double[] a = new double[LDA * LDA];
double[] b = new double[LDB * LDB];
double[] c = new double[LDC * LDC];
double[] x = new double[N];
double[] y = new double[N];
Console.WriteLine("");
Console.WriteLine("DDOT_TEST");
Console.WriteLine(" DDOT computes the dot product of vectors.");
Console.WriteLine("");
for (int i = 0; i < N; i++)
{
x[i] = i + 1;
}
for (int i = 0; i < N; i++)
{
y[i] = -(double)(i + 1);
}
for (int i = 0; i < N; i++)
{
for (int j = 0; j < N; j++)
{
a[i + j * LDA] = i + 1 + j + 1;
}
}
for (int i = 0; i < N; i++)
{
for (int j = 0; j < N; j++)
{
b[i + j * LDB] = i + 1 - (j + 1);
}
}
double sum1 = BLAS1D.ddot(N, x, 1, y, 1);
Console.WriteLine("");
Console.WriteLine(" Dot product of X and Y is " + sum1 + "");
//
// To multiply a ROW of a matrix A times a vector X, we need to
// specify the increment between successive entries of the row of A:
//
sum1 = BLAS1D.ddot(N, a, LDA, x, 1, 1);
Console.WriteLine("");
Console.WriteLine(" Product of row 2 of A and X is " + sum1 + "");
//
// Product of a column of A and a vector is simpler:
//
sum1 = BLAS1D.ddot(N, a, 1, x, 1, +0 + 1 * LDA);
Console.WriteLine("");
Console.WriteLine(" Product of column 2 of A and X is " + sum1 + "");
//
// Here's how matrix multiplication, c = a*b, could be done
// with DDOT:
//
for (int i = 0; i < N; i++)
{
for (int j = 0; j < N; j++)
{
c[i + j * LDC] = BLAS1D.ddot(N, a, LDA, b, 1, +i, +0 + j * LDB);
}
}
Console.WriteLine("");
Console.WriteLine(" Matrix product computed with DDOT:");
Console.WriteLine("");
for (int i = 0; i < N; i++)
{
string cout = "";
for (int j = 0; j < N; j++)
{
cout += " " + c[i + j * LDC].ToString(CultureInfo.InvariantCulture).PadLeft(14);
}
Console.WriteLine(cout);
}
}
public static void dgesl(double[] a, int lda, int n, int[] ipvt, ref double[] b, int job)
//****************************************************************************80
//
// Purpose:
//
// DGESL solves a real general linear system A * X = B.
//
// Discussion:
//
// DGESL can solve either of the systems A * X = B or A' * X = B.
//
// The system matrix must have been factored by DGECO or DGEFA.
//
// A division by zero will occur if the input factor contains a
// zero on the diagonal. Technically this indicates singularity
// but it is often caused by improper arguments or improper
// setting of LDA. It will not occur if the subroutines are
// called correctly and if DGECO has set 0.0 < RCOND
// or DGEFA has set INFO == 0.
//
// Licensing:
//
// This code is distributed under the GNU LGPL license.
//
// Modified:
//
// 16 May 2005
//
// Author:
//
// Original FORTRAN77 version by Jack Dongarra, Cleve Moler, Jim Bunch,
// Pete Stewart.
// C++ version by John Burkardt.
//
// Reference:
//
// Jack Dongarra, Cleve Moler, Jim Bunch and Pete Stewart,
// LINPACK User's Guide,
// SIAM, (Society for Industrial and Applied Mathematics),
// 3600 University City Science Center,
// Philadelphia, PA, 19104-2688.
// ISBN 0-89871-172-X
//
// Parameters:
//
// Input, double A[LDA*N], the output from DGECO or DGEFA.
//
// Input, int LDA, the leading dimension of A.
//
// Input, int N, the order of the matrix A.
//
// Input, int IPVT[N], the pivot vector from DGECO or DGEFA.
//
// Input/output, double B[N].
// On input, the right hand side vector.
// On output, the solution vector.
//
// Input, int JOB.
// 0, solve A * X = B;
// nonzero, solve A' * X = B.
//
{
int k;
int l;
double t;
switch (job)
{
//
// Solve A * X = B.
//
case 0:
{
for (k = 1; k <= n - 1; k++)
{
l = ipvt[k - 1];
t = b[l - 1];
if (l != k)
{
b[l - 1] = b[k - 1];
b[k - 1] = t;
}
BLAS1D.daxpy(n - k, t, a, 1, ref b, 1, +k + (k - 1) * lda, +k);
}
for (k = n; 1 <= k; k--)
{
b[k - 1] /= a[k - 1 + (k - 1) * lda];
t = -b[k - 1];
BLAS1D.daxpy(k - 1, t, a, 1, ref b, 1, +0 + (k - 1) * lda);
}
break;
}
//
default:
{
for (k = 1; k <= n; k++)
{
t = BLAS1D.ddot(k - 1, a, 1, b, 1, +0 + (k - 1) * lda);
b[k - 1] = (b[k - 1] - t) / a[k - 1 + (k - 1) * lda];
}
for (k = n - 1; 1 <= k; k--)
{
b[k - 1] += BLAS1D.ddot(n - k, a, 1, b, 1, +k + (k - 1) * lda, +k);
l = ipvt[k - 1];
if (l == k)
{
continue;
}
t = b[l - 1];
b[l - 1] = b[k - 1];
b[k - 1] = t;
}
break;
}
}
}
public static double dgbco(ref double[] abd, int lda, int n, int ml, int mu, ref int[] ipvt,
double[] z)
//****************************************************************************80
//
// Purpose:
//
// DGBCO factors a real band matrix and estimates its condition.
//
// Discussion:
//
// If RCOND is not needed, DGBFA is slightly faster.
//
// To solve A*X = B, follow DGBCO by DGBSL.
//
// To compute inverse(A)*C, follow DGBCO by DGBSL.
//
// To compute determinant(A), follow DGBCO by DGBDI.
//
// Example:
//
// If the original matrix is
//
// 11 12 13 0 0 0
// 21 22 23 24 0 0
// 0 32 33 34 35 0
// 0 0 43 44 45 46
// 0 0 0 54 55 56
// 0 0 0 0 65 66
//
// then for proper band storage,
//
// N = 6, ML = 1, MU = 2, 5 <= LDA and ABD should contain
//
// * * * + + + * = not used
// * * 13 24 35 46 + = used for pivoting
// * 12 23 34 45 56
// 11 22 33 44 55 66
// 21 32 43 54 65 *
//
// Band storage:
//
// If A is a band matrix, the following program segment
// will set up the input.
//
// ml = (band width below the diagonal)
// mu = (band width above the diagonal)
// m = ml + mu + 1
//
// do j = 1, n
// i1 = max ( 1, j-mu )
// i2 = min ( n, j+ml )
// do i = i1, i2
// k = i - j + m
// abd(k,j) = a(i,j)
// }
// }
//
// This uses rows ML+1 through 2*ML+MU+1 of ABD. In addition, the first
// ML rows in ABD are used for elements generated during the
// triangularization. The total number of rows needed in ABD is
// 2*ML+MU+1. The ML+MU by ML+MU upper left triangle and the ML by ML
// lower right triangle are not referenced.
//
// Licensing:
//
// This code is distributed under the GNU LGPL license.
//
// Modified:
//
// 07 June 2005
//
// Author:
//
// Original FORTRAN77 version by Jack Dongarra, Cleve Moler, Jim Bunch,
// Pete Stewart.
// C++ version by John Burkardt.
//
// Reference:
//
// Jack Dongarra, Cleve Moler, Jim Bunch and Pete Stewart,
// LINPACK User's Guide,
// SIAM, (Society for Industrial and Applied Mathematics),
// 3600 University City Science Center,
// Philadelphia, PA, 19104-2688.
// ISBN 0-89871-172-X
//
// Parameters:
//
// Input/output, double ABD[LDA*N]. On input, the matrix in band
// storage. The columns of the matrix are stored in the columns of ABD and
// the diagonals of the matrix are stored in rows ML+1 through 2*ML+MU+1
// of ABD. On output, an upper triangular matrix in band storage and
// the multipliers which were used to obtain it. The factorization can
// be written A = L*U where L is a product of permutation and unit lower
// triangular matrices and U is upper triangular.
//
// Input, int LDA, the leading dimension of the array ABD.
// 2*ML + MU + 1 <= LDA is required.
//
// Input, int N, the order of the matrix.
//
// Input, int ML, MU, the number of diagonals below and above the
// main diagonal. 0 <= ML < N, 0 <= MU < N.
//
// Output, int IPVT[N], the pivot indices.
//
// Workspace, double Z[N], a work vector whose contents are
// usually unimportant. If A is close to a singular matrix, then Z is an
// approximate null vector in the sense that
// norm(A*Z) = RCOND * norm(A) * norm(Z).
//
// Output, double DGBCO, an estimate of the reciprocal condition number RCOND
// of A. For the system A*X = B, relative perturbations in A and B of size
// EPSILON may cause relative perturbations in X of size EPSILON/RCOND.
// If RCOND is so small that the logical expression
// 1.0 + RCOND == 1.0D+00
// is true, then A may be singular to working precision. In particular,
// RCOND is zero if exact singularity is detected or the estimate underflows.
//
{
int i;
int j;
int k;
int lm;
double rcond;
double s;
double t;
//
// Compute the 1-norm of A.
//
double anorm = 0.0;
int l = ml + 1;
int is_ = l + mu;
for (j = 1; j <= n; j++)
{
anorm = Math.Max(anorm, BLAS1D.dasum(l, abd, 1, index: +is_ - 1 + (j - 1) * lda));
if (ml + 1 < is_)
{
is_ -= 1;
}
if (j <= mu)
{
l += 1;
}
if (n - ml <= j)
{
l -= 1;
}
}
//
// Factor.
//
DGBFA.dgbfa(ref abd, lda, n, ml, mu, ref ipvt);
//
// RCOND = 1/(norm(A)*(estimate of norm(inverse(A)))).
//
// Estimate = norm(Z)/norm(Y) where a*z = y and A'*Y = E.
//
// A' is the transpose of A. The components of E are
// chosen to cause maximum local growth in the elements of W where
// U'*W = E. The vectors are frequently rescaled to avoid
// overflow.
//
// Solve U' * W = E.
//
double ek = 1.0;
for (i = 1; i <= n; i++)
{
z[i - 1] = 0.0;
}
int m = ml + mu + 1;
int ju = 0;
for (k = 1; k <= n; k++)
{
if (z[k - 1] != 0.0)
{
ek *= typeMethods.r8_sign(-z[k - 1]);
}
if (Math.Abs(abd[m - 1 + (k - 1) * lda]) < Math.Abs(ek - z[k - 1]))
{
s = Math.Abs(abd[m - 1 + (k - 1) * lda]) / Math.Abs(ek - z[k - 1]);
for (i = 1; i <= n; i++)
{
z[i - 1] = s * z[i - 1];
}
ek = s * ek;
}
double wk = ek - z[k - 1];
double wkm = -ek - z[k - 1];
s = Math.Abs(wk);
double sm = Math.Abs(wkm);
if (abd[m - 1 + (k - 1) * lda] != 0.0)
{
wk /= abd[m - 1 + (k - 1) * lda];
wkm /= abd[m - 1 + (k - 1) * lda];
}
else
{
wk = 1.0;
wkm = 1.0;
}
ju = Math.Min(Math.Max(ju, mu + ipvt[k - 1]), n);
int mm = m;
if (k + 1 <= ju)
{
for (j = k + 1; j <= ju; j++)
{
mm -= 1;
sm += Math.Abs(z[j - 1] + wkm * abd[mm - 1 + (j - 1) * lda]);
z[j - 1] += wk * abd[mm - 1 + (j - 1) * lda];
s += Math.Abs(z[j - 1]);
}
if (s < sm)
{
t = wkm - wk;
wk = wkm;
mm = m;
for (j = k + 1; j <= ju; ju++)
{
mm -= 1;
z[j - 1] += t * abd[mm - 1 + (j - 1) * lda];
}
}
}
z[k - 1] = wk;
}
s = BLAS1D.dasum(n, z, 1);
for (i = 1; i <= n; i++)
{
z[i - 1] /= s;
}
//
// Solve L' * Y = W.
//
for (k = n; 1 <= k; k--)
{
lm = Math.Min(ml, n - k);
if (k < m)
{
z[k - 1] += BLAS1D.ddot(lm, abd, 1, z, 1, xIndex: +m + (k - 1) * lda, yIndex: +k);
}
switch (Math.Abs(z[k - 1]))
{
case > 1.0:
{
s = 1.0 / Math.Abs(z[k - 1]);
for (i = 1; i <= n; i++)
{
z[i - 1] = s * z[i - 1];
}
break;
}
}
l = ipvt[k - 1];
t = z[l - 1];
z[l - 1] = z[k - 1];
z[k - 1] = t;
}
s = BLAS1D.dasum(n, z, 1);
for (i = 1; i <= n; i++)
{
z[i - 1] /= s;
}
double ynorm = 1.0;
//
// Solve L * V = Y.
//
for (k = 1; k <= n; k++)
{
l = ipvt[k - 1];
t = z[l - 1];
z[l - 1] = z[k - 1];
z[k - 1] = t;
lm = Math.Min(ml, n - k);
if (k < n)
{
BLAS1D.daxpy(lm, t, abd, 1, ref z, 1, xIndex: +m + (k - 1) * lda, yIndex: +k);
}
switch (Math.Abs(z[k - 1]))
{
case > 1.0:
{
s = 1.0 / Math.Abs(z[k - 1]);
for (i = 1; i <= n; i++)
{
z[i - 1] = s * z[i - 1];
}
ynorm = s * ynorm;
break;
}
}
}
s = 1.0 / BLAS1D.dasum(n, z, 1);
for (i = 1; i <= n; i++)
{
z[i - 1] = s * z[i - 1];
}
ynorm = s * ynorm;
//
// Solve U * Z = W.
//
for (k = n; 1 <= k; k--)
{
if (Math.Abs(abd[m - 1 + (k - 1) * lda]) < Math.Abs(z[k - 1]))
{
s = Math.Abs(abd[m - 1 + (k - 1) * lda]) / Math.Abs(z[k - 1]);
for (i = 1; i <= n; i++)
{
z[i - 1] = s * z[i - 1];
}
ynorm = s * ynorm;
}
if (abd[m - 1 + (k - 1) * lda] != 0.0)
{
z[k - 1] /= abd[m - 1 + (k - 1) * lda];
}
else
{
z[k - 1] = 1.0;
}
lm = Math.Min(k, m) - 1;
int la = m - lm;
int lz = k - lm;
t = -z[k - 1];
BLAS1D.daxpy(lm, t, abd, 1, ref z, 1, xIndex: +la - 1 + (k - 1) * lda, yIndex: +lz - 1);
}
//
// Make ZNORM = 1.0.
//
s = 1.0 / BLAS1D.dasum(n, z, 1);
for (i = 1; i <= n; i++)
{
z[i - 1] = s * z[i - 1];
}
ynorm = s * ynorm;
if (anorm != 0.0)
{
rcond = ynorm / anorm;
}
else
{
rcond = 0.0;
}
return(rcond);
}
public static void dspsl(double[] ap, int n, int[] kpvt, ref double[] b)
//****************************************************************************80
//
// Purpose:
//
// DSPSL solves the real symmetric system factored by DSPFA.
//
// Discussion:
//
// To compute inverse(A) * C where C is a matrix with P columns:
//
// call dspfa ( ap, n, kpvt, info )
//
// if ( info /= 0 ) go to ...
//
// do j = 1, p
// call dspsl ( ap, n, kpvt, c(1,j) )
// end do
//
// A division by zero may occur if DSPCO has set RCOND == 0.0D+00
// or DSPFA has set INFO /= 0.
//
// Licensing:
//
// This code is distributed under the GNU LGPL license.
//
// Modified:
//
// 25 May 2005
//
// Author:
//
// Original FORTRAN77 version by Jack Dongarra, Cleve Moler, Jim Bunch,
// Pete Stewart.
// C++ version by John Burkardt.
//
// Reference:
//
// Jack Dongarra, Cleve Moler, Jim Bunch and Pete Stewart,
// LINPACK User's Guide,
// SIAM, (Society for Industrial and Applied Mathematics),
// 3600 University City Science Center,
// Philadelphia, PA, 19104-2688.
// ISBN 0-89871-172-X
//
// Parameters:
//
// Input, double AP[(N*(N+1))/2], the output from DSPFA.
//
// Input, int N, the order of the matrix.
//
// Input, int KPVT[N], the pivot vector from DSPFA.
//
// Input/output, double B[N]. On input, the right hand side.
// On output, the solution.
//
{
int kp;
double temp;
//
// Loop backward applying the transformations and D inverse to B.
//
int k = n;
int ik = n * (n - 1) / 2;
while (0 < k)
{
int kk = ik + k;
switch (kpvt[k - 1])
{
case >= 0:
{
//
// 1 x 1 pivot block.
//
if (k != 1)
{
kp = kpvt[k - 1];
//
// Interchange.
//
if (kp != k)
{
temp = b[k - 1];
b[k - 1] = b[kp - 1];
b[kp - 1] = temp;
}
//
// Apply the transformation.
//
BLAS1D.daxpy(k - 1, b[k - 1], ap, 1, ref b, 1, xIndex: +ik);
}
//
// Apply D inverse.
//
b[k - 1] /= ap[kk - 1];
k -= 1;
ik -= k;
break;
}
default:
{
//
// 2 x 2 pivot block.
//
int ikm1 = ik - (k - 1);
if (k != 2)
{
kp = Math.Abs(kpvt[k - 1]);
//
// Interchange.
//
if (kp != k - 1)
{
temp = b[k - 2];
b[k - 2] = b[kp - 1];
b[kp - 1] = temp;
}
//
// Apply the transformation.
//
BLAS1D.daxpy(k - 2, b[k - 1], ap, 1, ref b, 1, xIndex: +ik);
BLAS1D.daxpy(k - 2, b[k - 2], ap, 1, ref b, 1, xIndex: +ikm1);
}
//
// Apply D inverse.
//
int km1k = ik + k - 1;
kk = ik + k;
double ak = ap[kk - 1] / ap[km1k - 1];
int km1km1 = ikm1 + k - 1;
double akm1 = ap[km1km1 - 1] / ap[km1k - 1];
double bk = b[k - 1] / ap[km1k - 1];
double bkm1 = b[k - 2] / ap[km1k - 1];
double denom = ak * akm1 - 1.0;
b[k - 1] = (akm1 * bk - bkm1) / denom;
b[k - 2] = (ak * bkm1 - bk) / denom;
k -= 2;
ik = ik - (k + 1) - k;
break;
}
}
}
//
// Loop forward applying the transformations.
//
k = 1;
ik = 0;
while (k <= n)
{
switch (kpvt[k - 1])
{
case >= 0:
{
//
// 1 x 1 pivot block.
//
if (k != 1)
{
//
// Apply the transformation.
//
b[k - 1] += BLAS1D.ddot(k - 1, ap, 1, b, 1, xIndex: +ik);
kp = kpvt[k - 1];
//
// Interchange.
//
if (kp != k)
{
temp = b[k - 1];
b[k - 1] = b[kp - 1];
b[kp - 1] = temp;
}
}
ik += k;
k += 1;
break;
}
default:
{
//
// 2 x 2 pivot block.
//
if (k != 1)
{
//
// Apply the transformation.
//
b[k - 1] += BLAS1D.ddot(k - 1, ap, 1, b, 1, xIndex: +ik);
int ikp1 = ik + k;
b[k] += BLAS1D.ddot(k - 1, ap, 1, b, 1, xIndex: +ikp1);
kp = Math.Abs(kpvt[k - 1]);
//
// Interchange.
//
if (kp != k)
{
temp = b[k - 1];
b[k - 1] = b[kp - 1];
b[kp - 1] = temp;
}
}
ik = ik + k + k + 1;
k += 2;
break;
}
}
}
}
public static double dppco(ref double[] ap, int n, ref double[] z)
//****************************************************************************80
//
// Purpose:
//
// DPPCO factors a real symmetric positive definite matrix in packed form.
//
// Discussion:
//
// DPPCO also estimates the condition of the matrix.
//
// If RCOND is not needed, DPPFA is slightly faster.
//
// To solve A*X = B, follow DPPCO by DPPSL.
//
// To compute inverse(A)*C, follow DPPCO by DPPSL.
//
// To compute determinant(A), follow DPPCO by DPPDI.
//
// To compute inverse(A), follow DPPCO by DPPDI.
//
// Packed storage:
//
// The following program segment will pack the upper triangle of
// a symmetric matrix.
//
// k = 0
// do j = 1, n
// do i = 1, j
// k = k + 1
// ap[k-1] = a(i,j)
// }
// }
//
// Licensing:
//
// This code is distributed under the GNU LGPL license.
//
// Modified:
//
// 07 June 2005
//
// Author:
//
// Original FORTRAN77 version by Jack Dongarra, Cleve Moler, Jim Bunch,
// Pete Stewart.
// C++ version by John Burkardt.
//
// Reference:
//
// Jack Dongarra, Cleve Moler, Jim Bunch and Pete Stewart,
// LINPACK User's Guide,
// SIAM, (Society for Industrial and Applied Mathematics),
// 3600 University City Science Center,
// Philadelphia, PA, 19104-2688.
// ISBN 0-89871-172-X
//
// Parameters:
//
// Input/output, double AP[N*(N+1)/2]. On input, the packed
// form of a symmetric matrix A. The columns of the upper triangle are
// stored sequentially in a one-dimensional array. On output, an upper
// triangular matrix R, stored in packed form, so that A = R'*R.
// If INFO /= 0, the factorization is not complete.
//
// Input, int N, the order of the matrix.
//
// Output, double Z[N], a work vector whose contents are usually
// unimportant. If A is singular to working precision, then Z is an
// approximate null vector in the sense that
// norm(A*Z) = RCOND * norm(A) * norm(Z).
// If INFO /= 0, Z is unchanged.
//
// Output, double DPPCO, an estimate of the reciprocal condition number RCOND
// of A. For the system A*X = B, relative perturbations in A and B of size
// EPSILON may cause relative perturbations in X of size EPSILON/RCOND.
// If RCOND is so small that the logical expression
// 1.0 + RCOND == 1.0D+00
// is true, then A may be singular to working precision. In particular,
// RCOND is zero if exact singularity is detected or the estimate underflows.
//
{
int i;
int j;
int k;
double rcond;
double s;
double t;
//
// Find the norm of A.
//
int j1 = 1;
for (j = 1; j <= n; j++)
{
z[j - 1] = BLAS1D.dasum(j, ap, 1, index: +j1 - 1);
int ij = j1;
j1 += j;
for (i = 1; i <= j - 1; i++)
{
z[i - 1] += Math.Abs(ap[ij - 1]);
ij += 1;
}
}
double anorm = 0.0;
for (i = 1; i <= n; i++)
{
anorm = Math.Max(anorm, z[i - 1]);
}
//
// Factor.
//
int info = DPPFA.dppfa(ref ap, n);
if (info != 0)
{
rcond = 0.0;
return(rcond);
}
//
// RCOND = 1/(norm(A)*(estimate of norm(inverse(A)))).
//
// Estimate = norm(Z)/norm(Y) where A * Z = Y and A * Y = E.
//
// The components of E are chosen to cause maximum local
// growth in the elements of W where R'*W = E.
//
// The vectors are frequently rescaled to avoid overflow.
//
// Solve R' * W = E.
//
double ek = 1.0;
for (i = 1; i <= n; i++)
{
z[i - 1] = 0.0;
}
int kk = 0;
for (k = 1; k <= n; k++)
{
kk += k;
if (z[k - 1] != 0.0)
{
ek *= typeMethods.r8_sign(-z[k - 1]);
}
if (ap[kk - 1] < Math.Abs(ek - z[k - 1]))
{
s = ap[kk - 1] / Math.Abs(ek - z[k - 1]);
for (i = 1; i <= n; i++)
{
z[i - 1] = s * z[i - 1];
}
ek = s * ek;
}
double wk = ek - z[k - 1];
double wkm = -ek - z[k - 1];
s = Math.Abs(wk);
double sm = Math.Abs(wkm);
wk /= ap[kk - 1];
wkm /= ap[kk - 1];
int kj = kk + k;
if (k + 1 <= n)
{
for (j = k + 1; j <= n; j++)
{
sm += Math.Abs(z[j - 1] + wkm * ap[kj - 1]);
z[j - 1] += wk * ap[kj - 1];
s += Math.Abs(z[j - 1]);
kj += j;
}
if (s < sm)
{
t = wkm - wk;
wk = wkm;
kj = kk + k;
for (j = k + 1; j <= n; j++)
{
z[j - 1] += t * ap[kj - 1];
kj += j;
}
}
}
z[k - 1] = wk;
}
s = BLAS1D.dasum(n, z, 1);
for (i = 1; i <= n; i++)
{
z[i - 1] /= s;
}
//
// Solve R * Y = W.
//
for (k = n; 1 <= k; k--)
{
if (ap[kk - 1] < Math.Abs(z[k - 1]))
{
s = ap[kk - 1] / Math.Abs(z[k - 1]);
for (i = 1; i <= n; i++)
{
z[i - 1] = s * z[i - 1];
}
}
z[k - 1] /= ap[kk - 1];
kk -= k;
t = -z[k - 1];
BLAS1D.daxpy(k - 1, t, ap, 1, ref z, 1, xIndex: +kk);
}
s = BLAS1D.dasum(n, z, 1);
for (i = 1; i <= n; i++)
{
z[i - 1] /= s;
}
double ynorm = 1.0;
//
// Solve R' * V = Y.
//
for (k = 1; k <= n; k++)
{
z[k - 1] -= BLAS1D.ddot(k - 1, ap, 1, z, 1, xIndex: +kk);
kk += k;
if (ap[kk - 1] < Math.Abs(z[k - 1]))
{
s = ap[kk - 1] / Math.Abs(z[k - 1]);
for (i = 1; i <= n; i++)
{
z[i - 1] = s * z[i - 1];
}
ynorm = s * ynorm;
}
z[k - 1] /= ap[kk - 1];
}
s = 1.0 / BLAS1D.dasum(n, z, 1);
for (i = 1; i <= n; i++)
{
z[i - 1] = s * z[i - 1];
}
ynorm = s * ynorm;
//
// Solve R * Z = V.
//
for (k = n; 1 <= k; k--)
{
if (ap[kk - 1] < Math.Abs(z[k - 1]))
{
s = ap[kk - 1] / Math.Abs(z[k - 1]);
for (i = 1; i <= n; i++)
{
z[i - 1] = s * z[i - 1];
}
ynorm = s * ynorm;
}
z[k - 1] /= ap[kk - 1];
kk -= k;
t = -z[k - 1];
BLAS1D.daxpy(k - 1, t, ap, 1, ref z, 1, xIndex: +kk);
}
//
// Make ZNORM = 1.0.
//
s = 1.0 / BLAS1D.dasum(n, z, 1);
for (i = 1; i <= n; i++)
{
z[i - 1] = s * z[i - 1];
}
ynorm = s * ynorm;
if (anorm != 0.0)
{
rcond = ynorm / anorm;
}
else
{
rcond = 0.0;
}
return(rcond);
}
public static int dtrsl(double[] t, int ldt, int n, ref double[] b, int job)
//****************************************************************************80
//
// Purpose:
//
// DTRSL solves triangular linear systems.
//
// Discussion:
//
// DTRSL can solve T * X = B or T' * X = B where T is a triangular
// matrix of order N.
//
// Here T' denotes the transpose of the matrix T.
//
// Licensing:
//
// This code is distributed under the GNU LGPL license.
//
// Modified:
//
// 19 May 2005
//
// Author:
//
// Original FORTRAN77 version by Jack Dongarra, Cleve Moler, Jim Bunch,
// Pete Stewart.
// C++ version by John Burkardt.
//
// Reference:
//
// Jack Dongarra, Cleve Moler, Jim Bunch and Pete Stewart,
// LINPACK User's Guide,
// SIAM, (Society for Industrial and Applied Mathematics),
// 3600 University City Science Center,
// Philadelphia, PA, 19104-2688.
// ISBN 0-89871-172-X
//
// Parameters:
//
// Input, double T[LDT*N], the matrix of the system. The zero
// elements of the matrix are not referenced, and the corresponding
// elements of the array can be used to store other information.
//
// Input, int LDT, the leading dimension of the array T.
//
// Input, int N, the order of the matrix.
//
// Input/output, double B[N]. On input, the right hand side.
// On output, the solution.
//
// Input, int JOB, specifies what kind of system is to be solved:
// 00, solve T * X = B, T lower triangular,
// 01, solve T * X = B, T upper triangular,
// 10, solve T'* X = B, T lower triangular,
// 11, solve T'* X = B, T upper triangular.
//
// Output, int DTRSL, singularity indicator.
// 0, the system is nonsingular.
// nonzero, the index of the first zero diagonal element of T.
//
{
int info;
int j;
int jj;
double temp;
//
// Check for zero diagonal elements.
//
for (j = 1; j <= n; j++)
{
switch (t[j - 1 + (j - 1) * ldt])
{
case 0.0:
info = j;
return(info);
}
}
info = 0;
int kase = (job % 10) switch
{
//
// Determine the task and go to it.
//
0 => 1,
_ => 2
};
if (job % 100 / 10 != 0)
{
kase += 2;
}
switch (kase)
{
//
// Solve T * X = B for T lower triangular.
//
case 1:
{
b[0] /= t[0 + 0 * ldt];
for (j = 2; j <= n; j++)
{
temp = -b[j - 2];
BLAS1D.daxpy(n - j + 1, temp, t, 1, ref b, 1, xIndex: +(j - 1) + (j - 2) * ldt, yIndex: +j - 1);
b[j - 1] /= t[j - 1 + (j - 1) * ldt];
}
break;
}
//
// Solve T * X = B for T upper triangular.
//
case 2:
{
b[n - 1] /= t[n - 1 + (n - 1) * ldt];
for (jj = 2; jj <= n; jj++)
{
j = n - jj + 1;
temp = -b[j];
BLAS1D.daxpy(j, temp, t, 1, ref b, 1, xIndex: +0 + j * ldt);
b[j - 1] /= t[j - 1 + (j - 1) * ldt];
}
break;
}
//
// Solve T' * X = B for T lower triangular.
//
case 3:
{
b[n - 1] /= t[n - 1 + (n - 1) * ldt];
for (jj = 2; jj <= n; jj++)
{
j = n - jj + 1;
b[j - 1] -= BLAS1D.ddot(jj - 1, t, 1, b, 1, xIndex: +j + (j - 1) * ldt, yIndex: +j);
b[j - 1] /= t[j - 1 + (j - 1) * ldt];
}
break;
}
//
// Solve T' * X = B for T upper triangular.
//
case 4:
{
b[0] /= t[0 + 0 * ldt];
for (j = 2; j <= n; j++)
{
b[j - 1] -= BLAS1D.ddot(j - 1, t, 1, b, 1, xIndex: +0 + (j - 1) * ldt);
b[j - 1] /= t[j - 1 + (j - 1) * ldt];
}
break;
}
}
return(info);
}
}
public static void dsidi(ref double[] a, int lda, int n, int[] kpvt, ref double[] det,
ref int[] inert, double[] work, int job)
//****************************************************************************80
//
// Purpose:
//
// DSIDI computes the determinant, inertia and inverse of a real symmetric matrix.
//
// Discussion:
//
// DSIDI uses the factors from DSIFA.
//
// A division by zero may occur if the inverse is requested
// and DSICO has set RCOND == 0.0D+00 or DSIFA has set INFO /= 0.
//
// Variables not requested by JOB are not used.
//
// Licensing:
//
// This code is distributed under the GNU LGPL license.
//
// Modified:
//
// 25 May 2005
//
// Author:
//
// Original FORTRAN77 version by Jack Dongarra, Cleve Moler, Jim Bunch,
// Pete Stewart.
// C++ version by John Burkardt.
//
// Reference:
//
// Jack Dongarra, Cleve Moler, Jim Bunch and Pete Stewart,
// LINPACK User's Guide,
// SIAM, (Society for Industrial and Applied Mathematics),
// 3600 University City Science Center,
// Philadelphia, PA, 19104-2688.
// ISBN 0-89871-172-X
//
// Parameters:
//
// Input/output, double A(LDA,N). On input, the output from DSIFA.
// On output, the upper triangle of the inverse of the original matrix,
// if requested. The strict lower triangle is never referenced.
//
// Input, int LDA, the leading dimension of the array A.
//
// Input, int N, the order of the matrix.
//
// Input, int KPVT[N], the pivot vector from DSIFA.
//
// Output, double DET[2], the determinant of the original matrix,
// if requested.
// determinant = DET[0] * 10.0**DET[1]
// with 1.0D+00 <= abs ( DET[0] ) < 10.0D+00 or DET[0] = 0.0.
//
// Output, int INERT(3), the inertia of the original matrix,
// if requested.
// INERT(1) = number of positive eigenvalues.
// INERT(2) = number of negative eigenvalues.
// INERT(3) = number of zero eigenvalues.
//
// Workspace, double WORK[N].
//
// Input, int JOB, specifies the tasks.
// JOB has the decimal expansion ABC where
// If C /= 0, the inverse is computed,
// If B /= 0, the determinant is computed,
// If A /= 0, the inertia is computed.
// For example, JOB = 111 gives all three.
//
{
double d;
int k;
double t;
bool doinv = job % 10 != 0;
bool dodet = job % 100 / 10 != 0;
bool doert = job % 1000 / 100 != 0;
if (dodet || doert)
{
switch (doert)
{
case true:
inert[0] = 0;
inert[1] = 0;
inert[2] = 0;
break;
}
switch (dodet)
{
case true:
det[0] = 1.0;
det[1] = 0.0;
break;
}
t = 0.0;
for (k = 1; k <= n; k++)
{
d = a[k - 1 + (k - 1) * lda];
switch (kpvt[k - 1])
{
//
// 2 by 2 block.
//
// use det (d s) = (d/t * c - t) * t, t = abs ( s )
// (s c)
// to avoid underflow/overflow troubles.
//
// Take two passes through scaling. Use T for flag.
//
case <= 0 when t == 0.0:
t = Math.Abs(a[k - 1 + k * lda]);
d = d / t * a[k + k * lda] - t;
break;
case <= 0:
d = t;
t = 0.0;
break;
}
switch (doert)
{
case true:
switch (d)
{
case > 0.0:
inert[0] += 1;
break;
case < 0.0:
inert[1] += 1;
break;
case 0.0:
inert[2] += 1;
break;
}
break;
}
switch (dodet)
{
case true:
{
det[0] *= d;
if (det[0] != 0.0)
{
while (Math.Abs(det[0]) < 1.0)
{
det[0] *= 10.0;
det[1] -= 1.0;
}
while (10.0 <= Math.Abs(det[0]))
{
det[0] /= 10.0;
det[1] += 1.0;
}
}
break;
}
}
}
}
switch (doinv)
{
//
// Compute inverse(A).
//
case true:
{
k = 1;
while (k <= n)
{
int j;
int kstep;
switch (kpvt[k - 1])
{
case >= 0:
{
//
// 1 by 1.
//
a[k - 1 + (k - 1) * lda] = 1.0 / a[k - 1 + (k - 1) * lda];
switch (k)
{
case >= 2:
{
BLAS1D.dcopy(k - 1, a, 1, ref work, 1, xIndex: +0 + (k - 1) * lda);
for (j = 1; j <= k - 1; j++)
{
a[j - 1 + (k - 1) * lda] = BLAS1D.ddot(j, a, 1, work, 1, xIndex: +0 + (j - 1) * lda);
BLAS1D.daxpy(j - 1, work[j - 1], a, 1, ref a, 1, xIndex: +0 + (j - 1) * lda,
yIndex: +0 + (k - 1) * lda);
}
a[k - 1 + (k - 1) * lda] += BLAS1D.ddot(k - 1, work, 1, a, 1, yIndex: +0 + (k - 1) * lda);
break;
}
}
kstep = 1;
break;
}
//
default:
{
t = Math.Abs(a[k - 1 + k * lda]);
double ak = a[k - 1 + (k - 1) * lda] / t;
double akp1 = a[k + k * lda] / t;
double akkp1 = a[k - 1 + k * lda] / t;
d = t * (ak * akp1 - 1.0);
a[k - 1 + (k - 1) * lda] = akp1 / d;
a[k + k * lda] = ak / d;
a[k - 1 + k * lda] = -akkp1 / d;
switch (k)
{
case >= 2:
{
BLAS1D.dcopy(k - 1, a, 1, ref work, 1, xIndex: +0 + k * lda);
for (j = 1; j <= k - 1; j++)
{
a[j - 1 + k * lda] = BLAS1D.ddot(j, a, 1, work, 1, xIndex: +0 + (j - 1) * lda);
BLAS1D.daxpy(j - 1, work[j - 1], a, 1, ref a, 1, xIndex: +0 + (j - 1) * lda,
yIndex: +0 + k * lda);
}
a[k + k * lda] += BLAS1D.ddot(k - 1, work, 1, a, 1, yIndex: +0 + k * lda);
a[k - 1 + k * lda] += BLAS1D.ddot(k - 1, a, 1, a, 1, xIndex: +0 + (k - 1) * lda,
yIndex: +0 + k * lda);
BLAS1D.dcopy(k - 1, a, 1, ref work, 1, xIndex: +0 + (k - 1) * lda);
for (j = 1; j <= k - 1; j++)
{
a[j - 1 + (k - 1) * lda] = BLAS1D.ddot(j, a, 1, work, 1, xIndex: +0 + (j - 1) * lda);
BLAS1D.daxpy(j - 1, work[j - 1], a, 1, ref a, 1, xIndex: +0 + (j - 1) * lda,
yIndex: +0 + (k - 1) * lda);
}
a[k - 1 + (k - 1) * lda] += BLAS1D.ddot(k - 1, work, 1, a, 1, yIndex: +0 + (k - 1) * lda);
break;
}
}
kstep = 2;
break;
}
}
//
// Swap.
//
int ks = Math.Abs(kpvt[k - 1]);
if (ks != k)
{
BLAS1D.dswap(ks, ref a, 1, ref a, 1, xIndex: +0 + (ks - 1) * lda, yIndex: +0 + (k - 1) * lda);
int jb;
double temp;
for (jb = ks; jb <= k; jb++)
{
j = k + ks - jb;
temp = a[j - 1 + (k - 1) * lda];
a[j - 1 + (k - 1) * lda] = a[ks - 1 + (j - 1) * lda];
a[ks - 1 + (j - 1) * lda] = temp;
}
if (kstep != 1)
{
temp = a[ks - 1 + k * lda];
a[ks - 1 + k * lda] = a[k - 1 + k * lda];
a[k - 1 + k * lda] = temp;
}
}
k += kstep;
}
break;
}
}
}
public static int dpbfa(ref double[] abd, int lda, int n, int m)
//****************************************************************************80
//
// Purpose:
//
// DPBFA factors a symmetric positive definite matrix stored in band form.
//
// Discussion:
//
// DPBFA is usually called by DPBCO, but it can be called
// directly with a saving in time if RCOND is not needed.
//
// If A is a symmetric positive definite band matrix,
// the following program segment will set up the input.
//
// m = (band width above diagonal)
// do j = 1, n
// i1 = max ( 1, j-m )
// do i = i1, j
// k = i-j+m+1
// abd(k,j) = a(i,j)
// end do
// end do
//
// Licensing:
//
// This code is distributed under the GNU LGPL license.
//
// Modified:
//
// 23 May 2005
//
// Author:
//
// Original FORTRAN77 version by Jack Dongarra, Cleve Moler, Jim Bunch,
// Pete Stewart.
// C++ version by John Burkardt.
//
// Reference:
//
// Jack Dongarra, Cleve Moler, Jim Bunch and Pete Stewart,
// LINPACK User's Guide,
// SIAM, (Society for Industrial and Applied Mathematics),
// 3600 University City Science Center,
// Philadelphia, PA, 19104-2688.
// ISBN 0-89871-172-X
//
// Parameters:
//
// Input/output, double ABD[LDA*N]. On input, the matrix to be
// factored. The columns of the upper triangle are stored in the columns
// of ABD and the diagonals of the upper triangle are stored in the
// rows of ABD. On output, an upper triangular matrix R, stored in band
// form, so that A = R' * R.
//
// Input, int LDA, the leading dimension of the array ABD.
// M+1 <= LDA is required.
//
// Input, int N, the order of the matrix.
//
// Input, int M, the number of diagonals above the main diagonal.
//
// Output, int DPBFA, error indicator.
// 0, for normal return.
// K, if the leading minor of order K is not positive definite.
//
{
int info;
int j;
for (j = 1; j <= n; j++)
{
double s = 0.0;
int ik = m + 1;
int jk = Math.Max(j - m, 1);
int mu = Math.Max(m + 2 - j, 1);
int k;
for (k = mu; k <= m; k++)
{
double t = abd[k - 1 + (j - 1) * lda]
- BLAS1D.ddot(k - mu, abd, 1, abd, 1, xIndex: +ik - 1 + (jk - 1) * lda, yIndex: +mu - 1 + (j - 1) * lda);
t /= abd[m + (jk - 1) * lda];
abd[k - 1 + (j - 1) * lda] = t;
s += t * t;
ik -= 1;
jk += 1;
}
s = abd[m + (j - 1) * lda] - s;
switch (s)
{
case <= 0.0:
info = j;
return(info);
default:
abd[m + (j - 1) * lda] = Math.Sqrt(s);
break;
}
}
info = 0;
return(info);
}
public static void dspdi(ref double[] ap, int n, int[] kpvt, ref double[] det, ref int[] inert,
double[] work, int job)
//****************************************************************************80
//
// Purpose:
//
// DSPDI computes the determinant, inertia and inverse of a real symmetric matrix.
//
// Discussion:
//
// DSPDI uses the factors from DSPFA, where the matrix is stored in
// packed form.
//
// A division by zero will occur if the inverse is requested
// and DSPCO has set RCOND == 0.0D+00 or DSPFA has set INFO /= 0.
//
// Variables not requested by JOB are not used.
//
// Licensing:
//
// This code is distributed under the GNU LGPL license.
//
// Modified:
//
// 25 May 2005
//
// Author:
//
// Original FORTRAN77 version by Jack Dongarra, Cleve Moler, Jim Bunch,
// Pete Stewart.
// C++ version by John Burkardt.
//
// Reference:
//
// Jack Dongarra, Cleve Moler, Jim Bunch and Pete Stewart,
// LINPACK User's Guide,
// SIAM, (Society for Industrial and Applied Mathematics),
// 3600 University City Science Center,
// Philadelphia, PA, 19104-2688.
// ISBN 0-89871-172-X
//
// Parameters:
//
// Input/output, double AP[(N*(N+1))/2]. On input, the output from
// DSPFA. On output, the upper triangle of the inverse of the original
// matrix, stored in packed form, if requested. The columns of the upper
// triangle are stored sequentially in a one-dimensional array.
//
// Input, int N, the order of the matrix.
//
// Input, int KPVT[N], the pivot vector from DSPFA.
//
// Output, double DET[2], the determinant of the original matrix,
// if requested.
// determinant = DET[0] * 10.0**DET[1]
// with 1.0D+00 <= abs ( DET[0] ) < 10.0D+00 or DET[0] = 0.0.
//
// Output, int INERT[3], the inertia of the original matrix, if requested.
// INERT(1) = number of positive eigenvalues.
// INERT(2) = number of negative eigenvalues.
// INERT(3) = number of zero eigenvalues.
//
// Workspace, double WORK[N].
//
// Input, int JOB, has the decimal expansion ABC where:
// if A /= 0, the inertia is computed,
// if B /= 0, the determinant is computed,
// if C /= 0, the inverse is computed.
// For example, JOB = 111 gives all three.
//
{
double d;
int ik;
int ikp1;
int k;
int kk;
int kkp1;
double t;
bool doinv = job % 10 != 0;
bool dodet = job % 100 / 10 != 0;
bool doert = job % 1000 / 100 != 0;
if (dodet || doert)
{
switch (doert)
{
case true:
inert[0] = 0;
inert[1] = 0;
inert[2] = 0;
break;
}
switch (dodet)
{
case true:
det[0] = 1.0;
det[1] = 0.0;
break;
}
t = 0.0;
ik = 0;
for (k = 1; k <= n; k++)
{
kk = ik + k;
d = ap[kk - 1];
switch (kpvt[k - 1])
{
//
// 2 by 2 block
// use det (d s) = (d/t * c - t) * t, t = abs ( s )
// (s c)
// to avoid underflow/overflow troubles.
//
// Take two passes through scaling. Use T for flag.
//
case <= 0 when t == 0.0:
ikp1 = ik + k;
kkp1 = ikp1 + k;
t = Math.Abs(ap[kkp1 - 1]);
d = d / t * ap[kkp1] - t;
break;
case <= 0:
d = t;
t = 0.0;
break;
}
switch (doert)
{
case true:
switch (d)
{
case > 0.0:
inert[0] += 1;
break;
case < 0.0:
inert[1] += 1;
break;
case 0.0:
inert[2] += 1;
break;
}
break;
}
switch (dodet)
{
case true:
{
det[0] *= d;
if (det[0] != 0.0)
{
while (Math.Abs(det[0]) < 1.0)
{
det[0] *= 10.0;
det[1] -= 1.0;
}
while (10.0 <= Math.Abs(det[0]))
{
det[0] /= 10.0;
det[1] += 1.0;
}
}
break;
}
}
ik += k;
}
}
switch (doinv)
{
//
// Compute inverse(A).
//
case true:
{
k = 1;
ik = 0;
while (k <= n)
{
int km1 = k - 1;
kk = ik + k;
ikp1 = ik + k;
kkp1 = ikp1 + k;
int kstep;
int jk;
int j;
int ij;
switch (kpvt[k - 1])
{
case >= 0:
{
//
// 1 by 1.
//
ap[kk - 1] = 1.0 / ap[kk - 1];
switch (k)
{
case >= 2:
{
BLAS1D.dcopy(k - 1, ap, 1, ref work, 1, xIndex: +ik);
ij = 0;
for (j = 1; j <= k - 1; j++)
{
jk = ik + j;
ap[jk - 1] = BLAS1D.ddot(j, ap, 1, work, 1, xIndex: +ij);
BLAS1D.daxpy(j - 1, work[j - 1], ap, 1, ref ap, 1, xIndex: +ij, yIndex: +ik);
ij += j;
}
ap[kk - 1] += BLAS1D.ddot(k - 1, work, 1, ap, 1, yIndex: +ik);
break;
}
}
kstep = 1;
break;
}
default:
{
//
// 2 by 2.
//
t = Math.Abs(ap[kkp1 - 1]);
double ak = ap[kk - 1] / t;
double akp1 = ap[kkp1] / t;
double akkp1 = ap[kkp1 - 1] / t;
d = t * (ak * akp1 - 1.0);
ap[kk - 1] = akp1 / d;
ap[kkp1] = ak / d;
ap[kkp1 - 1] = -akkp1 / d;
switch (km1)
{
case >= 1:
{
BLAS1D.dcopy(km1, ap, 1, ref work, 1, xIndex: +ikp1);
ij = 0;
for (j = 1; j <= km1; j++)
{
int jkp1 = ikp1 + j;
ap[jkp1 - 1] = BLAS1D.ddot(j, ap, 1, work, 1, xIndex: +ij);
BLAS1D.daxpy(j - 1, work[j - 1], ap, 1, ref ap, 1, xIndex: +ij, yIndex: +ikp1);
ij += j;
}
ap[kkp1] += BLAS1D.ddot(km1, work, 1, ap, 1, yIndex: +ikp1);
ap[kkp1 - 1] += BLAS1D.ddot(km1, ap, 1, ap, 1, xIndex: +ik, yIndex: +ikp1);
BLAS1D.dcopy(km1, ap, 1, ref work, 1, xIndex: +ik);
ij = 0;
for (j = 1; j <= km1; j++)
{
jk = ik + j;
ap[jk - 1] = BLAS1D.ddot(j, ap, 1, work, 1, xIndex: +ij);
BLAS1D.daxpy(j - 1, work[j - 1], ap, 1, ref ap, 1, xIndex: +ij, yIndex: +ik);
ij += j;
}
ap[kk - 1] += BLAS1D.ddot(km1, work, 1, ap, 1, yIndex: +ik);
break;
}
}
kstep = 2;
break;
}
}
//
// Swap.
//
int ks = Math.Abs(kpvt[k - 1]);
if (ks != k)
{
int iks = ks * (ks - 1) / 2;
BLAS1D.dswap(ks, ref ap, 1, ref ap, 1, xIndex: +iks, yIndex: +ik);
int ksj = ik + ks;
double temp;
int jb;
for (jb = ks; jb <= k; jb++)
{
j = k + ks - jb;
jk = ik + j;
temp = ap[jk - 1];
ap[jk - 1] = ap[ksj - 1];
ap[ksj - 1] = temp;
ksj -= j - 1;
}
if (kstep != 1)
{
int kskp1 = ikp1 + ks;
temp = ap[kskp1 - 1];
ap[kskp1 - 1] = ap[kkp1 - 1];
ap[kkp1 - 1] = temp;
}
}
ik += k;
ik = kstep switch
{
2 => ik + k + 1,
_ => ik
};
k += kstep;
}
break;
}
}
}
}
public static void dposl(double[] a, int lda, int n, ref double[] b)
//****************************************************************************80
//
// Purpose:
//
// DPOSL solves a linear system factored by DPOCO or DPOFA.
//
// Discussion:
//
// To compute inverse(A) * C where C is a matrix with P columns:
//
// call dpoco ( a, lda, n, rcond, z, info )
//
// if ( rcond is not too small .and. info == 0 ) then
// do j = 1, p
// call dposl ( a, lda, n, c(1,j) )
// end do
// end if
//
// A division by zero will occur if the input factor contains
// a zero on the diagonal. Technically this indicates
// singularity but it is usually caused by improper subroutine
// arguments. It will not occur if the subroutines are called
// correctly and INFO == 0.
//
// Licensing:
//
// This code is distributed under the GNU LGPL license.
//
// Modified:
//
// 23 May 2005
//
// Author:
//
// Original FORTRAN77 version by Jack Dongarra, Cleve Moler, Jim Bunch,
// Pete Stewart.
// C++ version by John Burkardt.
//
// Reference:
//
// Jack Dongarra, Cleve Moler, Jim Bunch and Pete Stewart,
// LINPACK User's Guide,
// SIAM, (Society for Industrial and Applied Mathematics),
// 3600 University City Science Center,
// Philadelphia, PA, 19104-2688.
// ISBN 0-89871-172-X
//
// Parameters:
//
// Input, double A[LDA*N], the output from DPOCO or DPOFA.
//
// Input, int LDA, the leading dimension of the array A.
//
// Input, int N, the order of the matrix.
//
// Input/output, double B[N]. On input, the right hand side.
// On output, the solution.
//
{
int k;
double t;
//
// Solve R' * Y = B.
//
for (k = 1; k <= n; k++)
{
t = BLAS1D.ddot(k - 1, a, 1, b, 1, xIndex: +0 + (k - 1) * lda);
b[k - 1] = (b[k - 1] - t) / a[k - 1 + (k - 1) * lda];
}
//
// Solve R * X = Y.
//
for (k = n; 1 <= k; k--)
{
b[k - 1] /= a[k - 1 + (k - 1) * lda];
t = -b[k - 1];
BLAS1D.daxpy(k - 1, t, a, 1, ref b, 1, xIndex: +0 + (k - 1) * lda);
}
}
public static void dgbsl(double[] abd, int lda, int n, int ml, int mu, int[] ipvt,
ref double[] b, int job)
//****************************************************************************80
//
// Purpose:
//
// DGBSL solves a real banded system factored by DGBCO or DGBFA.
//
// Discussion:
//
// DGBSL can solve either A * X = B or A' * X = B.
//
// A division by zero will occur if the input factor contains a
// zero on the diagonal. Technically this indicates singularity
// but it is often caused by improper arguments or improper
// setting of LDA. It will not occur if the subroutines are
// called correctly and if DGBCO has set 0.0 < RCOND
// or DGBFA has set INFO == 0.
//
// To compute inverse(A) * C where C is a matrix with P columns:
//
// call dgbco ( abd, lda, n, ml, mu, ipvt, rcond, z )
//
// if ( rcond is too small ) then
// exit
// end if
//
// do j = 1, p
// call dgbsl ( abd, lda, n, ml, mu, ipvt, c(1,j), 0 )
// end do
//
// Licensing:
//
// This code is distributed under the GNU LGPL license.
//
// Modified:
//
// 23 May 2005
//
// Author:
//
// Original FORTRAN77 version by Jack Dongarra, Cleve Moler, Jim Bunch,
// Pete Stewart.
// C++ version by John Burkardt.
//
// Reference:
//
// Jack Dongarra, Cleve Moler, Jim Bunch and Pete Stewart,
// LINPACK User's Guide,
// SIAM, (Society for Industrial and Applied Mathematics),
// 3600 University City Science Center,
// Philadelphia, PA, 19104-2688.
// ISBN 0-89871-172-X
//
// Parameters:
//
// Input, double ABD[LDA*N], the output from DGBCO or DGBFA.
//
// Input, integer LDA, the leading dimension of the array ABD.
//
// Input, int N, the order of the matrix.
//
// Input, int ML, MU, the number of diagonals below and above the
// main diagonal. 0 <= ML < N, 0 <= MU < N.
//
// Input, int IPVT[N], the pivot vector from DGBCO or DGBFA.
//
// Input/output, double B[N]. On input, the right hand side.
// On output, the solution.
//
// Input, int JOB, job choice.
// 0, solve A*X=B.
// nonzero, solve A'*X=B.
//
{
int k;
int l;
int la;
int lb;
int lm;
double t;
int m = mu + ml + 1;
switch (job)
{
//
// JOB = 0, Solve A * x = b.
//
// First solve L * y = b.
//
case 0:
{
switch (ml)
{
case > 0:
{
for (k = 1; k <= n - 1; k++)
{
lm = Math.Min(ml, n - k);
l = ipvt[k - 1];
t = b[l - 1];
if (l != k)
{
b[l - 1] = b[k - 1];
b[k - 1] = t;
}
BLAS1D.daxpy(lm, t, abd, 1, ref b, 1, +m + (k - 1) * lda, k);
}
break;
}
}
//
// Now solve U * x = y.
//
for (k = n; 1 <= k; k--)
{
b[k - 1] /= abd[m - 1 + (k - 1) * lda];
lm = Math.Min(k, m) - 1;
la = m - lm;
lb = k - lm;
t = -b[k - 1];
BLAS1D.daxpy(lm, t, abd, 1, ref b, 1, +la - 1 + (k - 1) * lda, +lb - 1);
}
break;
}
//
default:
{
for (k = 1; k <= n; k++)
{
lm = Math.Min(k, m) - 1;
la = m - lm;
lb = k - lm;
t = BLAS1D.ddot(lm, abd, 1, b, 1, +la - 1 + (k - 1) * lda, +lb - 1);
b[k - 1] = (b[k - 1] - t) / abd[m - 1 + (k - 1) * lda];
}
switch (ml)
{
//
// Now solve L' * x = y.
//
case > 0:
{
for (k = n - 1; 1 <= k; k--)
{
lm = Math.Min(ml, n - k);
b[k - 1] += BLAS1D.ddot(lm, abd, 1, b, 1, +m + (k - 1) * lda, +k);
l = ipvt[k - 1];
if (l == k)
{
continue;
}
t = b[l - 1];
b[l - 1] = b[k - 1];
b[k - 1] = t;
}
break;
}
}
break;
}
}
}
public static int dppfa(ref double[] ap, int n)
//****************************************************************************80
//
// Purpose:
//
// DPPFA factors a real symmetric positive definite matrix in packed form.
//
// Discussion:
//
// DPPFA is usually called by DPPCO, but it can be called
// directly with a saving in time if RCOND is not needed.
//
// Packed storage:
//
// The following program segment will pack the upper
// triangle of a symmetric matrix.
//
// k = 0
// do j = 1, n
// do i = 1, j
// k = k + 1
// ap(k) = a(i,j)
// end do
// end do
//
// Licensing:
//
// This code is distributed under the GNU LGPL license.
//
// Modified:
//
// 24 May 2005
//
// Author:
//
// Original FORTRAN77 version by Jack Dongarra, Cleve Moler, Jim Bunch,
// Pete Stewart.
// C++ version by John Burkardt.
//
// Reference:
//
// Jack Dongarra, Cleve Moler, Jim Bunch and Pete Stewart,
// LINPACK User's Guide,
// SIAM, (Society for Industrial and Applied Mathematics),
// 3600 University City Science Center,
// Philadelphia, PA, 19104-2688.
// ISBN 0-89871-172-X
//
// Parameters:
//
// Input/output, double AP[N*(N+1)/2]. On input, the packed
// form of a symmetric matrix A. The columns of the upper triangle are
// stored sequentially in a one-dimensional array. On output, an upper
// triangular matrix R, stored in packed form, so that A = R'*R.
//
// Input, int N, the order of the matrix.
//
// Output, int DPPFA, error flag.
// 0, for normal return.
// K, if the leading minor of order K is not positive definite.
//
{
int j;
int info = 0;
int jj = 0;
for (j = 1; j <= n; j++)
{
double s = 0.0;
int kj = jj;
int kk = 0;
int k;
for (k = 1; k <= j - 1; k++)
{
kj += 1;
double t = ap[kj - 1] - BLAS1D.ddot(k - 1, ap, 1, ap, 1, xIndex: +kk, yIndex: +jj);
kk += k;
t /= ap[kk - 1];
ap[kj - 1] = t;
s += t * t;
}
jj += j;
s = ap[jj - 1] - s;
switch (s)
{
case <= 0.0:
info = j;
return(info);
default:
ap[jj - 1] = Math.Sqrt(s);
break;
}
}
return(info);
}
public static void dpbsl(double[] abd, int lda, int n, int m, ref double[] b)
//****************************************************************************80
//
// Purpose:
//
// DPBSL solves a real SPD band system factored by DPBCO or DPBFA.
//
// Discussion:
//
// The matrix is assumed to be a symmetric positive definite (SPD)
// band matrix.
//
// To compute inverse(A) * C where C is a matrix with P columns:
//
// call dpbco ( abd, lda, n, rcond, z, info )
//
// if ( rcond is too small .or. info /= 0) go to ...
//
// do j = 1, p
// call dpbsl ( abd, lda, n, c(1,j) )
// end do
//
// A division by zero will occur if the input factor contains
// a zero on the diagonal. Technically this indicates
// singularity but it is usually caused by improper subroutine
// arguments. It will not occur if the subroutines are called
// correctly and INFO == 0.
//
// Licensing:
//
// This code is distributed under the GNU LGPL license.
//
// Modified:
//
// 23 May 2005
//
// Author:
//
// Original FORTRAN77 version by Jack Dongarra, Cleve Moler, Jim Bunch,
// Pete Stewart.
// C++ version by John Burkardt.
//
// Reference:
//
// Jack Dongarra, Cleve Moler, Jim Bunch and Pete Stewart,
// LINPACK User's Guide,
// SIAM, (Society for Industrial and Applied Mathematics),
// 3600 University City Science Center,
// Philadelphia, PA, 19104-2688.
// ISBN 0-89871-172-X
//
// Parameters:
//
// Input, double ABD[LDA*N], the output from DPBCO or DPBFA.
//
// Input, int LDA, the leading dimension of the array ABD.
//
// Input, int N, the order of the matrix.
//
// Input, int M, the number of diagonals above the main diagonal.
//
// Input/output, double B[N]. On input, the right hand side.
// On output, the solution.
//
{
int k;
int la;
int lb;
int lm;
double t;
//
// Solve R'*Y = B.
//
for (k = 1; k <= n; k++)
{
lm = Math.Min(k - 1, m);
la = m + 1 - lm;
lb = k - lm;
t = BLAS1D.ddot(lm, abd, 1, b, 1, xIndex: +la - 1 + (k - 1) * lda, yIndex: +lb - 1);
b[k - 1] = (b[k - 1] - t) / abd[m + (k - 1) * lda];
}
//
// Solve R*X = Y.
//
for (k = n; 1 <= k; k--)
{
lm = Math.Min(k - 1, m);
la = m + 1 - lm;
lb = k - lm;
b[k - 1] /= abd[m + (k - 1) * lda];
t = -b[k - 1];
BLAS1D.daxpy(lm, t, abd, 1, ref b, 1, xIndex: +la - 1 + (k - 1) * lda, yIndex: +lb - 1);
}
}
public static double dpbco(ref double[] abd, int lda, int n, int m, ref double[] z)
//****************************************************************************80
//
// Purpose:
//
// DPBCO factors a real symmetric positive definite banded matrix.
//
// Discussion:
//
// DPBCO also estimates the condition of the matrix.
//
// If RCOND is not needed, DPBFA is slightly faster.
//
// To solve A*X = B, follow DPBCO by DPBSL.
//
// To compute inverse(A)*C, follow DPBCO by DPBSL.
//
// To compute determinant(A), follow DPBCO by DPBDI.
//
// Band storage:
//
// If A is a symmetric positive definite band matrix, the following
// program segment will set up the input.
//
// m = (band width above diagonal)
// do j = 1, n
// i1 = max (1, j-m)
// do i = i1, j
// k = i-j+m+1
// abd(k,j) = a(i,j)
// }
// }
//
// This uses M + 1 rows of A, except for the M by M upper left triangle,
// which is ignored.
//
// For example, if the original matrix is
//
// 11 12 13 0 0 0
// 12 22 23 24 0 0
// 13 23 33 34 35 0
// 0 24 34 44 45 46
// 0 0 35 45 55 56
// 0 0 0 46 56 66
//
// then N = 6, M = 2 and ABD should contain
//
// * * 13 24 35 46
// * 12 23 34 45 56
// 11 22 33 44 55 66
//
// Licensing:
//
// This code is distributed under the GNU LGPL license.
//
// Modified:
//
// 07 June 2005
//
// Author:
//
// Original FORTRAN77 version by Jack Dongarra, Cleve Moler, Jim Bunch,
// Pete Stewart.
// C++ version by John Burkardt.
//
// Reference:
//
// Jack Dongarra, Cleve Moler, Jim Bunch and Pete Stewart,
// LINPACK User's Guide,
// SIAM, (Society for Industrial and Applied Mathematics),
// 3600 University City Science Center,
// Philadelphia, PA, 19104-2688.
// ISBN 0-89871-172-X
//
// Parameters:
//
// Input/output, double ABD[LDA*N]. On input, the matrix to be
// factored. The columns of the upper triangle are stored in the columns
// of ABD and the diagonals of the upper triangle are stored in the rows
// of ABD. On output, an upper triangular matrix R, stored in band form,
// so that A = R'*R. If INFO /= 0, the factorization is not complete.
//
// Input, int LDA, the leading dimension of the array ABD.
// M+1 <= LDA is required.
//
// Input, int N, the order of the matrix.
//
// Input, int M, the number of diagonals above the main diagonal.
//
// Output, double Z[N], a work vector whose contents are usually
// unimportant. If A is singular to working precision, then Z is an
// approximate null vector in the sense that
// norm(A*Z) = RCOND * norm(A) * norm(Z).
// If INFO /= 0, Z is unchanged.
//
// Output, double DPBCO, an estimate of the reciprocal condition number
// RCOND. For the system A*X = B, relative perturbations in A and B of size
// EPSILON may cause relative perturbations in X of size EPSILON/RCOND.
// If RCOND is so small that the logical expression
// 1.0 + RCOND == 1.0D+00
// is true, then A may be singular to working precision. In particular,
// RCOND is zero if exact singularity is detected or the estimate underflows.
//
{
int i;
int j;
int k;
int la;
int lb;
int lm;
double rcond;
double s;
double t;
//
// Find the norm of A.
//
for (j = 1; j <= n; j++)
{
int l = Math.Min(j, m + 1);
int mu = Math.Max(m + 2 - j, 1);
z[j - 1] = BLAS1D.dasum(l, abd, 1, index: +mu - 1 + (j - 1) * lda);
k = j - l;
for (i = mu; i <= m; i++)
{
k += 1;
z[k - 1] += Math.Abs(abd[i - 1 + (j - 1) * lda]);
}
}
double anorm = 0.0;
for (i = 1; i <= n; i++)
{
anorm = Math.Max(anorm, z[i - 1]);
}
//
// Factor.
//
int info = DPBFA.dpbfa(ref abd, lda, n, m);
if (info != 0)
{
rcond = 0.0;
return(rcond);
}
//
// RCOND = 1/(norm(A)*(estimate of norm(inverse(A)))).
//
// Estimate = norm(Z)/norm(Y) where A*Z = Y and A*Y = E.
//
// The components of E are chosen to cause maximum local
// growth in the elements of W where R'*W = E.
//
// The vectors are frequently rescaled to avoid overflow.
//
// Solve R' * W = E.
//
double ek = 1.0;
for (i = 1; i <= n; i++)
{
z[i - 1] = 0.0;
}
for (k = 1; k <= n; k++)
{
if (z[k - 1] != 0.0)
{
ek *= typeMethods.r8_sign(-z[k - 1]);
}
if (abd[m + (k - 1) * lda] < Math.Abs(ek - z[k - 1]))
{
s = abd[m + (k - 1) * lda] / Math.Abs(ek - z[k - 1]);
for (i = 1; i <= n; i++)
{
z[i - 1] = s * z[i - 1];
}
ek = s * ek;
}
double wk = ek - z[k - 1];
double wkm = -ek - z[k - 1];
s = Math.Abs(wk);
double sm = Math.Abs(wkm);
wk /= abd[m + (k - 1) * lda];
wkm /= abd[m + (k - 1) * lda];
int j2 = Math.Min(k + m, n);
i = m + 1;
if (k + 1 <= j2)
{
for (j = k + 1; j <= j2; j++)
{
i -= 1;
sm += Math.Abs(z[j - 1] + wkm * abd[i - 1 + (j - 1) * lda]);
z[j - 1] += wk * abd[i - 1 + (j - 1) * lda];
s += Math.Abs(z[j - 1]);
}
if (s < sm)
{
t = wkm - wk;
wk = wkm;
i = m + 1;
for (j = k + 1; j <= j2; j++)
{
i -= 1;
z[j - 1] += t * abd[i - 1 + (j - 1) * lda];
}
}
}
z[k - 1] = wk;
}
s = BLAS1D.dasum(n, z, 1);
for (i = 1; i <= n; i++)
{
z[i - 1] /= s;
}
//
// Solve R * Y = W.
//
for (k = n; 1 <= k; k--)
{
if (abd[m + (k - 1) * lda] < Math.Abs(z[k - 1]))
{
s = abd[m + (k - 1) * lda] / Math.Abs(z[k - 1]);
for (i = 1; i <= n; i++)
{
z[i - 1] = s * z[i - 1];
}
}
z[k - 1] /= abd[m + (k - 1) * lda];
lm = Math.Min(k - 1, m);
la = m + 1 - lm;
lb = k - lm;
t = -z[k - 1];
BLAS1D.daxpy(lm, t, abd, 1, ref z, 1, xIndex: +la - 1 + (k - 1) * lda, yIndex: +lb - 1);
}
s = BLAS1D.dasum(n, z, 1);
for (i = 1; i <= n; i++)
{
z[i - 1] /= s;
}
double ynorm = 1.0;
//
// Solve R' * V = Y.
//
for (k = 1; k <= n; k++)
{
lm = Math.Min(k - 1, m);
la = m + 1 - lm;
lb = k - lm;
z[k - 1] -= BLAS1D.ddot(lm, abd, 1, z, 1, xIndex: +la - 1 + (k - 1) * lda, yIndex: +lb - 1);
if (abd[m + (k - 1) * lda] < Math.Abs(z[k - 1]))
{
s = abd[m + (k - 1) * lda] / Math.Abs(z[k - 1]);
for (i = 1; i <= n; i++)
{
z[i - 1] = s * z[i - 1];
}
ynorm = s * ynorm;
}
z[k - 1] /= abd[m + (k - 1) * lda];
}
s = 1.0 / BLAS1D.dasum(n, z, 1);
for (i = 1; i <= n; i++)
{
z[i - 1] = s * z[i - 1];
}
ynorm = s * ynorm;
//
// Solve R * Z = W.
//
for (k = n; 1 <= k; k--)
{
if (abd[m + (k - 1) * lda] < Math.Abs(z[k - 1]))
{
s = abd[m + (k - 1) * lda] / Math.Abs(z[k - 1]);
for (i = 1; i <= n; i++)
{
z[i - 1] = s * z[i - 1];
}
ynorm = s * ynorm;
}
z[k - 1] /= abd[m + (k - 1) * lda];
lm = Math.Min(k - 1, m);
la = m + 1 - lm;
lb = k - lm;
t = -z[k - 1];
BLAS1D.daxpy(lm, t, abd, 1, ref z, 1, xIndex: +la - 1 + (k - 1) * lda, yIndex: +lb - 1);
}
//
// Make ZNORM = 1.0.
//
s = 1.0 / BLAS1D.dasum(n, z, 1);
for (i = 1; i <= n; i++)
{
z[i - 1] = s * z[i - 1];
}
ynorm = s * ynorm;
if (anorm != 0.0)
{
rcond = ynorm / anorm;
}
else
{
rcond = 0.0;
}
return(rcond);
}