add sparse matrices to GenSVM and reorganize update functionality

1 files changed, 592 insertions, 0 deletions

diff --git a/src/gensvm_update.c b/src/gensvm_update.c
new file mode 100644
index 0000000..289f50c
--- /dev/null
+++ b/src/gensvm_update.c
@@ -0,0 +1,592 @@
+/**
+ * @file gensvm_update.c
+ * @author Gertjan van den Burg
+ * @date 2016-10-14
+ * @brief Functions for getting an update of the majorization algorithm
+
+ * Copyright (C)
+
+ This program is free software; you can redistribute it and/or
+ modify it under the terms of the GNU General Public License
+ as published by the Free Software Foundation; either version 2
+ of the License, or (at your option) any later version.
+
+ This program is distributed in the hope that it will be useful,
+ but WITHOUT ANY WARRANTY; without even the implied warranty of
+ MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
+ GNU General Public License for more details.
+
+ You should have received a copy of the GNU General Public License
+ along with this program; if not, write to the Free Software
+ Foundation, Inc., 59 Temple Place - Suite 330, Boston, MA  02111-1307, USA.
+
+ */
+
+#include "gensvm_update.h"
+
+/**
+ * @brief Calculate the value of omega for a single instance
+ *
+ * @details
+ * This function calculates the value of the @f$ \omega_i @f$ variable for a
+ * single instance, where
+ * @f[
+ * 	\omega_i = \frac{1}{p} \left( \sum_{j \neq y_i} h^p\left(
+ * 	\overline{q}_i^{(y_i j)} \right)  \right)^{1/p-1}
+ * @f]
+ * Note that his function uses the precalculated values from GenModel::H and
+ * GenModel::R to speed up the computation.
+ *
+ * @param[in] 	model 	GenModel structure with the current model
+ * @param[in] 	i 	index of the instance for which to calculate omega
+ * @returns 		the value of omega for instance i
+ *
+ */
+double gensvm_calculate_omega(struct GenModel *model, struct GenData *data,
+		long i)
+{
+	long j;
+	double h, omega = 0.0,
+	       p = model->p;
+
+	for (j=0; j<model->K; j++) {
+		if (j == (data->y[i]-1))
+			continue;
+		h = matrix_get(model->H, model->K, i, j);
+		omega += pow(h, p);
+	}
+	omega = (1.0/p)*pow(omega, 1.0/p - 1.0);
+
+	return omega;
+}
+
+/**
+ * @brief Check if we can do simple majorization for a given instance
+ *
+ * @details
+ * A simple majorization is possible if at most one of the Huberized hinge
+ * errors is nonzero for an instance. This is checked here. For this we
+ * compute the product of the Huberized error for all @f$j \neq y_i@f$ and
+ * check if strictly less than 2 are nonzero. See also the @ref update_math.
+ *
+ * @param[in] 	model 	GenModel structure with the current model
+ * @param[in] 	i 	index of the instance for which to check
+ * @returns 		whether or not we can do simple majorization
+ *
+ */
+bool gensvm_majorize_is_simple(struct GenModel *model, struct GenData *data,
+		long i)
+{
+	long j;
+	double h, value = 0;
+	for (j=0; j<model->K; j++) {
+		if (j == (data->y[i]-1))
+			continue;
+		h = matrix_get(model->H, model->K, i, j);
+		value += (h > 0) ? 1 : 0;
+		if (value > 1)
+			return false;
+	}
+	return true;
+}
+
+/**
+ * @brief Compute majorization coefficients for non-simple instance
+ *
+ * @details
+ * In this function we compute the majorization coefficients needed for an
+ * instance with a non-simple majorization (@f$\varepsilon_i = 0@f$). In this
+ * function, we distinguish a number of cases depending on the value of
+ * GenModel::p and the respective value of @f$\overline{q}_i^{(y_ij)}@f$. Note
+ * that the linear coefficient is of the form @f$b - a\overline{q}@f$, but
+ * often the second term is included in the definition of @f$b@f$, so it can
+ * be optimized out. The output argument \p b_aq contains this difference
+ * therefore in one go. More details on this function can be found in the @ref
+ * update_math. See also gensvm_calculate_ab_simple().
+ *
+ * @param[in] 	model 	GenModel structure with the current model
+ * @param[in] 	i 	index for the instance
+ * @param[in] 	j 	index for the class
+ * @param[out] 	*a 	output argument for the quadratic coefficient
+ * @param[out]  *b_aq 	output argument for the linear coefficient.
+ *
+ */
+void gensvm_calculate_ab_non_simple(struct GenModel *model, long i, long j,
+		double *a, double *b_aq)
+{
+	double q = matrix_get(model->Q, model->K, i, j);
+	double p = model->p;
+	double kappa = model->kappa;
+	const double a2g2 = 0.25*p*(2.0*p - 1.0)*pow((kappa+1.0)/2.0,p-2.0);
+
+	if (2.0 - model->p < 1e-2) {
+		if (q <= - kappa) {
+			*b_aq = 0.5 - kappa/2.0 - q;
+		} else if ( q <= 1.0) {
+			*b_aq = pow(1.0 - q, 3.0)/(2.0*pow(kappa + 1.0, 2.0));
+		} else {
+			*b_aq = 0;
+		}
+		*a = 1.5;
+	} else {
+		if (q <= (p + kappa - 1.0)/(p - 2.0)) {
+			*a = 0.25*pow(p, 2.0)*pow(0.5 - kappa/2.0 - q, p - 2.0);
+		} else if (q <= 1.0) {
+			*a = a2g2;
+		} else {
+			*a = 0.25*pow(p, 2.0)*pow((p/(p - 2.0))*(0.5 -
+						kappa/2.0 - q), p - 2.0);
+			*b_aq = (*a)*(2.0*q + kappa - 1.0)/(p - 2.0) +
+				0.5*p*pow(p/(p - 2.0)*(0.5 - kappa/2.0 - q),
+						p - 1.0);
+		}
+		if (q <= -kappa) {
+			*b_aq = 0.5*p*pow(0.5 - kappa/2.0 - q, p - 1.0);
+		} else if ( q <= 1.0) {
+			*b_aq = p*pow(1.0 - q, 2.0*p - 1.0)/pow(2*kappa+2.0, p);
+		}
+	}
+}
+
+/**
+ * @brief Compute majorization coefficients for simple instances
+ *
+ * @details
+ * In this function we compute the majorization coefficients needed for an
+ * instance with a simple majorization. This corresponds to the non-simple
+ * majorization for the case where GenModel::p equals 1. Due to this condition
+ * the majorization coefficients are quite simple to compute.  Note that the
+ * linear coefficient of the majorization is of the form @f$b -
+ * a\overline{q}@f$, but often the second term is included in the definition
+ * of @f$b@f$, so it can be optimized out. For more details see the @ref
+ * update_math, and gensvm_calculate_ab_non_simple().
+ *
+ * @param[in] 	model 	GenModel structure with the current model
+ * @param[in] 	i 	index for the instance
+ * @param[in] 	j 	index for the class
+ * @param[out] 	*a 	output argument for the quadratic coefficient
+ * @param[out] 	*b_aq 	output argument for the linear coefficient
+ *
+ */
+void gensvm_calculate_ab_simple(struct GenModel *model, long i, long j,
+		double *a, double *b_aq)
+{
+	double q = matrix_get(model->Q, model->K, i, j);
+
+	if (q <= - model->kappa) {
+		*a = 0.25/(0.5 - model->kappa/2.0 - q);
+		*b_aq = 0.5;
+	} else if (q <= 1.0) {
+		*a = 1.0/(2.0*model->kappa + 2.0);
+		*b_aq = (1.0 - q)*(*a);
+	} else {
+		*a = -0.25/(0.5 - model->kappa/2.0 - q);
+		*b_aq = 0;
+	}
+}
+
+/**
+ * @brief Compute the alpha_i and beta_i for an instance
+ *
+ * @details
+ * This computes the @f$\alpha_i@f$ value for an instance, and simultaneously
+ * updating the row of the B matrix corresponding to that
+ * instance (the @f$\boldsymbol{\beta}_i'@f$). The advantage of doing this at
+ * the same time is that we can compute the a and b values simultaneously in
+ * the gensvm_calculate_ab_simple() and gensvm_calculate_ab_non_simple()
+ * functions.
+ *
+ * The computation is done by first checking whether simple majorization is
+ * possible for this instance. If so, the @f$\omega_i@f$ value is set to 1.0,
+ * otherwise this value is computed. If simple majorization is possible, the
+ * coefficients a and b_aq are computed by gensvm_calculate_ab_simple(),
+ * otherwise they're computed by gensvm_calculate_ab_non_simple(). Next, the
+ * beta_i updated through the efficient BLAS daxpy function, and part of the
+ * value of @f$\alpha_i@f$ is computed. The final value of @f$\alpha_i@f$ is
+ * returned.
+ *
+ * @param[in] 		model 	GenModel structure with the current model
+ * @param[in] 		i 	index of the instance to update
+ * @param[out] 		beta	beta vector of linear coefficients (assumed to
+ * 				be allocated elsewhere, initialized here)
+ * @returns 			the @f$\alpha_i@f$ value of this instance
+ *
+ */
+double gensvm_get_alpha_beta(struct GenModel *model, struct GenData *data,
+		long i, double *beta)
+{
+	bool simple;
+	long j,
+	     K = model->K;
+	double omega, a, b_aq = 0.0,
+	       alpha = 0.0;
+	double *uu_row = NULL;
+	const double in = 1.0/((double) model->n);
+
+	simple = gensvm_majorize_is_simple(model, data, i);
+	omega = simple ? 1.0 : gensvm_calculate_omega(model, data, i);
+
+	Memset(beta, double, K-1);
+	for (j=0; j<K; j++) {
+		// skip the class y_i = k
+		if (j == (data->y[i]-1))
+			continue;
+
+		// calculate the a_ijk and (b_ijk - a_ijk q_i^(kj)) values
+		if (simple) {
+			gensvm_calculate_ab_simple(model, i, j, &a, &b_aq);
+		} else {
+			gensvm_calculate_ab_non_simple(model, i, j, &a, &b_aq);
+		}
+
+		// daxpy on beta and UU
+		// daxpy does: y = a*x + y
+		// so y = beta, UU_row = x, a = factor
+		b_aq *= model->rho[i] * omega * in;
+		uu_row = &model->UU[((data->y[i]-1)*K+j)*(K-1)];
+		cblas_daxpy(K-1, b_aq, uu_row, 1, beta, 1);
+
+		// increment Avalue
+		alpha += a;
+	}
+	alpha *= omega * model->rho[i] * in;
+	return alpha;
+}
+
+/**
+ * @brief Perform a single step of the majorization algorithm to update V
+ *
+ * @details
+ * This function contains the main update calculations of the algorithm. These
+ * calculations are necessary to find a new update V. The calculations exist of
+ * recalculating the majorization coefficients for all instances and all
+ * classes, and solving a linear system to find V.
+ *
+ * Because the function gensvm_get_update() is always called after a call to
+ * gensvm_get_loss() with the same GenModel::V, it is unnecessary to calculate
+ * the updated errors GenModel::Q and GenModel::H here too. This saves on
+ * computation time.
+ *
+ * In calculating the majorization coefficients we calculate the elements of a
+ * diagonal matrix A with elements
+ * @f[
+ * 	A_{i, i} = \frac{1}{n} \rho_i \sum_{j \neq k} \left[
+ * 		\varepsilon_i a_{ijk}^{(1)} + (1 - \varepsilon_i) \omega_i
+ * 		a_{ijk}^{(p)} \right],
+ * @f]
+ * where @f$ k = y_i @f$.
+ * Since this matrix is only used to calculate the matrix @f$ Z' A Z @f$, it
+ * is efficient to update a matrix ZAZ through consecutive rank 1 updates with
+ * a single element of A and the corresponding row of Z. The BLAS function
+ * dsyr is used for this.
+ *
+ * The B matrix is has rows
+ * @f[
+ * 	\boldsymbol{\beta}_i' = \frac{1}{n} \rho_i \sum_{j \neq k} \left[
+ * 		\varepsilon_i \left( b_{ijk}^{(1)} - a_{ijk}^{(1)}
+ * 			\overline{q}_i^{(kj)} \right) + (1 - \varepsilon_i)
+ * 		\omega_i \left( b_{ijk}^{(p)} - a_{ijk}^{(p)}
+ * 			\overline{q}_i^{(kj)} \right) \right]
+ * 		\boldsymbol{\delta}_{kj}'
+ * @f]
+ * This is also split into two cases, one for which @f$ \varepsilon_i = 1 @f$,
+ * and one for when it is 0. The 3D simplex difference matrix is used here, in
+ * the form of the @f$ \boldsymbol{\delta}_{kj}' @f$.
+ *
+ * Finally, the following system is solved
+ * @f[
+ * 	(\textbf{Z}'\textbf{AZ} + \lambda \textbf{J})\textbf{V} =
+ * 		(\textbf{Z}'\textbf{AZ}\overline{\textbf{V}} + \textbf{Z}'
+ * 		\textbf{B})
+ * @f]
+ * solving this system is done through dposv().
+ *
+ * @todo
+ * Consider using CblasColMajor everywhere
+ *
+ * @param[in,out] 	model 	model to be updated
+ * @param[in] 		data 	data used in model
+ * @param[in] 		work 	allocated workspace to use
+ */
+void gensvm_get_update(struct GenModel *model, struct GenData *data,
+		struct GenWork *work)
+{
+	int status;
+	long i, j;
+
+	long m = model->m;
+	long K = model->K;
+
+	gensvm_get_ZAZ_ZB(model, data, work);
+
+	// Calculate right-hand side of system we want to solve
+	// dsymm performs ZB := 1.0 * (ZAZ) * Vbar + 1.0 * ZB
+	// the right-hand side is thus stored in ZB after this call
+	// Note: LDB and LDC are second dimensions of the matrices due to
+	// Row-Major order
+	cblas_dsymm(CblasRowMajor, CblasLeft, CblasUpper, m+1, K-1, 1,
+			work->ZAZ, m+1, model->V, K-1, 1.0, work->ZB, K-1);
+
+	// Calculate left-hand side of system we want to solve
+	// Add lambda to all diagonal elements except the first one. Recall
+	// that ZAZ is of size m+1 and is symmetric.
+	for (i=m+2; i<=m*(m+2); i+=m+2)
+		work->ZAZ[i] += model->lambda;
+
+	// Lapack uses column-major order, so we transform the ZB matrix to
+	// correspond to this.
+	for (i=0; i<m+1; i++)
+		for (j=0; j<K-1; j++)
+			work->ZBc[j*(m+1)+i] = work->ZB[i*(K-1)+j];
+
+	// Solve the system using dposv. Note that above the upper triangular
+	// part has always been used in row-major order for ZAZ. This
+	// corresponds to the lower triangular part in column-major order.
+	status = dposv('L', m+1, K-1, work->ZAZ, m+1, work->ZBc, m+1);
+
+	// Use dsysv as fallback, for when the ZAZ matrix is not positive
+	// semi-definite for some reason (perhaps due to rounding errors).
+	// This step shouldn't be necessary but is included for safety.
+	if (status != 0) {
+		err("[GenSVM Warning]: Received nonzero status from "
+				"dposv: %i\n", status);
+		int *IPIV = Malloc(int, m+1);
+		double *WORK = Malloc(double, 1);
+		status = dsysv('L', m+1, K-1, work->ZAZ, m+1, IPIV, work->ZBc,
+				m+1, WORK, -1);
+
+		int LWORK = WORK[0];
+		WORK = Realloc(WORK, double, LWORK);
+		status = dsysv('L', m+1, K-1, work->ZAZ, m+1, IPIV, work->ZBc,
+				m+1, WORK, LWORK);
+		if (status != 0)
+			err("[GenSVM Warning]: Received nonzero "
+					"status from dsysv: %i\n", status);
+		free(WORK);
+		WORK = NULL;
+		free(IPIV);
+		IPIV = NULL;
+	}
+
+	// the solution is now stored in ZBc, in column-major order. Here we
+	// convert this back to row-major order
+	for (i=0; i<m+1; i++)
+		for (j=0; j<K-1; j++)
+			work->ZB[i*(K-1)+j] = work->ZBc[j*(m+1)+i];
+
+	// copy the old V to Vbar and the new solution to V
+	for (i=0; i<m+1; i++) {
+		for (j=0; j<K-1; j++) {
+			matrix_set(model->Vbar, K-1, i, j,
+					matrix_get(model->V, K-1, i, j));
+			matrix_set(model->V, K-1, i, j,
+					matrix_get(work->ZB, K-1, i, j));
+		}
+	}
+}
+
+/**
+ * @brief Calculate Z'*A*Z and Z'*B for dense matrices
+ *
+ * @details
+ * This function calculates the matrices Z'*A*Z and Z'*B for the case where Z
+ * is stored as a dense matrix. It calculates the Z'*A*Z product by
+ * constructing a matrix LZ = (A^(1/2) * Z), and calculating (LZ)'*(LZ) with
+ * the BLAS dsyrk function. The matrix Z'*B is calculated with successive
+ * rank-1 updates using the BLAS dger function. These functions came out as
+ * the most efficient way to do these computations in several simulation
+ * studies.
+ *
+ * @param[in] 		model 	a GenModel holding the current model
+ * @param[in] 		data 	a GenData with the data
+ * @param[in,out] 	work 	an allocated GenWork structure, contains
+ * 				updated ZAZ and ZB matrices on exit.
+ */
+void gensvm_get_ZAZ_ZB_dense(struct GenModel *model, struct GenData *data,
+		struct GenWork *work)
+{
+	long i;
+	double alpha, sqalpha;
+
+	long n = model->n;
+	long m = model->m;
+	long K = model->K;
+
+	// generate Z'*A*Z and Z'*B by rank 1 operations
+	for (i=0; i<n; i++) {
+		alpha = gensvm_get_alpha_beta(model, data, i, work->beta);
+
+		// calculate row of matrix LZ, which is a scalar
+		// multiplication of sqrt(alpha_i) and row z_i' of Z
+		// Note that we use the fact that the first column of Z is
+		// always 1, by only computing the product for m values and
+		// copying the first element over.
+		sqalpha = sqrt(alpha);
+		work->LZ[i*(m+1)] = sqalpha;
+		cblas_daxpy(m, sqalpha, &data->Z[i*(m+1)+1], 1,
+				&work->LZ[i*(m+1)+1], 1);
+
+		// rank 1 update of matrix Z'*B
+		// Note: LDA is the second dimension of ZB because of
+		// Row-Major order
+		cblas_dger(CblasRowMajor, m+1, K-1, 1, &data->Z[i*(m+1)], 1,
+				work->beta, 1, work->ZB, K-1);
+	}
+
+	// calculate Z'*A*Z by symmetric multiplication of LZ with itself
+	// (ZAZ = (LZ)' * (LZ)
+	cblas_dsyrk(CblasRowMajor, CblasUpper, CblasTrans, m+1, n, 1.0,
+			work->LZ, m+1, 0.0, work->ZAZ, m+1);
+}
+
+/**
+ * @brief Calculate Z'*A*Z and Z'*B for sparse matrices
+ *
+ * @details
+ * This function calculates the matrices Z'*A*Z and Z'*B for the case where Z 
+ * is stored as a CSR sparse matrix (GenSparse structure). It computes only 
+ * the products of the Z'*A*Z matrix that need to be computed, and updates the 
+ * Z'*B matrix row-wise for each non-zero element of a row of Z, using a BLAS 
+ * daxpy call.
+ *
+ * @sa
+ * gensvm_get_ZAZ_ZB()
+ * gensvm_get_ZAZ_ZB_dense()
+ *
+ * @param[in] 		model 	a GenModel holding the current model
+ * @param[in] 		data 	a GenData with the data
+ * @param[in,out] 	work 	an allocated GenWork structure, contains
+ * 				updated ZAZ and ZB matrices on exit.
+ */
+void gensvm_get_ZAZ_ZB_sparse(struct GenModel *model, struct GenData *data,
+		struct GenWork *work)
+{
+	long i, j, k, jj, kk, jj_start, jj_end, K,
+	     n_row = data->spZ->n_row,
+	     n_col = data->spZ->n_col;
+	double alpha, z_ij;
+
+	K = model->K;
+
+	int *Zia = data->spZ->ia;
+	int *Zja = data->spZ->ja;
+	double *vals = data->spZ->values;
+
+	for (i=0; i<n_row; i++) {
+		alpha = gensvm_get_alpha_beta(model, data, i, work->beta);
+
+		jj_start = Zia[i];
+		jj_end = Zia[i+1];
+
+		for (jj=jj_start; jj<jj_end; jj++) {
+			j = Zja[jj];
+			z_ij = vals[jj];
+			cblas_daxpy(K-1, z_ij, work->beta, 1,
+					&work->ZB[j*(K-1)], 1);
+			z_ij *= alpha;
+			for (kk=jj; kk<jj_end; kk++) {
+				k = Zja[kk];
+				matrix_add(work->ZAZ, n_col, j, k,
+						z_ij*vals[kk]);
+			}
+		}
+	}
+}
+
+
+void gensvm_get_ZAZ_ZB(struct GenModel *model, struct GenData *data,
+		struct GenWork *work)
+{
+	gensvm_reset_work(work);
+
+	if (data->spZ == NULL) {
+		gensvm_get_ZAZ_ZB_dense(model, data, work);
+	} else {
+		gensvm_get_ZAZ_ZB_sparse(model, data, work);
+	}
+}
+
+/**
+ * @brief Solve AX = B where A is symmetric positive definite.
+ *
+ * @details
+ * Solve a linear system of equations AX = B where A is symmetric positive
+ * definite. This function is a wrapper for the external  LAPACK routine
+ * dposv.
+ *
+ * @param[in] 		UPLO 	which triangle of A is stored
+ * @param[in] 		N 	order of A
+ * @param[in] 		NRHS 	number of columns of B
+ * @param[in,out] 	A 	double precision array of size (LDA, N). On
+ * 				exit contains the upper or lower factor of the
+ * 				Cholesky factorization of A.
+ * @param[in] 		LDA 	leading dimension of A
+ * @param[in,out] 	B 	double precision array of size (LDB, NRHS). On
+ * 				exit contains the N-by-NRHS solution matrix X.
+ * @param[in] 		LDB 	the leading dimension of B
+ * @returns 			info parameter which contains the status of the
+ * 				computation:
+ * 					- =0: 	success
+ * 					- <0: 	if -i, the i-th argument had
+ * 						an illegal value
+ * 					- >0: 	if i, the leading minor of A
+ * 						was not positive definite
+ *
+ * See the LAPACK documentation at:
+ * http://www.netlib.org/lapack/explore-html/dc/de9/group__double_p_osolve.html
+ */
+int dposv(char UPLO, int N, int NRHS, double *A, int LDA, double *B,
+		int LDB)
+{
+	extern void dposv_(char *UPLO, int *Np, int *NRHSp, double *A,
+			int *LDAp, double *B, int *LDBp, int *INFOp);
+	int INFO;
+	dposv_(&UPLO, &N, &NRHS, A, &LDA, B, &LDB, &INFO);
+	return INFO;
+}
+
+/**
+ * @brief Solve a system of equations AX = B where A is symmetric.
+ *
+ * @details
+ * Solve a linear system of equations AX = B where A is symmetric. This
+ * function is a wrapper for the external LAPACK routine dsysv.
+ *
+ * @param[in] 		UPLO 	which triangle of A is stored
+ * @param[in] 		N 	order of A
+ * @param[in] 		NRHS 	number of columns of B
+ * @param[in,out] 	A 	double precision array of size (LDA, N). On
+ * 				exit contains the block diagonal matrix D and
+ * 				the multipliers used to obtain the factor U or
+ * 				L from the factorization A = U*D*U**T or
+ * 				A = L*D*L**T.
+ * @param[in] 		LDA 	leading dimension of A
+ * @param[in] 		IPIV 	integer array containing the details of D
+ * @param[in,out] 	B 	double precision array of size (LDB, NRHS). On
+ * 				exit contains the N-by-NRHS matrix X
+ * @param[in] 		LDB 	leading dimension of B
+ * @param[out] 		WORK 	double precision array of size max(1,LWORK). On
+ * 				exit, WORK(1) contains the optimal LWORK
+ * @param[in] 		LWORK 	the length of WORK, can be used for determining
+ * 				the optimal blocksize for dsystrf.
+ * @returns 			info parameter which contains the status of the
+ * 				computation:
+ * 					- =0: 	success
+ * 					- <0: 	if -i, the i-th argument had an
+ * 						illegal value
+ * 					- >0: 	if i, D(i, i) is exactly zero,
+ * 						no solution can be computed.
+ *
+ * See the LAPACK documentation at:
+ * http://www.netlib.org/lapack/explore-html/d6/d0e/group__double_s_ysolve.html
+ */
+int dsysv(char UPLO, int N, int NRHS, double *A, int LDA, int *IPIV,
+		double *B, int LDB, double *WORK, int LWORK)
+{
+	extern void dsysv_(char *UPLO, int *Np, int *NRHSp, double *A,
+			int *LDAp, int *IPIV, double *B, int *LDBp,
+			double *WORK, int *LWORK, int *INFOp);
+	int INFO;
+	dsysv_(&UPLO, &N, &NRHS, A, &LDA, IPIV, B, &LDB, WORK, &LWORK, &INFO);
+	return INFO;
+}