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| <div class="subTitle">org.apache.mahout.math.solver</div> |
| <h2 title="Class LSMR" class="title">Class LSMR</h2> |
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| <pre>public final class <span class="typeNameLabel">LSMR</span> |
| extends <a href="http://docs.oracle.com/javase/7/docs/api/java/lang/Object.html?is-external=true" title="class or interface in java.lang">Object</a></pre> |
| <div class="block">Solves sparse least-squares using the LSMR algorithm. |
| <p/> |
| LSMR solves the system of linear equations A * X = B. If the system is inconsistent, it solves |
| the least-squares problem min ||b - Ax||_2. A is a rectangular matrix of dimension m-by-n, where |
| all cases are allowed: m=n, m>n, or m<n. B is a vector of length m. The matrix A may be dense |
| or sparse (usually sparse). |
| <p/> |
| Some additional configurable properties adjust the behavior of the algorithm. |
| <p/> |
| If you set lambda to a non-zero value then LSMR solves the regularized least-squares problem min |
| ||(B) - ( A )X|| ||(0) (lambda*I) ||_2 where LAMBDA is a scalar. If LAMBDA is not set, |
| the system is solved without regularization. |
| <p/> |
| You can also set aTolerance and bTolerance. These cause LSMR to iterate until a certain backward |
| error estimate is smaller than some quantity depending on ATOL and BTOL. Let RES = B - A*X be |
| the residual vector for the current approximate solution X. If A*X = B seems to be consistent, |
| LSMR terminates when NORM(RES) <= ATOL*NORM(A)*NORM(X) + BTOL*NORM(B). Otherwise, LSMR terminates |
| when NORM(A'*RES) <= ATOL*NORM(A)*NORM(RES). If both tolerances are 1.0e-6 (say), the final |
| NORM(RES) should be accurate to about 6 digits. (The final X will usually have fewer correct |
| digits, depending on cond(A) and the size of LAMBDA.) |
| <p/> |
| The default value for ATOL and BTOL is 1e-6. |
| <p/> |
| Ideally, they should be estimates of the relative error in the entries of A and B respectively. |
| For example, if the entries of A have 7 correct digits, set ATOL = 1e-7. This prevents the |
| algorithm from doing unnecessary work beyond the uncertainty of the input data. |
| <p/> |
| You can also set conditionLimit. In that case, LSMR terminates if an estimate of cond(A) exceeds |
| conditionLimit. For compatible systems Ax = b, conditionLimit could be as large as 1.0e+12 (say). |
| For least-squares problems, conditionLimit should be less than 1.0e+8. If conditionLimit is not |
| set, the default value is 1e+8. Maximum precision can be obtained by setting aTolerance = |
| bTolerance = conditionLimit = 0, but the number of iterations may then be excessive. |
| <p/> |
| Setting iterationLimit causes LSMR to terminate if the number of iterations reaches |
| iterationLimit. The default is iterationLimit = min(m,n). For ill-conditioned systems, a |
| larger value of ITNLIM may be needed. |
| <p/> |
| Setting localSize causes LSMR to run with rerorthogonalization on the last localSize v_k's. |
| (v-vectors generated by Golub-Kahan bidiagonalization) If localSize is not set, LSMR runs without |
| reorthogonalization. A localSize > max(n,m) performs reorthogonalization on all v_k's. |
| Reorthgonalizing only u_k or both u_k and v_k are not an option here. Details are discussed in |
| the SIAM paper. |
| <p/> |
| getTerminationReason() gives the reason for termination. ISTOP = 0 means X=0 is a solution. = 1 |
| means X is an approximate solution to A*X = B, according to ATOL and BTOL. = 2 means X |
| approximately solves the least-squares problem according to ATOL. = 3 means COND(A) seems to be |
| greater than CONLIM. = 4 is the same as 1 with ATOL = BTOL = EPS. = 5 is the same as 2 with ATOL |
| = EPS. = 6 is the same as 3 with CONLIM = 1/EPS. = 7 means ITN reached ITNLIM before the other |
| stopping conditions were satisfied. |
| <p/> |
| getIterationCount() gives ITN = the number of LSMR iterations. |
| <p/> |
| getResidualNorm() gives an estimate of the residual norm: NORMR = norm(B-A*X). |
| <p/> |
| getNormalEquationResidual() gives an estimate of the residual for the normal equation: NORMAR = |
| NORM(A'*(B-A*X)). |
| <p/> |
| getANorm() gives an estimate of the Frobenius norm of A. |
| <p/> |
| getCondition() gives an estimate of the condition number of A. |
| <p/> |
| getXNorm() gives an estimate of NORM(X). |
| <p/> |
| LSMR uses an iterative method. For further information, see D. C.-L. Fong and M. A. Saunders |
| LSMR: An iterative algorithm for least-square problems Draft of 03 Apr 2010, to be submitted to |
| SISC. |
| <p/> |
| David Chin-lung Fong clfong@stanford.edu Institute for Computational and Mathematical |
| Engineering Stanford University |
| <p/> |
| Michael Saunders saunders@stanford.edu Systems Optimization Laboratory Dept of |
| MS&E, Stanford University. -----------------------------------------------------------------------</div> |
| </li> |
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| <th class="colOne" scope="col">Constructor and Description</th> |
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| <tr class="altColor"> |
| <td class="colOne"><code><span class="memberNameLink"><a href="../../../../../org/apache/mahout/math/solver/LSMR.html#LSMR--">LSMR</a></span>()</code> |
| <div class="block">LSMR uses an iterative method to solve a linear system.</div> |
| </td> |
| </tr> |
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| <h3>Method Summary</h3> |
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| <caption><span id="t0" class="activeTableTab"><span>All Methods</span><span class="tabEnd"> </span></span><span id="t2" class="tableTab"><span><a href="javascript:show(2);">Instance Methods</a></span><span class="tabEnd"> </span></span><span id="t4" class="tableTab"><span><a href="javascript:show(8);">Concrete Methods</a></span><span class="tabEnd"> </span></span></caption> |
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| <th class="colLast" scope="col">Method and Description</th> |
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| <tr id="i0" class="altColor"> |
| <td class="colFirst"><code>double</code></td> |
| <td class="colLast"><code><span class="memberNameLink"><a href="../../../../../org/apache/mahout/math/solver/LSMR.html#getANorm--">getANorm</a></span>()</code> </td> |
| </tr> |
| <tr id="i1" class="rowColor"> |
| <td class="colFirst"><code>double</code></td> |
| <td class="colLast"><code><span class="memberNameLink"><a href="../../../../../org/apache/mahout/math/solver/LSMR.html#getAtolerance--">getAtolerance</a></span>()</code> </td> |
| </tr> |
| <tr id="i2" class="altColor"> |
| <td class="colFirst"><code>double</code></td> |
| <td class="colLast"><code><span class="memberNameLink"><a href="../../../../../org/apache/mahout/math/solver/LSMR.html#getBtolerance--">getBtolerance</a></span>()</code> </td> |
| </tr> |
| <tr id="i3" class="rowColor"> |
| <td class="colFirst"><code>double</code></td> |
| <td class="colLast"><code><span class="memberNameLink"><a href="../../../../../org/apache/mahout/math/solver/LSMR.html#getCondition--">getCondition</a></span>()</code> </td> |
| </tr> |
| <tr id="i4" class="altColor"> |
| <td class="colFirst"><code>int</code></td> |
| <td class="colLast"><code><span class="memberNameLink"><a href="../../../../../org/apache/mahout/math/solver/LSMR.html#getIterationCount--">getIterationCount</a></span>()</code> </td> |
| </tr> |
| <tr id="i5" class="rowColor"> |
| <td class="colFirst"><code>double</code></td> |
| <td class="colLast"><code><span class="memberNameLink"><a href="../../../../../org/apache/mahout/math/solver/LSMR.html#getLambda--">getLambda</a></span>()</code> </td> |
| </tr> |
| <tr id="i6" class="altColor"> |
| <td class="colFirst"><code>double</code></td> |
| <td class="colLast"><code><span class="memberNameLink"><a href="../../../../../org/apache/mahout/math/solver/LSMR.html#getNormalEquationResidual--">getNormalEquationResidual</a></span>()</code> </td> |
| </tr> |
| <tr id="i7" class="rowColor"> |
| <td class="colFirst"><code>double</code></td> |
| <td class="colLast"><code><span class="memberNameLink"><a href="../../../../../org/apache/mahout/math/solver/LSMR.html#getResidualNorm--">getResidualNorm</a></span>()</code> </td> |
| </tr> |
| <tr id="i8" class="altColor"> |
| <td class="colFirst"><code>double</code></td> |
| <td class="colLast"><code><span class="memberNameLink"><a href="../../../../../org/apache/mahout/math/solver/LSMR.html#getXNorm--">getXNorm</a></span>()</code> </td> |
| </tr> |
| <tr id="i9" class="rowColor"> |
| <td class="colFirst"><code>void</code></td> |
| <td class="colLast"><code><span class="memberNameLink"><a href="../../../../../org/apache/mahout/math/solver/LSMR.html#setAtolerance-double-">setAtolerance</a></span>(double aTolerance)</code> </td> |
| </tr> |
| <tr id="i10" class="altColor"> |
| <td class="colFirst"><code>void</code></td> |
| <td class="colLast"><code><span class="memberNameLink"><a href="../../../../../org/apache/mahout/math/solver/LSMR.html#setBtolerance-double-">setBtolerance</a></span>(double bTolerance)</code> </td> |
| </tr> |
| <tr id="i11" class="rowColor"> |
| <td class="colFirst"><code>void</code></td> |
| <td class="colLast"><code><span class="memberNameLink"><a href="../../../../../org/apache/mahout/math/solver/LSMR.html#setConditionLimit-double-">setConditionLimit</a></span>(double conditionLimit)</code> </td> |
| </tr> |
| <tr id="i12" class="altColor"> |
| <td class="colFirst"><code>void</code></td> |
| <td class="colLast"><code><span class="memberNameLink"><a href="../../../../../org/apache/mahout/math/solver/LSMR.html#setIterationLimit-int-">setIterationLimit</a></span>(int iterationLimit)</code> </td> |
| </tr> |
| <tr id="i13" class="rowColor"> |
| <td class="colFirst"><code>void</code></td> |
| <td class="colLast"><code><span class="memberNameLink"><a href="../../../../../org/apache/mahout/math/solver/LSMR.html#setLocalSize-int-">setLocalSize</a></span>(int localSize)</code> </td> |
| </tr> |
| <tr id="i14" class="altColor"> |
| <td class="colFirst"><code><a href="../../../../../org/apache/mahout/math/Vector.html" title="interface in org.apache.mahout.math">Vector</a></code></td> |
| <td class="colLast"><code><span class="memberNameLink"><a href="../../../../../org/apache/mahout/math/solver/LSMR.html#solve-org.apache.mahout.math.Matrix-org.apache.mahout.math.Vector-">solve</a></span>(<a href="../../../../../org/apache/mahout/math/Matrix.html" title="interface in org.apache.mahout.math">Matrix</a> A, |
| <a href="../../../../../org/apache/mahout/math/Vector.html" title="interface in org.apache.mahout.math">Vector</a> b)</code> </td> |
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| <h4>LSMR</h4> |
| <pre>public LSMR()</pre> |
| <div class="block">LSMR uses an iterative method to solve a linear system. For further information, see D. C.-L. |
| Fong and M. A. Saunders LSMR: An iterative algorithm for least-square problems Draft of 03 Apr |
| 2010, to be submitted to SISC. |
| <p/> |
| 08 Dec 2009: First release version of LSMR. 09 Apr 2010: Updated documentation and default |
| parameters. 14 Apr 2010: Updated documentation. 03 Jun 2010: LSMR with local |
| reorthogonalization (full reorthogonalization is also implemented) |
| <p/> |
| David Chin-lung Fong clfong@stanford.edu Institute for Computational and |
| Mathematical Engineering Stanford University |
| <p/> |
| Michael Saunders saunders@stanford.edu Systems Optimization Laboratory Dept of |
| MS&E, Stanford University. -----------------------------------------------------------------------</div> |
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| <pre>public <a href="../../../../../org/apache/mahout/math/Vector.html" title="interface in org.apache.mahout.math">Vector</a> solve(<a href="../../../../../org/apache/mahout/math/Matrix.html" title="interface in org.apache.mahout.math">Matrix</a> A, |
| <a href="../../../../../org/apache/mahout/math/Vector.html" title="interface in org.apache.mahout.math">Vector</a> b)</pre> |
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