/**
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* @license
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* Copyright 2020 Google LLC. All Rights Reserved.
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* Licensed under the Apache License, Version 2.0 (the "License");
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* you may not use this file except in compliance with the License.
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* You may obtain a copy of the License at
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*
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* http://www.apache.org/licenses/LICENSE-2.0
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*
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* Unless required by applicable law or agreed to in writing, software
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* distributed under the License is distributed on an "AS IS" BASIS,
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* WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
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* See the License for the specific language governing permissions and
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* limitations under the License.
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* =============================================================================
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*/
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/// <amd-module name="@tensorflow/tfjs-core/dist/ops/linalg/gram_schmidt" />
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import { Tensor1D, Tensor2D } from '../../tensor';
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/**
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* Gram-Schmidt orthogonalization.
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*
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* ```js
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* const x = tf.tensor2d([[1, 2], [3, 4]]);
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* let y = tf.linalg.gramSchmidt(x);
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* y.print();
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* console.log('Orthogonalized:');
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* y.dot(y.transpose()).print(); // should be nearly the identity matrix.
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* console.log('First row direction maintained:');
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* const data = await y.array();
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* console.log(data[0][1] / data[0][0]); // should be nearly 2.
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* ```
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*
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* @param xs The vectors to be orthogonalized, in one of the two following
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* formats:
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* - An Array of `tf.Tensor1D`.
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* - A `tf.Tensor2D`, i.e., a matrix, in which case the vectors are the rows
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* of `xs`.
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* In each case, all the vectors must have the same length and the length
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* must be greater than or equal to the number of vectors.
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* @returns The orthogonalized and normalized vectors or matrix.
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* Orthogonalization means that the vectors or the rows of the matrix
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* are orthogonal (zero inner products). Normalization means that each
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* vector or each row of the matrix has an L2 norm that equals `1`.
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*
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* @doc {heading:'Operations', subheading:'Linear Algebra', namespace:'linalg'}
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*/
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declare function gramSchmidt_(xs: Tensor1D[] | Tensor2D): Tensor1D[] | Tensor2D;
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export declare const gramSchmidt: typeof gramSchmidt_;
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export {};
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