These are a particular example of the principle that purely abstract or intellectual methods are not patentable. For example, an abstract shortcut method of division would be excluded from patentability by Art. 52(2)(a) and Art. 52(3). However, a calculating machine constructed to operate accordingly (e.g. by executing a program designed to carry out the method) would not be excluded. Electrical filters designed according to a particular mathematical method would also not be excluded.

Furthermore, a method for analysing the cyclical behaviour of a curve relating two parameters, which are not further specified, to one another is a mathematical method as such, excluded from patentability by Art. 52(2)(a) and Art. 52(3), unless it uses technical means, for example, if it is computer-implemented.

A claim directed to a technical process in which a mathematical method is used, thus being restricted to a particular application of the mathematical method in a technical field, does not seek protection for the mathematical method as such. For instance, a method of encoding audio information in a communication system may aim to reduce distortion induced by channel noise. Although the idea underlying such a method may be considered to reside in a mathematical method, the encoding method as a whole is not a mathematical method as such, and hence is not excluded from patentability by Art. 52(2)(a) and Art. 52(3). Similarly, a method of encrypting/decrypting or signing electronic communications may be regarded as a technical method, even if it is essentially based on a mathematical method (see T 1326/06).

A procedural step (e.g. a mathematical algorithm) may contribute to the technical character of a claimed method only if it serves an adequately defined technical purpose of the method. In particular, specific technical applications of computer-implemented simulation methods, even if involving mathematical formulae, are to be regarded as modern technical methods which form an essential part of the fabrication process. Such simulation methods cannot be denied a technical effect merely on the ground that they do not yet incorporate the physical end product. However, the meta-specification of an undefined technical purpose (for example, the simulation of a "technical system"), could not be considered adequate (T 1227/05).

In a mathematical method for processing data, although defining the origin of the data records, i.e. what the data represents, may imply technical aspects, it does not necessarily confer technical character upon the method. For example, in a mathematical method for classifying data records, the classification algorithm would not derive a technical character from specifying that the data records are assembled from events in a telecommunications network if the classification is not performed for a technical purpose. What is also decisive is whether a technical effect is achieved by the functional nature of the data irrespective of its cognitive content (see T 1194/97, T 1161/04). For example, a mathematical method for processing data representing an image stored as an electric signal by a computer-implemented method and providing as its result a certain change in the image (e.g. restoring the image if it is distorted) is considered as being used in a technical process (T 208/84 and T 1161/04).

The increased speed or efficiency of a method based on improved algorithms is not sufficient on its own to establish a technical character of the method (see T 1227/05). Characteristics such as speed and efficiency are inherent in both technical and non-technical methods. For example, if a sequence of auction steps leads to price determination more quickly than some other auction method, that does not necessarily imply that the auction steps contribute to the technical character of the method (see T 258/03).

Mathematical methods play an important role in the solution of technical problems in all fields of technology. However, they are excluded from patentability under Art. 52(2)(a) when claimed as such (Art. 52(3)).

The exclusion applies if a claim is directed to a purely abstract mathematical method and the claim does not require any technical means. For instance, a method for performing a Fast Fourier Transform on abstract data which does not specify the use of any technical means is a mathematical method as such. A purely abstract mathematical object or concept, e.g. a particular type of geometric object or of graph with nodes and edges, is not a method but is nevertheless not an invention in the sense of Art. 52(1) because it lacks a technical character.

If a claim is directed either to a method involving the use of technical means (e.g. a computer) or to a device, its subject-matter has a technical character as a whole and is thus not excluded from patentability under Art. 52(2) and (3).

Merely specifying the technical nature of the data or parameters of the mathematical method may not be sufficient to define an invention in the sense of Art. 52(1), as the resulting method may still fall under the excluded category of methods for performing mental acts as such (Art. 52(2)(c) and (3), see G-II, 3.5.1).

Once it is established that the claimed subject-matter as a whole is not excluded from patentability under Art. 52(2) and (3) and is thus an invention in the sense of Art. 52(1), it is examined in respect of the other requirements of patentability, in particular novelty and inventive step (G‑I, 1).

For the assessment of inventive step, all features which contribute to the technical character of the invention must be taken into account (G-VII, 5.4). When the claimed invention is based on a mathematical method, it is assessed whether the mathematical method contributes to the technical character of the invention.

A mathematical method may contribute to the technical character of an invention, i.e. contribute to producing a **technical effect** that serves a technical purpose, by its application to a field of technology and/or by being adapted to a specific technical implementation. The criteria for assessing these two situations are explained below.

When assessing the contribution made by a mathematical method to the technical character of an invention, it must be taken into account whether the method, in the context of the invention, serves a **technical purpose** (T 1227/05, T 1358/09).

Examples of technical purposes which may be served by a mathematical method are:

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controlling a specific technical system or process, e.g. an X-ray apparatus or a steel cooling process;

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determining from measurements a required number of passes of a compaction machine to achieve a desired material density;

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digital audio, image or video enhancement or analysis, e.g. de-noising, detecting persons in a digital image, estimating the quality of a transmitted digital audio signal;

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separation of sources in speech signals; speech recognition, e.g. mapping a speech input to a text output;

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encoding data for reliable and/or efficient transmission or storage (and corresponding decoding), e.g. error-correction coding of data for transmission over a noisy channel, compression of audio, image, video or sensor data;

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encrypting/decrypting or signing electronic communications; generating keys in an RSA cryptographic system;

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optimising load distribution in a computer network;

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determining the energy expenditure of a subject by processing data obtained from physiological sensors; deriving the body temperature of a subject from data obtained from an ear temperature detector;

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providing a genotype estimate based on an analysis of DNA samples, as well as providing a confidence interval for this estimate so as to quantify its reliability;

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providing a medical diagnosis by an automated system processing physiological measurements;

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simulating the behaviour of an adequately defined class of technical items, or specific technical processes, under technically relevant conditions (see G-II, 3.3.2).

A **generic** purpose such as "controlling a technical system" is not sufficient to confer a technical character to the mathematical method. The technical purpose must be a **specific** one.

Furthermore, the mere fact that a mathematical method may serve a technical purpose is not sufficient, either. The claim is to be functionally **limited** to the technical purpose, either explicitly or implicitly. This can be achieved by establishing a sufficient link between the technical purpose and the mathematical method steps, for example, by specifying how the input and the output of the sequence of mathematical steps relate to the technical purpose so that the mathematical method is causally linked to a technical effect. See G-VII, 5.4.2.4 for a worked-out example.

Defining the nature of the data input to a mathematical method does not necessarily imply that the mathematical method contributes to the technical character of the invention (T 2035/11, T 1029/06, T 1161/04). Whether a technical purpose is served by the mathematical method is primarily determined by the direct technical relevance of the results it provides.

A mathematical method may also contribute to the technical character of the invention independently of any technical application when the claim is directed to a **specific technical implementation** of the mathematical method and the mathematical method is particularly **adapted** for that implementation in that its design is motivated by technical considerations of the **internal functioning **of the computer (T 1358/09). For instance, the adaptation of a polynomial reduction algorithm to exploit word-size shifts matched to the word size of the computer hardware is based on such technical considerations and can contribute to producing the technical effect of an efficient hardware implementation of said algorithm.

If the mathematical method does not serve a technical purpose and the claimed technical implementation does not go beyond a generic technical implementation, the mathematical method does not contribute to the technical character of the invention. In such a case, it is not sufficient that the mathematical method is algorithmically more efficient than prior-art mathematical methods (see G-II, 3.6).